Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory

*Kayupe Kikodio Patrick*

#### **Abstract**

Bessel functions form an important class of special functions and are applied almost everywhere in mathematical physics. They are also called cylindrical functions, or cylindrical harmonics. This chapter is devoted to the construction of the generalized coherent state (GCS) and the theory of Bessel wavelets. The GCS is built by replacing the coefficient *z<sup>n</sup>=n*!, *z*∈ of the canonical CS by the cylindrical Bessel functions. Then, the Paley-Wiener space *PW*<sup>1</sup> is discussed in the framework of a set of GCS related to the cylindrical Bessel functions and to the Legendre oscillator. We prove that the kernel of the finite Fourier transform (FFT) of *L*<sup>2</sup> functions supported on ½ � �1, 1 form a set of GCS. Otherwise, the wavelet transform is the special case of CS associated respectively with the Weyl-Heisenberg group (which gives the canonical CS) and with the affine group on the line. We recall the wavelet theory on **R**. As an application, we discuss the continuous Bessel wavelet. Thus, coherent state transformation (CST) and continuous Bessel wavelet transformation (CBWT) are defined. This chapter is mainly devoted to the application of the Bessel function.

**Keywords:** coherent state, Hankel transformation, Bessel wavelet transformation

#### **1. Introduction**

Coherent state (CS) was originally introduced by Schrödinger in 1926 as a Gaussian wavepacket to describe the evolution of a harmonic oscillator [1].

The notion of coherence associated with these states of physics was first noticed by Glauber [2, 3] and then introduced by Klauder [4, 5]. Because of their important properties these states were then generalized to other systems either from a physical or mathematical point of view. As the electromagnetic field in free space can be regarded as a superposition of many classical modes, each one governed by the equation of simple harmonic oscillator, the CS became significant as the tool for connecting quantum and classical optics. For a review of all of these generalizations see [6–9].

Four main methods are well used in the literature to build CS, the so-called Schrödinger, Klauder-Perelomov, Barut-Girardello and Gazeau-Klauder approaches. The second and third approaches are based directly on the Lie algebra symmetries with their corresponding generators, the first is only established by means of an appropriate infinite superposition of wave functions associated with the harmonic oscillator whatever the Lie algebra symmetries. In [10–12] the authors introduced a new family of CS as a suitable superposition of the associated Bessel functions and in [13–15] the authors also use the generating function approach to construct a new type CS associated with Hermite polynomials and the associated Legendre functions, respectively. The important fact is that we do not use algebraic and group approaches (Barut-Girardello and Klauder-Perelomov) to construct generalized coherent states (GCS).

We first discuss GCS associated with a one-dimensional Schrödinger operator [16, 17] by following the work in [18, 19]. We build a family of GCS through superpositions of the corresponding eigenstates, say *ψn*, *n* ∈ , which are expressed in terms of the Legendre polynomial *Pn*ð Þ *<sup>x</sup>* [16]. The role of coefficients *<sup>z</sup>n<sup>=</sup>* ffiffiffiffi *<sup>n</sup>*! <sup>p</sup> of the canonical CS is played by

$$\mathfrak{D}\_n(\xi) \coloneqq i^n \left( \frac{\pi(2n+1)}{2\xi} \right)^{\frac{1}{2}} J\_{n+\frac{1}{2}}(\xi), \quad n = 0, 1, 2, \dots, \tag{1}$$

where *<sup>ξ</sup>*<sup>∈</sup> and *Jn*þ<sup>1</sup> 2 ð Þ*:* denotes the cylindrical Bessel function [20]. When *n* ¼ 0, Eq. (1) becomes

$$\mathfrak{O}\_0 = \mathcal{J}\_0(\xi) = \frac{\sin(\xi)}{\xi} \tag{2}$$

wavelet, by translation and dilation. This tool permits the representation of *L*<sup>2</sup>

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

functions in a basis well localized in time and in freqency. Wavelets are special functions with special properties which may not be satisfied by other functions. In the current context, our objective is to make a link between the construction of GCS and the theory of wavelets. Therefore, we will talk about coherent state transformation (CST) and the continuous Bessel wavelet transformation (CBWT). The rest of this chapter is organized as follows: Section 2 is devoted to the generalized CS formalism that we are going to use. In Section 3, we briefly introduce the Paley-Wiener space *PW*<sup>Ω</sup> and some notions on Legendre's Hamiltonian. We give in Section 4 a summary concept on the continuous wavelet transform on . In Section 5, we have constructed a class of GCS related to the Bessel cylindrical function for the legendre Hamiltonian. In Section 6, we discuss the theory of CBWT

*<sup>σ</sup>*ð Þ <sup>þ</sup>

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup> , *<sup>w</sup>*<sup>0</sup> <sup>&</sup>gt;0, (4)

, for arbitrary *x*∈ X

*<sup>n</sup>* C *<sup>n</sup>*ð Þ *x* C *<sup>n</sup>*ð Þ*y* , *x*, *y*∈ X *:* (6)

<sup>2</sup> <sup>&</sup>lt; <sup>þ</sup> <sup>∞</sup> (5)

where we show as an example that the function *f* ∈*L*<sup>2</sup>

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

**2. Generalized coherent states formalism**

*<sup>n</sup>*¼<sup>0</sup> be a satisfactory orthogonal basis of <sup>N</sup><sup>2</sup>

X∞ *n*¼0

*<sup>L</sup>*2ð Þ <sup>X</sup> ,*<sup>μ</sup>* . Define the kernel

*n*¼0

*<sup>ϑ</sup><sup>x</sup>* <sup>≔</sup> ð Þ <sup>N</sup> ð Þ *<sup>x</sup>* �1*=*2X<sup>∞</sup>

h i *<sup>ϑ</sup>x*, *<sup>ϑ</sup><sup>x</sup>* <sup>¼</sup> <sup>N</sup> ð Þ *<sup>x</sup>* �<sup>1</sup>X<sup>∞</sup>

*K x*ð Þ , *<sup>y</sup>* <sup>≔</sup> <sup>X</sup><sup>∞</sup>

kernel Hilbert space and N ð Þ *x* ≔ *K x*ð Þ , *x* , *x*∈ X . Define

*ρ*�<sup>1</sup> *<sup>n</sup>* j j C *<sup>n</sup>*ð Þ *x*

*ρ*�<sup>1</sup>

Then, the expression *K x*ð Þ , *<sup>y</sup>* is a reproducing kernel, <sup>N</sup><sup>2</sup> is the corresponding

*n*¼0

*ρ*�<sup>1</sup>

*n*¼0

*<sup>ρ</sup>*�1*=*<sup>2</sup> *<sup>n</sup>* <sup>C</sup> *<sup>n</sup>*ð Þ *<sup>x</sup> <sup>φ</sup>n:*

*<sup>n</sup>* C *<sup>n</sup>*ð Þ *x* C *<sup>n</sup>*ð Þ¼ *x* 1,

such that Ð

space and let N<sup>2</sup> ⊂*L*<sup>2</sup>

where *ρ<sup>n</sup>* ≔ ∥C *<sup>n</sup>*∥<sup>2</sup>

Therefore,

**265**

f g <sup>C</sup> *<sup>n</sup>* <sup>∞</sup>

the chapter.

*f t*ð Þ: <sup>¼</sup> <sup>2</sup>*w*<sup>0</sup> � *<sup>t</sup>*

2 *w*<sup>2</sup>

2

Legesgue's measure on . Finally in Section 7. we gives some concluding remarks on

We follow the generalization of canonical coherent states (CCS) introduced in [18, 19]. The definition of CS as a set of vectors associated with a reproducing kernel is general, it encompasses all the situations encountered in the physical literature. For applications we will work with normalized vectors. Let ð Þ X , *μ* be a measure

<sup>þ</sup> *f t*ð Þ*dσ*ðÞ¼ *<sup>t</sup>* 0 is the mother wavelet where *<sup>d</sup>σ*ð Þ*<sup>t</sup>* is an appropriate

ð Þ X , *μ* be a sub-closed space of infinite dimension. Let


where J0ð Þ*:* denotes the spherical Bessel function of order 0. The choosen coefficients (1) and eigenfunctions (27) (see below) have been used in ([21], p. 1625). We proceed by determining the wavefunctions of these GCS in a closed form. The latter gives the kernel of the associated CS transform which makes correspondence between the quantum states Hilbert space *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup> *dx* � � of the Legendre oscillator and a subspace of a Hilbert space of square integrable functions with respect to a suitable measure on the real line. We show that the kernel *eix<sup>ξ</sup>*, *ξ*∈ , of the *L*<sup>2</sup> -functions that are supported in ½ � �1, 1 form a set of GCS.

There are in literature several approach to introducce Bessel Wavelets. We refer for instence to [22, 23]. Note that, for ½ � �1, 1 ∍ *x* ↦ cosð Þ *y=n* , *n*∈ , the Legendre polynomial *Pn*ð Þ *x* and the Bessel function of order 0 are related by the Hansen's limit

$$\lim\_{n \to \infty} P\_n \left( \cos \frac{\mathcal{Y}}{n} \right) = \int\_0^\pi e^{i\boldsymbol{\gamma} \cos \phi} d\phi = J\_0(\boldsymbol{\chi}),$$

and the integral

$$\int\_0^\infty J\_0(\mathbf{y}) J\_0(\mathbf{y}) d\mathbf{y} = \frac{\pi}{2}. \tag{3}$$

Note that in [22, 23] the authors have introduced the Bessel wavelet based on the Hankel transform. The notion of wavelets was first introduced by J. Morlet a French petroleum engineer at ELF-Aquitaine, in connection with his study of seismic traces. The mathematical foundations were given by A. Grossmann and J. Morlet [24]. Harmonic analyst Y. Meyer and other mathematicians understood the importance of this theory and they recognized many classical results within (see [25–27]). Classical wavelets have several applications ranging from geophysical and acoustic signal analysis to quantum theory and pure mathematics. A wavelet base is a family of functions obtained from a function known as mother

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

wavelet, by translation and dilation. This tool permits the representation of *L*<sup>2</sup> functions in a basis well localized in time and in freqency. Wavelets are special functions with special properties which may not be satisfied by other functions. In the current context, our objective is to make a link between the construction of GCS and the theory of wavelets. Therefore, we will talk about coherent state transformation (CST) and the continuous Bessel wavelet transformation (CBWT).

The rest of this chapter is organized as follows: Section 2 is devoted to the generalized CS formalism that we are going to use. In Section 3, we briefly introduce the Paley-Wiener space *PW*<sup>Ω</sup> and some notions on Legendre's Hamiltonian. We give in Section 4 a summary concept on the continuous wavelet transform on . In Section 5, we have constructed a class of GCS related to the Bessel cylindrical function for the legendre Hamiltonian. In Section 6, we discuss the theory of CBWT where we show as an example that the function *f* ∈*L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup>

$$f(t) \colon= \frac{2w\_0 - t^2}{2\left(w\_0^2 + t^2\right)^{5/2}}, \quad w\_0 > 0,\tag{4}$$

such that Ð <sup>þ</sup> *f t*ð Þ*dσ*ðÞ¼ *<sup>t</sup>* 0 is the mother wavelet where *<sup>d</sup>σ*ð Þ*<sup>t</sup>* is an appropriate Legesgue's measure on . Finally in Section 7. we gives some concluding remarks on the chapter.

#### **2. Generalized coherent states formalism**

We follow the generalization of canonical coherent states (CCS) introduced in [18, 19]. The definition of CS as a set of vectors associated with a reproducing kernel is general, it encompasses all the situations encountered in the physical literature. For applications we will work with normalized vectors. Let ð Þ X , *μ* be a measure space and let N<sup>2</sup> ⊂*L*<sup>2</sup> ð Þ X , *μ* be a sub-closed space of infinite dimension. Let f g <sup>C</sup> *<sup>n</sup>* <sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup> be a satisfactory orthogonal basis of <sup>N</sup><sup>2</sup> , for arbitrary *x*∈ X

$$\sum\_{n=0}^{\infty} \rho\_n^{-1} |\mathfrak{G}\_n(\mathbf{x})|^2 < +\infty \tag{5}$$

where *ρ<sup>n</sup>* ≔ ∥C *<sup>n</sup>*∥<sup>2</sup> *<sup>L</sup>*2ð Þ <sup>X</sup> ,*<sup>μ</sup>* . Define the kernel

$$K(\boldsymbol{x}, \boldsymbol{y}) \coloneqq \sum\_{n=0}^{\infty} \rho\_n^{-1} \mathfrak{G}\_n(\boldsymbol{x}) \overline{\mathfrak{G}\_n(\boldsymbol{y})}, \ \boldsymbol{x}, \boldsymbol{y} \in \mathcal{K}. \tag{6}$$

Then, the expression *K x*ð Þ , *<sup>y</sup>* is a reproducing kernel, <sup>N</sup><sup>2</sup> is the corresponding kernel Hilbert space and N ð Þ *x* ≔ *K x*ð Þ , *x* , *x*∈ X . Define

$$\mathfrak{G}\_{\mathfrak{x}} \coloneqq \left(\mathcal{J}'(\mathfrak{x})\right)^{-1/2} \sum\_{n=0}^{\infty} \rho\_n^{-1/2} \overline{\mathfrak{G}\_n(\mathfrak{x})} \rho\_n.$$

Therefore,

$$
\langle \theta\_{\mathbf{x}}, \theta\_{\mathbf{x}} \rangle = \mathcal{J}'(\mathbf{x})^{-1} \sum\_{n=0}^{\infty} \rho\_n^{-1} \theta\_n(\mathbf{x}) \overline{\theta\_n(\mathbf{x})} = 1,
$$

introduced a new family of CS as a suitable superposition of the associated Bessel functions and in [13–15] the authors also use the generating function approach to construct a new type CS associated with Hermite polynomials and the associated Legendre functions, respectively. The important fact is that we do not use algebraic and group approaches (Barut-Girardello and Klauder-Perelomov) to construct gen-

We first discuss GCS associated with a one-dimensional Schrödinger operator

2 *Jn*þ<sup>1</sup> 2

<sup>O</sup><sup>0</sup> <sup>¼</sup> <sup>J</sup>0ð Þ¼ *<sup>ξ</sup>* sin ð Þ*<sup>ξ</sup>*

where J0ð Þ*:* denotes the spherical Bessel function of order 0. The choosen coefficients (1) and eigenfunctions (27) (see below) have been used in ([21], p. 1625). We proceed by determining the wavefunctions of these GCS in a closed form. The latter gives the kernel of the associated CS transform which makes correspondence between the quantum states Hilbert space *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup>

Legendre oscillator and a subspace of a Hilbert space of square integrable functions

There are in literature several approach to introducce Bessel Wavelets. We refer for instence to [22, 23]. Note that, for ½ � �1, 1 ∍ *x* ↦ cosð Þ *y=n* , *n*∈ , the Legendre polynomial *Pn*ð Þ *x* and the Bessel function of order 0 are related by the


*iy* cos *<sup>ϕ</sup>d<sup>ϕ</sup>* <sup>¼</sup> *<sup>J</sup>*0ð Þ*<sup>y</sup>* ,

2

with respect to a suitable measure on the real line. We show that the kernel

lim*<sup>n</sup>*!<sup>∞</sup>*Pn* cos *<sup>y</sup>*

*n* � �

> ð<sup>∞</sup> 0

J. Morlet [24]. Harmonic analyst Y. Meyer and other mathematicians

¼ ð*π* 0 *e*

*<sup>J</sup>*0ð Þ*<sup>y</sup> <sup>J</sup>*0ð Þ*<sup>y</sup> dy* <sup>¼</sup> *<sup>π</sup>*

Note that in [22, 23] the authors have introduced the Bessel wavelet based on the

understood the importance of this theory and they recognized many classical results within (see [25–27]). Classical wavelets have several applications ranging from geophysical and acoustic signal analysis to quantum theory and pure mathematics. A wavelet base is a family of functions obtained from a function known as mother

Hankel transform. The notion of wavelets was first introduced by J. Morlet a French petroleum engineer at ELF-Aquitaine, in connection with his study of seismic traces. The mathematical foundations were given by A. Grossmann and

*<sup>n</sup>*! <sup>p</sup> of

ð Þ*ξ* , *n* ¼ 0, 1, 2, … , (1)

*<sup>ξ</sup>* (2)

*:* (3)

*dx* � � of the

ð Þ*:* denotes the cylindrical Bessel function [20]. When *n* ¼ 0,

[16, 17] by following the work in [18, 19]. We build a family of GCS through superpositions of the corresponding eigenstates, say *ψn*, *n* ∈ , which are expressed in terms of the Legendre polynomial *Pn*ð Þ *<sup>x</sup>* [16]. The role of coefficients *<sup>z</sup>n<sup>=</sup>* ffiffiffiffi

> *<sup>n</sup> π*ð Þ 2*n* þ 1 2*ξ* � �<sup>1</sup>

eralized coherent states (GCS).

the canonical CS is played by

where *<sup>ξ</sup>*<sup>∈</sup> and *Jn*þ<sup>1</sup>

Eq. (1) becomes

*Wavelet Theory*

*eix<sup>ξ</sup>*, *ξ*∈ , of the *L*<sup>2</sup>

and the integral

Hansen's limit

**264**

O*n*ð Þ*ξ* ≔ *i*

2

and

$$\mathcal{W}: \mathcal{H} \to \mathfrak{N}^2 \quad \text{with} \quad \mathcal{W}\phi = \mathcal{H}^{1/2} \langle \theta\_{\mathfrak{x}}, \phi \rangle.$$

is an isometry. For *ϕ*, *ψ* ∈ H , whe have

$$
\langle \phi, \psi \rangle\_{\mathcal{H}} = \langle \mathcal{W}\phi, \mathcal{W}\psi \rangle\_{\mathcal{H}^2} = \int\_{\mathcal{X}} \overline{\mathcal{W}\phi(\mathbf{x})} \mathcal{W}\psi(\mathbf{x}) d\mu(\mathbf{x}) \tag{7}
$$

$$=\int\_{\mathcal{X}} \langle \phi, \theta\_{\mathbf{x}} \rangle \langle \theta\_{\mathbf{x}}, \psi \rangle \mathcal{A}'(\mathbf{x}) d\mu(\mathbf{x}),\tag{8}$$

with f g *ϕ<sup>n</sup>*

*π*�1*ezz*, *z*∈ .

**overview**

*F* are verified:

3.as ∣*z*∣ ! þ∞

*PW<sup>p</sup>*

*and we set*

**267**

<sup>Ω</sup> <sup>¼</sup> *<sup>f</sup>* <sup>∈</sup>*L*<sup>2</sup>

The Paley-Wiener *PW<sup>p</sup>*

to denote the Paley-Wiener space *PW*<sup>2</sup>

�

∞

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

**3.1 The Paley-wiener space** *PW***<sup>Ω</sup>**

general overview on this notion ([29], pp. 45–47).

1.For all *ε*> 0 there exists *C<sup>ε</sup>* such that

2.There exists *C* >0 such that

*<sup>n</sup>*¼<sup>0</sup> being an orthonormal basis of eigenstates of the quantum har-

monic oscillator. Then, the space <sup>N</sup><sup>2</sup> is the *Fock space* <sup>F</sup>ð Þ and <sup>N</sup> ð Þ¼ *<sup>z</sup>*

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

**3. The Paley-wiener space** *PW***<sup>Ω</sup> and the Legendre Hamiltonian: a brief**

The Paley-Wiener space is made up of all integer functions of exponential type whose restrictions on the real line is square integrable. We give in this Section a

**Definition 3.** *Consider F as an entire function. Then, F is an entire function of*

<sup>∣</sup>*F z*ð Þ∣ ≤ *Ae<sup>B</sup>*∣*z*<sup>∣</sup>

<sup>Ω</sup> <sup>¼</sup> lim*<sup>r</sup>*!þ<sup>∞</sup>sup log *M r*ð Þ

and where *M r*ð Þ¼ sup∣*z*∣¼*<sup>r</sup>*∣*F z*ð Þ∣. The following conditions on an entire function

∣*F z*ð Þ∣ ≤*Cεe*

<sup>∣</sup>*F z*ð Þ∣ ≤*Ce*Ω∣*z*<sup>∣</sup>

<sup>∣</sup>*F z*ð Þ<sup>∣</sup> <sup>¼</sup> *o e*<sup>Ω</sup>∣*z*<sup>∣</sup> � �*:*

Then cleary, 3ð Þ) ð Þ)2 ð Þ)1 *F* is of exponential type at most Ω. **Definition 4.** *Let* Ω >0 *and* 1≤*p*≤ ∞*. The Paley-Wiener space PW<sup>p</sup>*

> �Ω *g y*ð Þ*e*

∥*f* ∥*PW<sup>p</sup>* Ω

that are supported in ½ � �Ω, Ω . We will be interested in the case *p* ¼ 2, in which *PW<sup>Ω</sup>*

ð Þ : *f x*ð Þ¼ <sup>ð</sup><sup>Ω</sup>

*r*

ð Þ Ωþ*ε* ∣*z*∣ ;

;

�*ixydy*, where *<sup>g</sup>* <sup>∈</sup> *Lp*ð�Ω, <sup>Ω</sup>Þg

<sup>Ω</sup> is the image via the *Fourier transform* of the *Lp*-function

*<sup>ω</sup>*. From the *Plancherel formula* we have

¼ 2*π*∥*g*∥*L<sup>p</sup> :* (18)

*:* (15)

(16)

<sup>Ω</sup> *is defined as*

(17)

*exponential type if there exists constants A*, *B* >0 *such that, for all z*∈

Note that, if *F* satisfy Definition 3, we call Ω the type of *F* where

and

$$\int\_{\mathcal{X}} |\mathfrak{G}\_{\mathbf{x}}\rangle \langle \mathfrak{G}\_{\mathbf{x}}| \mathcal{N}(\mathbf{x}) d\mu(\mathbf{x}) = I\_{\mathcal{A}^{\mathbb{R}}},\tag{9}$$

where N ð Þ *x* is a positive weight function.

**Definition 1.** *Let* H *be a Hilbert space with* dim H ¼ ∞ *and* f g *φ<sup>n</sup>* ∞ *<sup>n</sup>*¼<sup>0</sup> *be an orthonormal basis of* H *.The generalized coherent state (GCS) labeled by point x* ∈ X *are defined as the ket-vector ϑ<sup>x</sup>* ∈ H *, such that*

$$\mathfrak{G}\_{\mathbf{x}} \coloneqq \left(\mathcal{J}'(\mathbf{x})\right)^{-1/2} \sum\_{n=0}^{\infty} \rho\_n^{-1/2} \overline{\mathfrak{G}\_n(\mathbf{x})} \rho\_n. \tag{10}$$

*By definition, it is straightforward to show that* h i *ϑx*, *ϑ<sup>x</sup>* <sup>H</sup> ¼ 1.

**Definition 2.** *For each function f* ∈ H *, the coherent state transform (CST) associated to the set* ð Þ *ϑ<sup>x</sup> <sup>x</sup>*<sup>∈</sup> <sup>X</sup> *is the isometric map*

$$\mathcal{W}[f](\mathbf{x}) \coloneqq \left(\mathcal{N}(\mathbf{x})\right)^{1/2} \langle f|\mathfrak{d}\_{\mathbf{x}}\rangle\_{\mathcal{M}'}.\tag{11}$$

*Thereby, we have a resolution of the identity of* H *which can be expressed in Dirac's bra-ket notation as*

$$\mathcal{I}\_{\mathcal{M}} = \int\_{\mathcal{X}} T\_{\mathbf{x}} \mathcal{N}(\mathbf{x}) d\mu(\mathbf{x}) \tag{12}$$

*where the rank one operator Tx* ≔ j i *ϑ<sup>x</sup>* h j *ϑ<sup>x</sup>* : H ! H *is define by*

$$f \mapsto T\_{\mathfrak{x}}[f] = \langle \mathfrak{g}\_{\mathfrak{x}}[f] \mathfrak{g}\_{\mathfrak{x}} \rangle$$

N ð Þ *x appears as a weight function.*

Next, the reproducing kernel has the additional property of being square integrable, i.e.,

$$\int\_{\mathcal{X}} K(\mathbf{x}, \mathbf{z}) K(\mathbf{z}, \mathbf{y}) \mathcal{N}(\mathbf{z}) d\mu(\mathbf{z}) = K(\mathbf{x}, \mathbf{y}).\tag{13}$$

Note that the formula (10) can be considered as generalization of the series expansion of the CCS [28].

$$\mathfrak{G}\_x = \sqrt{\pi} e^{-\frac{x^2}{2}} \sum\_{k=0}^{\infty} \frac{z^n}{\sqrt{n!}} \phi\_n, \ z \in \mathbb{C} \tag{14}$$

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

with f g *ϕ<sup>n</sup>* ∞ *<sup>n</sup>*¼<sup>0</sup> being an orthonormal basis of eigenstates of the quantum harmonic oscillator. Then, the space <sup>N</sup><sup>2</sup> is the *Fock space* <sup>F</sup>ð Þ and <sup>N</sup> ð Þ¼ *<sup>z</sup> π*�1*ezz*, *z*∈ .

#### **3. The Paley-wiener space** *PW***<sup>Ω</sup> and the Legendre Hamiltonian: a brief overview**

#### **3.1 The Paley-wiener space** *PW***<sup>Ω</sup>**

and

*Wavelet Theory*

and

<sup>W</sup> : <sup>H</sup> ! <sup>N</sup><sup>2</sup> *with* <sup>W</sup> *<sup>ϕ</sup>* <sup>¼</sup> <sup>N</sup> <sup>1</sup>*=*<sup>2</sup>

ð X

is an isometry. For *ϕ*, *ψ* ∈ H , whe have

h i *ϕ*, *ψ* <sup>H</sup> ¼ h i W *ϕ*, W *ψ* <sup>N</sup><sup>2</sup> ¼

¼ ð X

ð X

**Definition 1.** *Let* H *be a Hilbert space with* dim H ¼ ∞ *and* f g *φ<sup>n</sup>*

*<sup>ϑ</sup><sup>x</sup>* <sup>≔</sup> ð Þ <sup>N</sup> ð Þ *<sup>x</sup>* �1*=*2X<sup>∞</sup>

*By definition, it is straightforward to show that* h i *ϑx*, *ϑ<sup>x</sup>* <sup>H</sup> ¼ 1.

*1*<sup>H</sup> ¼ ð X

*<sup>ϑ</sup><sup>z</sup>* <sup>¼</sup> ffiffiffi *<sup>π</sup>* <sup>p</sup> *<sup>e</sup>* �*zz* 2 X∞ *k*¼0

*where the rank one operator Tx* ≔ j i *ϑ<sup>x</sup>* h j *ϑ<sup>x</sup>* : H ! H *is define by*

W ½ � *f* ð Þ *x* ≔ ð Þ N ð Þ *x*

*orthonormal basis of* H *.The generalized coherent state (GCS) labeled by point x* ∈ X

*n*¼0

**Definition 2.** *For each function f* ∈ H *, the coherent state transform (CST) associ-*

*Thereby, we have a resolution of the identity of* H *which can be expressed in Dirac's*

*f* ↦ *Tx*½ �¼ *f* h i *ϑx*j*f ϑx:*

Next, the reproducing kernel has the additional property of being square

Note that the formula (10) can be considered as generalization of the series

*zn* ffiffiffiffi

1*=*2

where N ð Þ *x* is a positive weight function.

*are defined as the ket-vector ϑ<sup>x</sup>* ∈ H *, such that*

*ated to the set* ð Þ *ϑ<sup>x</sup> <sup>x</sup>*<sup>∈</sup> <sup>X</sup> *is the isometric map*

N ð Þ *x appears as a weight function.*

ð X

*bra-ket notation as*

integrable, i.e.,

**266**

expansion of the CCS [28].

h i *ϑx*, *ϕ*

h i *ϕ*, *ϑ<sup>x</sup>* h i *ϑx*, *ψ* N ð Þ *x dμ*ð Þ *x* , (8)

j i *ϑ<sup>x</sup>* h j *ϑ<sup>x</sup>* N ð Þ *x dμ*ð Þ¼ *x I*<sup>H</sup> , (9)

W *ϕ*ð Þ *x* W *ψ*ð Þ *x dμ*ð Þ *x* (7)

∞ *<sup>n</sup>*¼<sup>0</sup> *be an*

*<sup>ρ</sup>*�1*=*<sup>2</sup> *<sup>n</sup>* <sup>C</sup> *<sup>n</sup>*ð Þ *<sup>x</sup> <sup>φ</sup>n:* (10)

h i *f*j*ϑ<sup>x</sup>* <sup>H</sup> *:* (11)

*Tx*N ð Þ *x dμ*ð Þ *x* (12)

*<sup>n</sup>*! <sup>p</sup> *<sup>ϕ</sup>n*, *<sup>z</sup>*<sup>∈</sup> (14)

*K x*ð Þ , *z K z*ð Þ , *y* N ð Þ*z dμ*ð Þ¼ *z K x*ð Þ , *y :* (13)

The Paley-Wiener space is made up of all integer functions of exponential type whose restrictions on the real line is square integrable. We give in this Section a general overview on this notion ([29], pp. 45–47).

**Definition 3.** *Consider F as an entire function. Then, F is an entire function of exponential type if there exists constants A*, *B* >0 *such that, for all z*∈

$$|F(z)| \le Ae^{B|x|}. \tag{15}$$

Note that, if *F* satisfy Definition 3, we call Ω the type of *F* where

$$\Omega = \lim\_{r \to +\infty} \sup \frac{\log M(r)}{r} \tag{16}$$

and where *M r*ð Þ¼ sup∣*z*∣¼*<sup>r</sup>*∣*F z*ð Þ∣. The following conditions on an entire function *F* are verified:

1.For all *ε*> 0 there exists *C<sup>ε</sup>* such that

$$|F(z)| \le C\_\varepsilon e^{(\Omega + \varepsilon)|x|};$$

2.There exists *C* >0 such that

$$|F(z)| \le C e^{\Omega |z|};$$

3.as ∣*z*∣ ! þ∞

$$|F(x)| = o\left(e^{\Omega|x|}\right).$$

Then cleary, 3ð Þ) ð Þ)2 ð Þ)1 *F* is of exponential type at most Ω. **Definition 4.** *Let* Ω >0 *and* 1≤*p*≤ ∞*. The Paley-Wiener space PW<sup>p</sup>* <sup>Ω</sup> *is defined as*

$$P\mathcal{W}\_{\Omega}^{p} = \left\{ f \in L^{2}(\mathbb{R}) : f(\mathbf{x}) = \int\_{-\Omega}^{\Omega} g(\mathbf{y}) e^{-i\mathbf{x}\mathbf{y}} d\mathbf{y}, \text{ where } \mathbf{g} \in L^{p}(-\Omega, \Omega) \right\} \tag{17}$$

*and we set*

$$\|f\|\_{PW\_{\Omega}^p} = 2\pi \|\mathbf{g}\|\_{L^p} \,. \tag{18}$$

The Paley-Wiener *PW<sup>p</sup>* <sup>Ω</sup> is the image via the *Fourier transform* of the *Lp*-function that are supported in ½ � �Ω, Ω . We will be interested in the case *p* ¼ 2, in which *PW<sup>Ω</sup>* to denote the Paley-Wiener space *PW*<sup>2</sup> *<sup>ω</sup>*. From the *Plancherel formula* we have

$$\|f\|\_{PW\_{\Omega}^2} = \|\hat{\mathfrak{g}}\|\_{PW\_{\Omega}^2} = 2\pi\|\mathfrak{g}\|\_{L^2} = \|\hat{f}\|\_{L^2} = \|f\|\_{L^2}.\tag{19}$$

Hence, by polarization, for *f*, *φ* ∈*PW*Ω,

$$
\langle f, \rho \rangle\_{\text{PW}\_{\Omega}} = \langle f, \rho \rangle\_{L^2}. \tag{20}
$$

in terms of the Legendre polynomial *Pn*ð Þ*:* , which form an orthonormal basis

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>2</sup> ð Þ 2*n* þ 1 ð Þ 2*n* þ 3

The generalized position operator on the Hilbert space H connected with the Legendre polynomials *Pn*ð Þ *x* is an operator of multiplication by argument *Xψ<sup>n</sup>* ¼

self-adjoint operator on the Hilbert space H (see [30–32]). The momentum operator *P* by the way described in ([17], p. 126) acts on the basis elements in H , by the

� �*ψ<sup>n</sup>* <sup>¼</sup> <sup>2</sup>*<sup>i</sup>*

The creation and annihilation operators (25) are define by relations

2

**4. Wavelet theory on and the reproduction of kernels**

C *<sup>ψ</sup>* ≔

dilatation and translation, one gets affine transformation

<sup>p</sup> ð Þ *<sup>X</sup>* � *iP* ; *<sup>a</sup>*� <sup>¼</sup> <sup>1</sup>

We briefly describe below some basis definitions and properties of the one-dimensional wavelet transform on þ, we refer to [22, 23, 33]. In the Hilbert

∞ð

j j *ψ ξ* ^ð Þ <sup>2</sup>

where *ψ*^ being the Hankel transform of *ψ*. Not every vector in N satisfies the above condition. A vector *ψ* satisfying (33) is called a mother wavelet. Combining

�∞

*Xψn*ð Þ¼ *x bnψ<sup>n</sup>*þ<sup>1</sup>ð Þþ *x bn*�<sup>1</sup>*ψ<sup>n</sup>*�<sup>1</sup>ð Þ *x* , (30)

� �*:* The usual commutator of operator *X* and *P* on

ffiffi 2

<sup>p</sup> *bnψ<sup>n</sup>*þ<sup>1</sup> and *<sup>a</sup>*�*ψ<sup>n</sup>* <sup>¼</sup> ffiffi

ð Þ , *dx* , the function *ψ* satisfying the so-called admissibility condition

*y* ¼ ð Þ *b*, *a x* � *ax* þ *b*, *a*>0, *b*∈ , *x*∈ þ*:* (34)

ð Þ <sup>2</sup>*<sup>n</sup>* � <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>3</sup> *<sup>ψ</sup>n:* (31)

2

*<sup>ξ</sup> <sup>d</sup>ξ*<sup>&</sup>lt; <sup>∞</sup>, (33)

p ð Þ *X* þ *iP* , (32)

<sup>p</sup> *bn*�<sup>1</sup>*ψ<sup>n</sup>*�1. They satisfy

*xψn*ð Þ¼ *x bn*�1*ψn*�1ð Þþ *x bnψn*þ1ð Þ *x* , *ψ*�1ð Þ¼ *x* 0, *ψ*0ð Þ¼ *x* 1, (28)

*dx* � �. These functions satisfy

, *n* ≥0*:* (29)

*<sup>n</sup>*¼<sup>0</sup>1*=bn* ¼ þ∞, *<sup>X</sup>* is a

*<sup>n</sup>*¼<sup>0</sup> in the Hilbert space <sup>H</sup> <sup>≔</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup>

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

s

whee *bn* are coefficients defined by Eq. (29). Because P<sup>∞</sup>

*<sup>n</sup>* � *<sup>b</sup>*<sup>2</sup> *n*�1

*<sup>a</sup>*<sup>þ</sup> <sup>¼</sup> <sup>1</sup> ffiffi 2

*bn* ¼

*xψn:* Taking into account of the relation (28), then

formula *Pψ<sup>n</sup>* ¼ *i bnψ<sup>n</sup>*þ<sup>1</sup> � *bn*�<sup>1</sup>*ψ<sup>n</sup>*�<sup>1</sup>

½ � *<sup>X</sup>*, *<sup>P</sup> <sup>ψ</sup><sup>n</sup>* <sup>¼</sup> <sup>2</sup>*i b*<sup>2</sup>

these operators act as *<sup>a</sup>*þ*ψ<sup>n</sup>* <sup>¼</sup> ffiffi

*a*�, *a*<sup>þ</sup> ½ �¼�*i X*½ � , *P* , the commutation relations.

the basis elements reads as

space <sup>N</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup>

**269**

f g *<sup>ψ</sup><sup>n</sup>* � j i *<sup>n</sup>* <sup>∞</sup>

the recurrence relations

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

with coefficients

Theorem 1.1 Let *F* be an entire function and Ω >0. Then the following are equivalent

• *F*∣ ∈*L*<sup>2</sup> ð Þ and

$$|F(z)| = o\left(e^{\Omega|z|}\right) \text{ as } |z| \to +\infty,\tag{21}$$

• there exists *f* ∈*L*<sup>2</sup> ð Þ with *supp*^*<sup>f</sup>* <sup>⊆</sup>½ � �Ω, <sup>Ω</sup> such that

$$F(z) = \frac{1}{2\pi} \int\_{\mathbb{R}} \hat{f}(\xi) e^{i\varepsilon\xi} d\xi. \tag{22}$$

The function *f* ∈ *PW*<sup>Ω</sup> if and only if *f* ∈*L*<sup>2</sup> ð Þ and *f* ¼ *F*∣ (that is, *f* is the restriction to the real line of a function *F*), where *F* is an entire function of exponential type such that <sup>∣</sup>*F z*ð Þ<sup>∣</sup> <sup>¼</sup> *o e*Ω∣*z*<sup>∣</sup> � � for <sup>∣</sup>*z*<sup>∣</sup> ! þ∞.

Theorem 1.2 The Paley-Wiener space *PW*<sup>Ω</sup> is a Hilbert space with reproducing kernel w.r.t the inner product (20). Its reproducing kernel is the function

$$K(\boldsymbol{x}, \boldsymbol{y}) = \frac{\boldsymbol{\Omega}}{\pi} \text{sinc}(\boldsymbol{\Omega}(\boldsymbol{x} - \boldsymbol{y})),\tag{23}$$

*where sinct* ¼ *sint=t*. Hence, for every *f* ∈*PW*<sup>Ω</sup>

$$f(\mathbf{x}) = \frac{\Omega}{\pi} \int\_{\mathbb{R}} f(\mathbf{y}) \text{sinc}(\Omega(\mathbf{x} - \mathbf{y})) d\mathbf{y},\tag{24}$$

where *x*∈ .

#### **3.2 The Legendre Hamiltonian**

The Legendre polynomials *Pn*ð Þ *x* and the Legendre function *ψn*ð Þ *x* are similar to the Hermite polynomials and the Hermite function in standard quantum mechanics. Based on the work of Borzov and Demaskinsky [16, 17] the Legendre Hamiltonian has the form

$$H = X^2 + P^2 = a^+a^- + a^-a^+,\tag{25}$$

where *X* and *P* denotes respectively the position and momentum operators, *a*<sup>þ</sup> and *a*� are the creation and annihilation operators. The eigenvalues of operators *H* are equal to

$$
\lambda\_0 = \frac{2}{3}, \quad \lambda\_n = \frac{n(n+1) - \frac{1}{2}}{\left(n + \frac{3}{2}\right)\left(n - \frac{1}{2}\right)}, n = 1, 2, 3, \dots \quad , \tag{26}
$$

and the corresponding eigenfunctions reads

$$\psi\_n(\mathbf{x}) = \sqrt{2n + 1} P\_n(\mathbf{x}), \quad n = 0, 1, 2, 3, \dots \quad , \tag{27}$$

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

in terms of the Legendre polynomial *Pn*ð Þ*:* , which form an orthonormal basis f g *<sup>ψ</sup><sup>n</sup>* � j i *<sup>n</sup>* <sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup> in the Hilbert space <sup>H</sup> <sup>≔</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup> *dx* � �. These functions satisfy the recurrence relations

$$a\infty\mu\_n(\mathbf{x}) = b\_{n-1}\mu\_{n-1}(\mathbf{x}) + b\_n\mu\_{n+1}(\mathbf{x}), \quad \mu\_{-1}(\mathbf{x}) = \mathbf{0}, \ \mu\_0(\mathbf{x}) = \mathbf{1},\tag{28}$$

with coefficients

∥*f* ∥*PW*<sup>2</sup>

Hence, by polarization, for *f*, *φ* ∈*PW*Ω,

The function *f* ∈ *PW*<sup>Ω</sup> if and only if *f* ∈*L*<sup>2</sup>

exponential type such that <sup>∣</sup>*F z*ð Þ<sup>∣</sup> <sup>¼</sup> *o e*Ω∣*z*<sup>∣</sup> � � for <sup>∣</sup>*z*<sup>∣</sup> ! þ∞.

*where sinct* ¼ *sint=t*. Hence, for every *f* ∈*PW*<sup>Ω</sup>

*f x*ð Þ¼ <sup>Ω</sup> *π* ð 

equivalent

*Wavelet Theory*

• *F*∣ ∈*L*<sup>2</sup>

ð Þ and

• there exists *f* ∈*L*<sup>2</sup>

where *x*∈ .

**3.2 The Legendre Hamiltonian**

*<sup>λ</sup>*<sup>0</sup> <sup>¼</sup> <sup>2</sup> 3

and the corresponding eigenfunctions reads

*<sup>ψ</sup>n*ð Þ¼ *<sup>x</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffi

Hamiltonian has the form

are equal to

**268**

<sup>Ω</sup> ¼ ∥^*g*∥*PW*<sup>2</sup>

Theorem 1.1 Let *F* be an entire function and Ω >0. Then the following are

ð Þ with *supp*^*<sup>f</sup>* <sup>⊆</sup>½ � �Ω, <sup>Ω</sup> such that

*F z*ð Þ¼ <sup>1</sup> 2*π* ð ^*f*ð Þ*<sup>ξ</sup> <sup>e</sup> izξ*

restriction to the real line of a function *F*), where *F* is an entire function of

kernel w.r.t the inner product (20). Its reproducing kernel is the function

Ω

*K x*ð Þ¼ , *y*

Theorem 1.2 The Paley-Wiener space *PW*<sup>Ω</sup> is a Hilbert space with reproducing

The Legendre polynomials *Pn*ð Þ *x* and the Legendre function *ψn*ð Þ *x* are similar

where *X* and *P* denotes respectively the position and momentum operators, *a*<sup>þ</sup> and *a*� are the creation and annihilation operators. The eigenvalues of operators *H*

2

2

to the Hermite polynomials and the Hermite function in standard quantum mechanics. Based on the work of Borzov and Demaskinsky [16, 17] the Legendre

, *<sup>λ</sup><sup>n</sup>* <sup>¼</sup> *n n*ð Þ� <sup>þ</sup> <sup>1</sup> <sup>1</sup>

*<sup>n</sup>* <sup>þ</sup> <sup>3</sup> 2 � � *<sup>n</sup>* � <sup>1</sup>

<sup>Ω</sup> <sup>¼</sup> <sup>2</sup>*π*∥*g*∥*L*<sup>2</sup> <sup>¼</sup> <sup>∥</sup>^*<sup>f</sup>* <sup>∥</sup>*L*<sup>2</sup> <sup>¼</sup> <sup>∥</sup>*<sup>f</sup>* <sup>∥</sup>*L*<sup>2</sup> *:* (19)

h i *f*, *φ PW*<sup>Ω</sup> ¼ h i *f*, *φ <sup>L</sup>*<sup>2</sup> *:* (20)

<sup>∣</sup>*F z*ð Þ<sup>∣</sup> <sup>¼</sup> *o e*<sup>Ω</sup>∣*z*<sup>∣</sup> � � as <sup>∣</sup>*z*<sup>∣</sup> ! þ∞, (21)

*dξ:* (22)

ð Þ and *f* ¼ *F*∣ (that is, *f* is the

*<sup>π</sup> sinc*ð Þ <sup>Ω</sup>ð Þ *<sup>x</sup>* � *<sup>y</sup>* , (23)

*f y*ð Þ*sinc*ð Þ Ωð Þ *x* � *y dy*, (24)

*<sup>H</sup>* <sup>¼</sup> *<sup>X</sup>*<sup>2</sup> <sup>þ</sup> *<sup>P</sup>*<sup>2</sup> <sup>¼</sup> *<sup>a</sup>*þ*a*� <sup>þ</sup> *<sup>a</sup>*�*a*þ, (25)

<sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup> *Pn*ð Þ *<sup>x</sup>* , *<sup>n</sup>* <sup>¼</sup> 0, 1, 2, 3, *::* , (27)

� � , *<sup>n</sup>* <sup>¼</sup> 1, 2, 3, … , (26)

$$b\_n = \sqrt{\frac{\left(n+1\right)^2}{\left(2n+1\right)\left(2n+3\right)}}, \quad n \ge 0. \tag{29}$$

The generalized position operator on the Hilbert space H connected with the Legendre polynomials *Pn*ð Þ *x* is an operator of multiplication by argument *Xψ<sup>n</sup>* ¼ *xψn:* Taking into account of the relation (28), then

$$X\psi\_n(\mathbf{x}) = b\_n \psi\_{n+1}(\mathbf{x}) + b\_{n-1} \psi\_{n-1}(\mathbf{x}),\tag{30}$$

whee *bn* are coefficients defined by Eq. (29). Because P<sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>1*=bn* ¼ þ∞, *<sup>X</sup>* is a self-adjoint operator on the Hilbert space H (see [30–32]). The momentum operator *P* by the way described in ([17], p. 126) acts on the basis elements in H , by the formula *Pψ<sup>n</sup>* ¼ *i bnψ<sup>n</sup>*þ<sup>1</sup> � *bn*�<sup>1</sup>*ψ<sup>n</sup>*�<sup>1</sup> � �*:* The usual commutator of operator *X* and *P* on the basis elements reads as

$$[X,P]\varphi\_n = 2i(b\_n^2 - b\_{n-1}^2)\varphi\_n = \frac{2i}{(2n-1)(2n+1)(2n+3)}\varphi\_n. \tag{31}$$

The creation and annihilation operators (25) are define by relations

$$a^{+} = \frac{1}{\sqrt{2}}(X - iP); \quad a^{-} = \frac{1}{\sqrt{2}}(X + iP), \tag{32}$$

these operators act as *<sup>a</sup>*þ*ψ<sup>n</sup>* <sup>¼</sup> ffiffi 2 <sup>p</sup> *bnψ<sup>n</sup>*þ<sup>1</sup> and *<sup>a</sup>*�*ψ<sup>n</sup>* <sup>¼</sup> ffiffi 2 <sup>p</sup> *bn*�<sup>1</sup>*ψ<sup>n</sup>*�1. They satisfy *a*�, *a*<sup>þ</sup> ½ �¼�*i X*½ � , *P* , the commutation relations.

#### **4. Wavelet theory on and the reproduction of kernels**

We briefly describe below some basis definitions and properties of the one-dimensional wavelet transform on þ, we refer to [22, 23, 33]. In the Hilbert space <sup>N</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ð Þ , *dx* , the function *ψ* satisfying the so-called admissibility condition

$$\mathcal{H}\_{\psi} := \int\_{-\infty}^{\infty} \frac{|\hat{\boldsymbol{\mu}}(\xi)|^2}{\xi} d\xi < \infty,\tag{33}$$

where *ψ*^ being the Hankel transform of *ψ*. Not every vector in N satisfies the above condition. A vector *ψ* satisfying (33) is called a mother wavelet. Combining dilatation and translation, one gets affine transformation

$$y = (b, a)\mathbf{x} \equiv a\mathbf{x} + b, \ a > \mathbf{0}, \ b \in \mathbb{R}, \ \mathbf{x} \in \mathbb{R}\_+.\tag{34}$$

Thus f g ð Þ *b*, *a* ≕*Gaff* ¼ � ð Þ 0, ∞ , the affine group of the line. Specifically, for each pair ð Þ *a*, *b* of the real numbers, with *a*>0, from translations and dilatations of the function *ψ*, we obtain a family of wavelets *ψa*,*<sup>b</sup>* � �∈ N as

$$
\psi\_{a,b}(\mathbf{x}) = \frac{1}{\sqrt{a}} \boldsymbol{\upmu} \left( \frac{\mathbf{x} - b}{a} \right), \qquad \boldsymbol{\upmu}\_{1,0} = \boldsymbol{\upmu}.\tag{35}
$$

*f x*ð Þ¼ <sup>1</sup> C *<sup>ψ</sup>* ð

*a* >0

ð

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

*b*∈

reproducing kernel associated to the signal is

S *<sup>f</sup>*ð Þ *a*, *b ψa*,*<sup>b</sup>*

<sup>C</sup> *<sup>ψ</sup>* k k*<sup>f</sup>* <sup>2</sup> <sup>¼</sup>

*K<sup>ψ</sup> b*, *a*, *b*<sup>0</sup>

and *ψ*<sup>0</sup> are two mother wavelets such that *ψ*<sup>0</sup> h i j*ψ* 6¼ 0, then

ð ð

*<sup>f</sup>*ð Þ *b*, *a ψ<sup>a</sup>*,*<sup>b</sup>*

�<sup>∗</sup> þ *ψ<sup>a</sup>*,*<sup>b</sup>* � � � *<sup>ψ</sup>*<sup>0</sup>

*dbda*

, *<sup>a</sup>*<sup>0</sup> � � <sup>¼</sup> <sup>1</sup>

1 *ψ*<sup>0</sup> h i j*ψ*

The repoducing kernel Hilbert space <sup>N</sup> <sup>⊂</sup>*L*<sup>2</sup> � <sup>∗</sup>

*Kψ*,*ψ*<sup>0</sup> *b*, *a*; *b*<sup>0</sup>

forms with respect to the mother wavelet *ψ*<sup>0</sup>

The formula (41) generalizes to

�<sup>∗</sup> þ S <sup>0</sup>

ð ð

representation theory [7, 33, 35].

indexed by point *ξ*∈ .

**271**

**5.1 GCS for the Legendre Hamiltonian**

By replacing the coefficients *z<sup>n</sup>=* ffiffiffiffi

*<sup>f</sup>* <sup>¼</sup> <sup>1</sup> *ψ*<sup>0</sup> h i j*ψ* *dadb*

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

The function S *<sup>f</sup>* is the continuous wavelet transform of the signal *f*. The parameter 1*=a* represents the signal frequency of *f* and *b* its time. The conservation

�<sup>þ</sup>

Then, the transform <sup>S</sup> *<sup>f</sup>* is a fonction in the Hilbert space *<sup>L</sup>*<sup>2</sup> � <sup>∗</sup>

, *<sup>a</sup>*<sup>0</sup> � � <sup>¼</sup> <sup>1</sup>

<sup>S</sup> *<sup>f</sup>*ð Þ *<sup>b</sup>*, *<sup>a</sup>* � � �

C *<sup>ψ</sup>*

*a*,*b* � � � *dbda*

*<sup>a</sup>*<sup>2</sup> , *where* <sup>S</sup> <sup>0</sup>

þ

. Then, we have

*ψa*,*b*j*ψ*<sup>0</sup> *a*0 ,*b*0

*<sup>n</sup>*! <sup>p</sup> of the canonical CS by the function <sup>O</sup>*n*ð Þ*<sup>ξ</sup>* in

The vector *ψ*<sup>0</sup> is called the analyzing wavelet and *ψ* the reconstructing wavelet.

C *<sup>ψ</sup>*C *<sup>ψ</sup>*<sup>0</sup> � �<sup>1</sup> 2

is the integral kernel of a unitary map between N*<sup>ψ</sup>*<sup>0</sup> and N*ψ*. The properties of the wavelet transform can be understood in terms of the unitary irreductible representation of the one-dilensional affine group.It is important to note that the Wavelets built on the basis of the group representation theory have all the properties of CS. There is a wole body of work devoted to the study of CS arising from group

**5. Application 1: GCS for the Legendre Hamiltonian and CS transform**

(1) as mentioned in the introduction. We construct in this section a class of GCS

which satisfies the square integrability condition (13) with respect to the measure *dbda=a*2. The corresponding reproducing kernel Hilbert space N*<sup>ψ</sup>* , one see that this is the space of all signal transforms, corresponding to the mother wavelet *ψ*. If *ψ*

� <sup>2</sup> *dbda*

*ψa*,*b*j*ψ<sup>a</sup>*<sup>0</sup> ,*b*0

of the energy of the signal is due to the resolution of the identity (37), so

ð ð

*<sup>a</sup>*<sup>2</sup> , *where* <sup>S</sup> *<sup>f</sup>*ð Þ¼ *<sup>a</sup>*, *<sup>b</sup> <sup>ψ</sup>a*,*b*j*<sup>f</sup>* � �*:* (41)

*<sup>a</sup>*<sup>2</sup> *:* (42)

� �*:* (43)

*<sup>a</sup>*<sup>2</sup> <sup>¼</sup> *<sup>I</sup>*N, (44)

*<sup>a</sup>*,*<sup>b</sup>*j*<sup>f</sup>* � �*:* (45)

*<sup>f</sup>* ð Þ¼ *a*, *b ψ*<sup>0</sup>

� �, consisting of all signal trans-

D E (46)

<sup>þ</sup> , *dbda a*2 � �*:* The

Here *a* is the parameter of dilation (or scale) and *b* is the parameter of translation (or position). It is then easily cheked that

$$\left\|\left|\boldsymbol{\varphi}\_{a,b}(\mathbf{x})\right\|\right\|\_{\mathfrak{N}}^2 = \left\|\left|\boldsymbol{\varphi}(\mathbf{x})\right\|\right\|\_{\mathfrak{N}}^2, \quad \text{for} \quad \text{all} \quad a > \mathbf{0} \quad \text{and} \ b \in \mathbb{R}. \tag{36}$$

Moreover, in terms of the Dirac's bracket notation it is an easy to show that the resolution of the identity

$$\frac{1}{\mathcal{R}\_{\Psi}} \int\_{\mathbb{R} \times \mathbb{R}\_{+}^{\*}} |\boldsymbol{\nu}\_{a,b}\rangle \langle \boldsymbol{\nu}\_{a,b}| \frac{d\boldsymbol{b}da}{a} = I\_{\mathfrak{N}} \tag{37}$$

holds for these vectors (in the weak sense). Here *I*<sup>N</sup> is the identity operator on N. The *continuous wavelet transform* of an arbitrary vector (signal) *f* ∈ N at the scale *a* and the position *b* is given by

$$\mathcal{F}\_f(a,b) = \int\_0^\infty f(t)\nu\_{a,b}(t)dt. \tag{38}$$

The wavelet transform S *<sup>f</sup>*ð Þ *a*, *b* has several properties [34]:

• It is linear in the sense that:

$$
\mathcal{F}\_{af\_1+\beta f\_2}(a,b) = a\mathcal{F}\_{f\_1}(a,b) + \beta\mathcal{F}\_{f\_2}(a,b), \quad \forall a,\beta \in \mathbb{R} \quad and \quad f\_1, f\_2 \in L^2(\mathbb{R}\_+).
$$

• It is translation invariant:

$$\mathcal{F}\_{\mathfrak{r}\_{\emptyset}f}(a,b) = \mathcal{F}\_f(a,b-b')$$

where *τb*<sup>0</sup> refers to the translation of the function *f* by *b*<sup>0</sup> given

$$(\pi\_b f)(\mathfrak{x}) = f\left(\mathfrak{x} - b'\right).$$

• It is dilatation-invariant, in the sense that, if *f* satisfies the invariance dilatation property *f x*ð Þ¼ *λf rx* ð Þ for some *λ*,*r*> 0 fixed then

$$\mathcal{F}\_f(a,b) = \lambda \mathcal{F}\_f(ra,rb). \tag{39}$$

As in Fourier or Hilbert analysis, wavelet analysis provides a Plancherel type relation which permits itself the reconstruction of the analyzed function from its wavelet transform. More precisely we have

$$\langle f, \mathbf{g} \rangle = \frac{1}{\mathcal{G}\_{\boldsymbol{\Psi}}} \int\_{a > 0} \int\_{b \in \mathbb{R}} \mathcal{G}\_f(a, b) \overline{\mathcal{G}\_{\boldsymbol{\S}}(a, b)} \frac{d adb}{a^2}, \quad \forall f, \mathbf{g} \in L^2(\mathbb{R}) \tag{40}$$

which in turns to reconstruct the analyzed function *f* in the *L*<sup>2</sup> - sense from its wavelet transform as

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

$$f(x) = \frac{1}{\mathcal{G}\_{\varphi}} \int\_{a > 0} \int\_{b \in \mathbb{R}} \mathcal{G}\_f(a, b) \varphi\_{a, b} \frac{da db}{a^2}, \quad where \quad \mathcal{G}\_f(a, b) = \langle \varphi\_{a, b} | f \rangle. \tag{41}$$

The function S *<sup>f</sup>* is the continuous wavelet transform of the signal *f*. The parameter 1*=a* represents the signal frequency of *f* and *b* its time. The conservation of the energy of the signal is due to the resolution of the identity (37), so

$$\left|\mathcal{H}\_{\Psi}\left||f\right|\right|^{2} = \int\int\_{\mathbb{R}\times\mathbb{R}\_{+}} \left|\mathcal{F}\_{f}(b,a)\right|^{2} \frac{dbda}{a^{2}}.\tag{42}$$

Then, the transform <sup>S</sup> *<sup>f</sup>* is a fonction in the Hilbert space *<sup>L</sup>*<sup>2</sup> � <sup>∗</sup> <sup>þ</sup> , *dbda a*2 � �*:* The reproducing kernel associated to the signal is

$$K\_{\mathbb{V}}\left(b,a,b',a'\right) = \frac{1}{\mathfrak{G}\_{\mathbb{V}}} \langle \psi\_{a,b} | \psi\_{a',b'} \rangle. \tag{43}$$

which satisfies the square integrability condition (13) with respect to the measure *dbda=a*2. The corresponding reproducing kernel Hilbert space N*<sup>ψ</sup>* , one see that this is the space of all signal transforms, corresponding to the mother wavelet *ψ*. If *ψ* and *ψ*<sup>0</sup> are two mother wavelets such that *ψ*<sup>0</sup> h i j*ψ* 6¼ 0, then

$$\frac{1}{\langle \boldsymbol{\eta}^{\prime} | \boldsymbol{\eta}^{\prime} \rangle} \Bigg\{ \int\_{\mathbb{R} \times \mathbb{R}\_{+}^{\prime}} | \boldsymbol{\eta}\_{a,b} \rangle \langle \boldsymbol{\eta}\_{a,b}^{\prime} | \frac{dbda}{a^{2}} = I\_{\mathfrak{N}},\tag{44}$$

The formula (41) generalizes to

$$f = \frac{1}{\langle \boldsymbol{\mu}' | \boldsymbol{\nu} \rangle} \left[ \int\_{\mathbb{R} \times \mathbb{R}\_+^\*} \mathcal{G}\_f'(b, a) \boldsymbol{\mu}\_{a, b} \frac{dbda}{a^2}, \quad \text{where} \quad \mathcal{G}\_f'(a, b) = \langle \boldsymbol{\nu}\_{a, b}' | \boldsymbol{f} \rangle. \tag{45} \right]$$

The vector *ψ*<sup>0</sup> is called the analyzing wavelet and *ψ* the reconstructing wavelet. The repoducing kernel Hilbert space <sup>N</sup> <sup>⊂</sup>*L*<sup>2</sup> � <sup>∗</sup> þ � �, consisting of all signal transforms with respect to the mother wavelet *ψ*<sup>0</sup> . Then, we have

$$K\_{\boldsymbol{\Psi},\boldsymbol{\Psi}'}\left(\boldsymbol{b},a;\boldsymbol{b}',a'\right) = \frac{1}{\left[\boldsymbol{\xi}\boldsymbol{\xi}\_{\boldsymbol{\Psi}}\boldsymbol{\xi}\boldsymbol{\xi}\_{\boldsymbol{\Psi}'}\right]^{\frac{1}{2}}}\left\langle\boldsymbol{\psi}\_{a,b}|\boldsymbol{\psi}'\_{a',b'}\right\rangle\tag{46}$$

is the integral kernel of a unitary map between N*<sup>ψ</sup>*<sup>0</sup> and N*ψ*. The properties of the wavelet transform can be understood in terms of the unitary irreductible representation of the one-dilensional affine group.It is important to note that the Wavelets built on the basis of the group representation theory have all the properties of CS. There is a wole body of work devoted to the study of CS arising from group representation theory [7, 33, 35].

#### **5. Application 1: GCS for the Legendre Hamiltonian and CS transform**

#### **5.1 GCS for the Legendre Hamiltonian**

By replacing the coefficients *z<sup>n</sup>=* ffiffiffiffi *<sup>n</sup>*! <sup>p</sup> of the canonical CS by the function <sup>O</sup>*n*ð Þ*<sup>ξ</sup>* in (1) as mentioned in the introduction. We construct in this section a class of GCS indexed by point *ξ*∈ .

Thus f g ð Þ *b*, *a* ≕*Gaff* ¼ � ð Þ 0, ∞ , the affine group of the line. Specifically, for each pair ð Þ *a*, *b* of the real numbers, with *a*>0, from translations and dilatations of

> *x* � *b a* � �

Here *a* is the parameter of dilation (or scale) and *b* is the parameter of transla-

Moreover, in terms of the Dirac's bracket notation it is an easy to show that the

� � *<sup>ψ</sup><sup>a</sup>*,*<sup>b</sup>* � � � *dbda*

> ð<sup>∞</sup> 0

<sup>S</sup> *<sup>τ</sup>b*0*<sup>f</sup>*ð Þ¼ *<sup>a</sup>*, *<sup>b</sup>* <sup>S</sup> *<sup>f</sup> <sup>a</sup>*, *<sup>b</sup>* � *<sup>b</sup>*<sup>0</sup> � �

*<sup>τ</sup><sup>b</sup>* ð Þ <sup>0</sup>*<sup>f</sup>* ð Þ¼ *<sup>x</sup> f x* � *<sup>b</sup>*<sup>0</sup> � �*:*

As in Fourier or Hilbert analysis, wavelet analysis provides a Plancherel type relation which permits itself the reconstruction of the analyzed function from its

<sup>S</sup> *<sup>f</sup>*ð Þ *<sup>a</sup>*, *<sup>b</sup>* <sup>S</sup> *<sup>g</sup>* ð Þ *<sup>a</sup>*, *<sup>b</sup> dadb*

which in turns to reconstruct the analyzed function *f* in the *L*<sup>2</sup>

• It is dilatation-invariant, in the sense that, if *f* satisfies the invariance dilatation

holds for these vectors (in the weak sense). Here *I*<sup>N</sup> is the identity operator on N. The *continuous wavelet transform* of an arbitrary vector (signal) *f* ∈ N at the scale *a*

� �∈ N as

, *ψ*1,0 ¼ *ψ:* (35)

*<sup>a</sup>* <sup>¼</sup> *<sup>I</sup>*<sup>N</sup> (37)

*f t*ð Þ*ψ<sup>a</sup>*,*<sup>b</sup>*ð Þ*t dt:* (38)

ð Þ <sup>þ</sup> *:*

ð Þ (40)


ð Þ *<sup>a</sup>*, *<sup>b</sup>* , <sup>∀</sup>*α*, *<sup>β</sup>* <sup>∈</sup> *and f* <sup>1</sup>, *<sup>f</sup>* <sup>2</sup> <sup>∈</sup>*L*<sup>2</sup>

S *<sup>f</sup>*ð Þ¼ *a*, *b λ*S *<sup>f</sup>*ð Þ *ra*,*rb :* (39)

*<sup>a</sup>*<sup>2</sup> , <sup>∀</sup>*f*, *<sup>g</sup>* <sup>∈</sup>*L*<sup>2</sup>

<sup>N</sup>, *for all a*>0 *and b*∈ *:* (36)

the function *ψ*, we obtain a family of wavelets *ψa*,*<sup>b</sup>*

tion (or position). It is then easily cheked that

*<sup>ψ</sup>a*,*b*ð Þ *<sup>x</sup>* � � � � 2

resolution of the identity

*Wavelet Theory*

and the position *b* is given by

• It is linear in the sense that:

• It is translation invariant:

ð Þ¼ *a*, *b α*S *<sup>f</sup>* <sup>1</sup>

S *<sup>α</sup> <sup>f</sup>* <sup>1</sup>þ*<sup>β</sup> <sup>f</sup>* <sup>2</sup>

*ψa*,*b*ð Þ¼ *x*

<sup>N</sup> ¼ k k *ψ*ð Þ *x*

ð ð

�<sup>∗</sup> þ *ψ<sup>a</sup>*,*<sup>b</sup>* �

S *<sup>f</sup>*ð Þ¼ *a*, *b*

The wavelet transform S *<sup>f</sup>*ð Þ *a*, *b* has several properties [34]:

ð Þþ *a*, *b β*S *<sup>f</sup>* <sup>2</sup>

where *τb*<sup>0</sup> refers to the translation of the function *f* by *b*<sup>0</sup> given

property *f x*ð Þ¼ *λf rx* ð Þ for some *λ*,*r*> 0 fixed then

wavelet transform. More precisely we have

*a* >0

ð

*b* ∈

h i *<sup>f</sup>*, *<sup>g</sup>* <sup>¼</sup> <sup>1</sup>

wavelet transform as

**270**

C *<sup>ψ</sup>* ð

1 C *<sup>ψ</sup>*

1 ffiffiffi *<sup>a</sup>* <sup>p</sup> *<sup>ψ</sup>*

2

**Definition 5.** *The GCS labeled by points ξ*∈ *is defined by the following superposition*

$$\theta\_{\xi} = \mathcal{A}'(\xi)^{-1/2} \sum\_{n=0}^{\infty} \mathcal{D}\_n(\xi) \mu\_n, \quad \xi \in \mathbb{R} \tag{47}$$

*here* N ð Þ*ξ is a normalization factor, the function* O*n*ð Þ*ξ* ≔ Φ*n*ð Þ*ξ ρ* �1*=*<sup>2</sup> *<sup>n</sup>* , *with*

$$\Phi\_n(\xi) = i^n \sqrt{\frac{\pi}{2\xi}} J\_{n+\frac{1}{2}}(\xi),\tag{48}$$

*<sup>φ</sup>* <sup>↦</sup> *<sup>T</sup>ξ*½ �¼ *<sup>φ</sup> ϑξ*j*<sup>φ</sup>* � �*ϑξ:* (56)

*dμ ξ*ð Þ¼ *σ ξ*ð Þ*dξ*, (57)

*ξ*

ð Þ*<sup>ξ</sup> σ ξ*ð Þ *<sup>d</sup><sup>ξ</sup> ξ*

� �*Tm*,*<sup>n</sup>:* (59)

*Tm*,*<sup>n</sup>* (58)

*δ<sup>m</sup>*,*<sup>n</sup>:* (60)

*δ<sup>m</sup>*,*<sup>n</sup>*, (61)

*δ<sup>m</sup>*,*<sup>n</sup>:* (62)

**Proof.** We need to determine the function *σ ξ*ð Þ. Let

(56). According to (56) and by writing

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*, *m*¼0

Hence, we need *σ ξ*ð Þ such that

ð<sup>∞</sup> �∞ *Jn*þ<sup>1</sup> 2 ð Þ*<sup>ξ</sup> Jm*þ<sup>1</sup> 2

We make appeal to the integral ([36], p. 211):

1 *y Jm*þ<sup>1</sup> 2 ð Þ *cy Jn*þ<sup>1</sup> 2

with condition *c*> 0. Then, for parameters *c* ¼ 1, we have

ð 

ð<sup>∞</sup> �∞

> ð<sup>∞</sup> �∞

1 *ξ Jm*þ<sup>1</sup> 2 ð Þ*<sup>ξ</sup> Jn*þ<sup>1</sup> 2

*<sup>n</sup>*¼<sup>0</sup>*Tn*,*<sup>n</sup>* <sup>¼</sup> <sup>1</sup><sup>H</sup> , in other words

*Plots of the probability distribution P n*ð Þ , *ξ versus ξ for various values of n.*

<sup>¼</sup> <sup>X</sup><sup>∞</sup> *n*, *m*¼0

further to P<sup>∞</sup>

**Figure 1.**

**273**

*π* 2 ð Þ �<sup>1</sup> *<sup>n</sup> i* *π* 2 ð Þ �<sup>1</sup> *<sup>n</sup> i n*þ*m*

where *σ ξ*ð Þ is an auxiliary function. Let us writte *Tm*,*<sup>n</sup>* ≔ j i *ψ <sup>m</sup> ψ<sup>n</sup>* h j, defined as in

*Tξdμ ξ*ð Þ

*Jm*þ<sup>1</sup> 2 ð Þ*<sup>ξ</sup> Jn*þ<sup>1</sup> 2 ð Þ*ξ*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>ρ</sup>*ð Þ *<sup>m</sup> <sup>ρ</sup>*ð Þ *<sup>n</sup>* <sup>p</sup> *σ ξ*ð Þ *<sup>d</sup><sup>ξ</sup>*

> �∞ *Jm*þ<sup>1</sup> 2 ð Þ*<sup>ξ</sup> Jn*þ<sup>1</sup> 2

*<sup>ξ</sup>* <sup>¼</sup> <sup>2</sup>

ð Þ *cy dy* <sup>¼</sup> <sup>2</sup>

ð Þ*<sup>ξ</sup> <sup>d</sup><sup>ξ</sup>* <sup>¼</sup> <sup>2</sup>

By comparing (62) with (66) we obtain finally the desired weight function *σ ξ*ð Þ¼ 1*=π:* Therefore, the measure (57) has the form (55) [37]. Indeed (59) reduces

According to this construction, the state *ϑξ* form an overcomplete basis in the Hilbert space <sup>H</sup> (**Figure 1**). □

*π*ð Þ 2*n* þ 1

2*n* þ 1

2*n* þ 1

*Tξdμ ξ*ð Þ¼ **1**<sup>H</sup> *:* (63)

!

ð 

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

ð<sup>∞</sup> �∞

ð Þ*<sup>ξ</sup> σ ξ*ð Þ *<sup>d</sup><sup>ξ</sup>*

*<sup>n</sup>*þ*<sup>m</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ <sup>2</sup>*<sup>m</sup>* <sup>þ</sup> <sup>1</sup> ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup> <sup>ð</sup><sup>∞</sup>

*where Jn*þ1*=*2ð Þ*: is the cylindrical Bessel function* ([20], p. 626):

$$J\_{n+\frac{1}{2}}(z) = \sum\_{s=0}^{\infty} \frac{(-1)^s}{s!(s+n+1/2)!} \left(\frac{z}{2}\right)^{2s+n+\frac{1}{2}}, \ z \in \mathbb{C} \tag{49}$$

*and ρ<sup>n</sup> are positive numbers given by*

$$\rho\_n = \frac{1}{2n+1}, \quad n = 0, 1, 2, \dots \quad , \tag{50}$$

*and* f g *<sup>ψ</sup><sup>n</sup> is an orthonormal basis of the Hilbert space* <sup>H</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup> *dx* � � *defined in* (27).

**Proposition 1.** *The normalization factor defined by the GCS (47) reads as*

$$\mathcal{A}'(\xi) = \mathbf{1},\tag{51}$$

*for every ξ*∈ .

**Proof.** From (47) and by using the orthonormality relation of basis elements f g *ψ<sup>n</sup>* þ∞ *<sup>n</sup>*¼<sup>0</sup> in (27), then

$$
\langle \vartheta\_{\xi} | \vartheta\_{\xi} \rangle = \pi (\xi \mathcal{J}'(\xi))^{-1} \sum\_{n=0}^{\infty} \left( n + \frac{1}{2} \right) I\_{n + \frac{1}{2}}(\xi) I\_{n + \frac{1}{2}}(\xi) . \tag{52}
$$

In order to identify the above series, we make appeal to the formula ([36], p. 591):

$$\sum\_{n=0}^{\infty} \left( n + \frac{1}{2} \right) I\_{n + \frac{1}{2}}(\xi) I\_{n + \frac{1}{2}}(\xi) = \pi^{-1} \xi,\tag{53}$$

we then obtain the result (51) by using the GCS condition *ϑξ*j*ϑξ* � � <sup>¼</sup> 1. □ **Proposition 2.** *The GCS defined in* (47) *satisfy the following resolution of the identity*

$$\int\_{\mathbb{R}} T\_{\xi} d\mu(\xi) = \mathbf{1}\_{\mathcal{M}'},\tag{54}$$

*(in the weak sense) in terms of an acceptable measure*

$$d\mu(\xi) = \frac{1}{\pi} d\xi,\tag{55}$$

*where dξ the Lebesgue's measure on* . *The rank one operator T<sup>ξ</sup>* ¼ *ϑξ* � � � *ϑξ* � � � : H ! H *is define as*

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

$$
\rho \mapsto T\_{\xi}[\rho] = \left\langle \theta\_{\xi} | \rho \right\rangle \theta\_{\xi}. \tag{56}
$$

**Proof.** We need to determine the function *σ ξ*ð Þ. Let

$$d\mu(\xi) = \sigma(\xi)d\xi,\tag{57}$$

where *σ ξ*ð Þ is an auxiliary function. Let us writte *Tm*,*<sup>n</sup>* ≔ j i *ψ <sup>m</sup> ψ<sup>n</sup>* h j, defined as in (56). According to (56) and by writing

$$\int\_{\mathbb{R}} T\_{\xi} d\mu(\xi)$$

$$= \sum\_{n,m=0}^{\infty} \frac{\pi}{2} (-1)^{n} i^{n+m} \left( \int\_{-\infty}^{\infty} \frac{J\_{m+\frac{1}{2}}(\xi) J\_{n+\frac{1}{2}}(\xi)}{\sqrt{\rho(m)\rho(n)}} \sigma(\xi) \frac{d\xi}{\xi} \right) T\_{m,n} \tag{58}$$

$$=\sum\_{n,m=0}^{\infty} \frac{\pi}{2}(-1)^{n}i^{n+m}\sqrt{(2m+1)(2n+1)}\left(\int\_{-\infty}^{\infty}J\_{m+\frac{1}{2}}(\xi)J\_{n+\frac{1}{2}}(\xi)\sigma(\xi)\frac{d\xi}{\xi}\right)T\_{m,n}.\tag{59}$$

Hence, we need *σ ξ*ð Þ such that

**Definition 5.** *The GCS labeled by points ξ*∈ *is defined by the following superposition*

O*n*ð Þ*ξ ψn*, *ξ*∈ (47)

, *n* ¼ 0, 1, 2, … , (50)

N ð Þ¼ *ξ* 1, (51)

ð Þ*ξ* , (48)

�1*=*<sup>2</sup> *<sup>n</sup>* , *with*

, *z*∈ (49)

*dx* � �

ð Þ*ξ :* (52)

*ξ*, (53)

� � <sup>¼</sup> 1. □

*n*¼0

*n* ffiffiffiffiffi *π* 2*ξ* r

ð Þ �<sup>1</sup> *<sup>s</sup> s*!ð Þ *s* þ *n* þ 1*=*2 !

*and* f g *<sup>ψ</sup><sup>n</sup> is an orthonormal basis of the Hilbert space* <sup>H</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup>

**Proposition 1.** *The normalization factor defined by the GCS (47) reads as*

**Proof.** From (47) and by using the orthonormality relation of basis elements

*n*¼0

**Proposition 2.** *The GCS defined in* (47) *satisfy the following resolution of the identity*

In order to identify the above series, we make appeal to the formula ([36],

*Jn*þ<sup>1</sup> 2 ð Þ*<sup>ξ</sup> Jn*þ<sup>1</sup> 2

*<sup>d</sup>μ ξ*ð Þ¼ <sup>1</sup>

*where dξ the Lebesgue's measure on* . *The rank one operator T<sup>ξ</sup>* ¼ *ϑξ*

we then obtain the result (51) by using the GCS condition *ϑξ*j*ϑξ*

ð 

*(in the weak sense) in terms of an acceptable measure*

*n* þ 1 2 � �

*Jn*þ<sup>1</sup> 2 ð Þ*<sup>ξ</sup> Jn*þ<sup>1</sup> 2

ð Þ¼ *<sup>ξ</sup> <sup>π</sup>*�<sup>1</sup>

*Tξdμ ξ*ð Þ¼ **1**<sup>H</sup> , (54)

*<sup>π</sup> <sup>d</sup>ξ*, (55)

� � � *ϑξ* � �

� : H !

*Jn*þ<sup>1</sup> 2

> *z* 2 � �2*s*þ*n*þ<sup>1</sup> 2

*ϑξ* <sup>¼</sup> <sup>N</sup> ð Þ*<sup>ξ</sup>* �1*=*2X<sup>∞</sup>

*here* N ð Þ*ξ is a normalization factor, the function* O*n*ð Þ*ξ* ≔ Φ*n*ð Þ*ξ ρ*

Φ*n*ð Þ¼ *ξ i*

*where Jn*þ1*=*2ð Þ*: is the cylindrical Bessel function* ([20], p. 626):

ð Þ¼ *<sup>z</sup>* <sup>X</sup><sup>∞</sup>

*s*¼0

*<sup>ρ</sup><sup>n</sup>* <sup>¼</sup> <sup>1</sup> 2*n* þ 1

� � <sup>¼</sup> *π ξ*ð Þ <sup>N</sup> ð Þ*<sup>ξ</sup>* �<sup>1</sup>X<sup>∞</sup>

*n* þ 1 2 � �

X∞ *n*¼0

*Jn*þ<sup>1</sup> 2

*and ρ<sup>n</sup> are positive numbers given by*

*ϑξ*j*ϑξ*

*defined in* (27).

*Wavelet Theory*

f g *ψ<sup>n</sup>* þ∞

p. 591):

H *is define as*

**272**

*for every ξ*∈ .

*<sup>n</sup>*¼<sup>0</sup> in (27), then

$$\int\_{-\infty}^{\infty} J\_{n+\frac{1}{2}}(\xi) J\_{m+\frac{1}{2}}(\xi) \sigma(\xi) \frac{d\xi}{\xi} = \frac{2}{\pi (2n+1)} \delta\_{m,n} \,. \tag{60}$$

We make appeal to the integral ([36], p. 211):

$$\int\_{-\infty}^{\infty} \frac{1}{\mathcal{Y}} J\_{m+\frac{1}{2}}(c\mathcal{Y}) J\_{n+\frac{1}{2}}(c\mathcal{Y}) d\mathcal{Y} = \frac{2}{2n+1} \delta\_{m,n},\tag{61}$$

with condition *c*> 0. Then, for parameters *c* ¼ 1, we have

$$\int\_{-\infty}^{\infty} \frac{1}{\xi} J\_{m+\frac{1}{2}}(\xi) J\_{n+\frac{1}{2}}(\xi) d\xi = \frac{2}{2n+1} \delta\_{m,n}.\tag{62}$$

By comparing (62) with (66) we obtain finally the desired weight function *σ ξ*ð Þ¼ 1*=π:* Therefore, the measure (57) has the form (55) [37]. Indeed (59) reduces further to P<sup>∞</sup> *<sup>n</sup>*¼<sup>0</sup>*Tn*,*<sup>n</sup>* <sup>¼</sup> <sup>1</sup><sup>H</sup> , in other words

$$\int\_{\mathbb{R}} T\_{\xi} d\mu(\xi) = \mathbf{1}\_{\mathcal{A}^{\rho}}.\tag{63}$$

According to this construction, the state *ϑξ* form an overcomplete basis in the Hilbert space <sup>H</sup> (**Figure 1**). □

**Figure 1.** *Plots of the probability distribution P n*ð Þ , *ξ versus ξ for various values of n.*

When the GCS (47) describes a quantum system, the probability of finding the state *ψ<sup>n</sup>* in some normalized state *ϑξ* of the state Hilbert space H is given by *P n*ð Þ , *ξ* ≔ *ψn*j*ϑξ* �� � � � � 2 *:* For the GCS (47) the probability distribution function is given by

$$P(n,\xi) = \frac{\pi(2n+1)}{2|\xi|} \left| J\_{n+\frac{1}{2}}(\xi) \right|^2, \ \xi \in \mathbb{R}\_+^\*. \tag{64}$$

#### **5.2 Coherent state transform**

To discuss coherent state transforms (CST), we will start by establishing the kernel of this transformation by giving the closed form of the GCS (47).

**Proposition 3.** *For all x*∈½ � �1, 1 *, the wave functions of GCS in (47) can be written as*

$$\vartheta\_{\xi}(\mathbf{x}) = e^{-i\mathbf{x}\xi},\tag{65}$$

**Proof.** The result follows immediately by using the formula ([20], p. 647):

ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> !! , *<sup>ξ</sup>* <sup>≪</sup> <sup>1</sup> (71)

ð Þ*ξ* , *n* ¼ 0, 1, 2, … , (72)

*π η*ð Þ � *<sup>ξ</sup>* sin ð Þ *<sup>η</sup>* � *<sup>ξ</sup>* , (73)

� � (74)

*<sup>η</sup>* � *<sup>ξ</sup>* , (75)

<sup>2</sup> , (77)

�*i<sup>ξ</sup>xdx*, *ξ*∈ *:* (78)

ð Þ*ξ :* (79)

*Jn*þ<sup>1</sup> 2 ð Þ*ξ*

n o; *<sup>n</sup>* <sup>∈</sup> 0,

*dx* � � spanned by eigenstates

<sup>J</sup>*n*ð Þ*<sup>ξ</sup>* <sup>≈</sup> *<sup>ξ</sup><sup>n</sup>*

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

ffiffiffiffiffi *π* 2*ξ* r

*Jn*þ<sup>1</sup> 2

is the spherical Bessel function [20]. This ends the proof. □ The careful reader has certainly recognized in (70) the expression of nonlinear

> ffiffiffiffiffi *ηξ* p

*<sup>ξ</sup>* <sup>p</sup> <sup>¼</sup> sin ð Þ *<sup>η</sup>* � *<sup>ξ</sup>*

*dx* � �*, the CST is the unitary map*

ð1 �1 *e* �*ixξ φ*ð Þ *x dx*

*dx* � � <sup>¼</sup> *PW*1, � (76)

J*n*ð Þ¼ *ξ*

Let us note that, in view of the formula ([36], p. 667):

*n* þ 1 2 � � *Jn*þ<sup>1</sup>

Wiener Hilbert space *PW*1. Then, the family ½ � *<sup>π</sup>*ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> *<sup>=</sup><sup>ξ</sup>* <sup>1</sup>*=*<sup>2</sup>

ð Þ , *dμ* as.

<sup>W</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup>

2 ð Þ*<sup>η</sup> Jn*þ<sup>1</sup> 2 ð Þ¼ *ξ*

the reproducing kernel arising from GCS (47) can be written as

*K*ð Þ *η*, *ξ* ≔ *ϑη*j*ϑξ*

denotes the Dyson's sine kernel, which is the reproducing kernel of the Paley-

Once we have a closed form of GCS, we can look for the associated CST, this

h i *φ*j*ξ* <sup>H</sup> ¼

*Pn*ð Þ *x e*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *π*ð Þ 2*n* þ 1 2*ξ*

*Jn*þ<sup>1</sup> 2

*Jn*þ<sup>1</sup> 2 ð Þ*ξ* ffiffi

2 ð Þ*η* ffiffi *η* p

where

coherent states [38].

X∞ *n*¼0

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

¼ *π* X∞ *n*¼0

forms an orthonormal basis of *PW*<sup>1</sup> [39].

**Proposition 4.** *For <sup>φ</sup>*<sup>∈</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup>

**Corollary 2.** *The following integral*

ð Þ �*<sup>i</sup> <sup>n</sup>* ffiffi *<sup>ξ</sup>* <sup>p</sup> *Jn*þ<sup>1</sup> 2

f g *<sup>ψ</sup><sup>n</sup>* in (27) onto *PW*<sup>1</sup> <sup>⊂</sup>*L*<sup>2</sup>

*defined by means of* (65) *as*

*for all ξ*∈ .

*holds*.

**275**

should exactly be

transform should map the space <sup>H</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup>

<sup>W</sup> ½ � *<sup>φ</sup>* ð Þ¼ *<sup>ξ</sup>* ð Þ <sup>N</sup> ð Þ*<sup>ξ</sup>* <sup>1</sup>*=*<sup>2</sup>

ð Þ¼ *<sup>ξ</sup>* <sup>1</sup>

<sup>W</sup> *<sup>ψ</sup><sup>n</sup>* ½ �ð Þ¼ � *<sup>ξ</sup>* ð Þ*<sup>i</sup> <sup>n</sup>*

ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup>

ð1 �1

**Proof.** From (75), the image of the basis vector f g *ψ<sup>n</sup>* under the transform W

s

*n* þ 1 2 � �*Jn*þ<sup>1</sup>

*for all ξ*∈ .

**Proof.** We start by the following expression

$$\vartheta\_{\xi}(\mathbf{x}) = \mathcal{A}'(\xi)^{-1/2}\mathfrak{S}(\mathbf{x}, \xi),\tag{66}$$

where the series

$$\mathfrak{S}(\mathfrak{x},\xi) \coloneqq \sum\_{n=0}^{\infty} \mathfrak{D}\_n(\xi) \wp\_n(\mathfrak{x}),\tag{67}$$

with the function O*n*ð Þ¼ *ξ* Φ*n*ð Þ*ξ ρ* �1*=*<sup>2</sup> *<sup>n</sup>* , mentioned in Definition 5. To do this, we start by replacing the function Φ*n*ð Þ*ξ* and the positive sequences *ρ<sup>n</sup>* by their expressions in (48) and (50) thus Eq. (67) reads

$$\mathfrak{S}(\mathbf{x},\xi) = \sqrt{\frac{\pi}{2\xi}} \sum\_{n=0}^{\infty} (-1)^{n} i^{n} \sqrt{2n+1} \mathfrak{J}\_{n+\frac{1}{2}}(\xi) \wp\_{n}(\mathbf{x}).\tag{68}$$

Making use the explicit expression (27) of the eigenstates *ψn*ð Þ *x* , then the sum (68) becomes

$$\mathfrak{S}(\mathfrak{x},\xi) = \sqrt{\frac{2\pi}{\xi}} \sum\_{n=0}^{\infty} (-1)^n i^n \binom{n+\frac{1}{2}}{2} J\_{n+\frac{1}{2}}(\xi) P\_n(\mathfrak{x}).\tag{69}$$

We now appeal to the Gegenbauer's expansion of the plane wave in Gegenbauer polynomials and Bessel functions ([38], p. 116):

$$\mathbf{e}^{\mathbf{i}\xi \mathbf{x}} = \Gamma(\boldsymbol{\gamma}) (\xi \mathbf{2})^{-\gamma} \sum\_{\mathbf{n}=\mathbf{0}}^{\infty} \mathbf{i}^{\mathbf{n}} (\mathbf{n} + \boldsymbol{\gamma}) \mathbf{J}\_{\mathbf{n}+\boldsymbol{\gamma}}(\xi) \mathbf{C}\_{\mathbf{n}}^{\boldsymbol{\gamma}}(\mathbf{x})$$

Then, for *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*=*2, *<sup>y</sup>* <sup>¼</sup> *<sup>x</sup>* and by using the identity <sup>Γ</sup>ð Þ¼ <sup>1</sup>*=*<sup>2</sup> ffiffiffi *π* p , we arrive at (65). □

**Corollary 1.** *When the variable ξ* ≪ 1*, the GCS in* (47) *becomes*

$$\vartheta\_{\xi} \approx \mathcal{J}'(\xi)^{-1/2} \sum\_{n=0}^{\infty} \frac{\sqrt{2\pi}(-i\xi)^n}{\sqrt{2^{2n+1}(2n+1)}\Gamma\left(n+\frac{1}{2}\right)} \mathcal{W}\_n. \tag{70}$$

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

**Proof.** The result follows immediately by using the formula ([20], p. 647):

$$\, \_\mathcal{J} \mathcal{J}\_n(\xi) \approx \frac{\xi^n}{(2n+1)!!}, \quad \xi \ll 1 \tag{71}$$

where

When the GCS (47) describes a quantum system, the probability of finding the

<sup>2</sup>j j *<sup>ξ</sup> Jn*þ<sup>1</sup>

To discuss coherent state transforms (CST), we will start by establishing the

*ϑξ*ð Þ¼ *x e*

*ϑξ*ð Þ¼ *<sup>x</sup>* <sup>N</sup> ð Þ*<sup>ξ</sup>* �1*=*<sup>2</sup>

<sup>S</sup>ð Þ *<sup>x</sup>*, *<sup>ξ</sup>* <sup>≔</sup> <sup>X</sup><sup>∞</sup>

ffiffiffiffiffi *π* 2*ξ* <sup>r</sup> <sup>X</sup><sup>∞</sup>

ffiffiffiffiffi 2*π ξ* <sup>s</sup> <sup>X</sup><sup>∞</sup>

ei*<sup>ξ</sup>*<sup>x</sup> <sup>¼</sup> <sup>Γ</sup>ð Þ*<sup>γ</sup>* ð Þ *<sup>ξ</sup>*<sup>2</sup> �*<sup>γ</sup>* <sup>X</sup><sup>∞</sup>

*ϑξ* <sup>≈</sup> <sup>N</sup> ð Þ*<sup>ξ</sup>* �1*=*2X<sup>∞</sup>

*n*¼0

*n*¼0

Then, for *<sup>γ</sup>* <sup>¼</sup> <sup>1</sup>*=*2, *<sup>y</sup>* <sup>¼</sup> *<sup>x</sup>* and by using the identity <sup>Γ</sup>ð Þ¼ <sup>1</sup>*=*<sup>2</sup> ffiffiffi

**Corollary 1.** *When the variable ξ* ≪ 1*, the GCS in* (47) *becomes*

*n*¼0

q

*n*¼0

we start by replacing the function Φ*n*ð Þ*ξ* and the positive sequences *ρ<sup>n</sup>* by their

ð Þ �<sup>1</sup> *<sup>n</sup> i*

ð Þ �<sup>1</sup> *<sup>n</sup> i <sup>n</sup> <sup>n</sup>* <sup>þ</sup> 1 2 � �

n¼0 i

(65). □

ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> ð Þ �*i<sup>ξ</sup> <sup>n</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2<sup>2</sup>*n*þ<sup>1</sup>

ð Þ 2*n* þ 1

<sup>Γ</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2 � �

Making use the explicit expression (27) of the eigenstates *ψn*ð Þ *x* , then the sum

We now appeal to the Gegenbauer's expansion of the plane wave in Gegenbauer

*<sup>n</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup> *Jn*þ<sup>1</sup>

**Proposition 3.** *For all x*∈½ � �1, 1 *, the wave functions of GCS in (47) can be written as*

�*ixξ*

kernel of this transformation by giving the closed form of the GCS (47).

� � �

2 ð Þ*ξ*

*:* For the GCS (47) the probability distribution function is given by

, *ξ*∈ <sup>∗</sup>

<sup>þ</sup> *:* (64)

, (65)

Sð Þ *x*, *ξ* , (66)

O*n*ð Þ*ξ ψn*ð Þ *x* , (67)

ð Þ*ξ ψn*ð Þ *x :* (68)

ð Þ*ξ Pn*ð Þ *x :* (69)

*π* p , we arrive at

*ψn:* (70)

�1*=*<sup>2</sup> *<sup>n</sup>* , mentioned in Definition 5. To do this,

2

*Jn*þ<sup>1</sup> 2

<sup>n</sup>ð Þ x

<sup>n</sup>ð Þ <sup>n</sup> <sup>þ</sup> *<sup>γ</sup>* Jnþ*<sup>γ</sup>* ð Þ*<sup>ξ</sup>* <sup>C</sup>*<sup>γ</sup>*

� � � 2

state *ψ<sup>n</sup>* in some normalized state *ϑξ* of the state Hilbert space H is given by

*P n*ð Þ¼ , *<sup>ξ</sup> <sup>π</sup>*ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup>

**Proof.** We start by the following expression

with the function O*n*ð Þ¼ *ξ* Φ*n*ð Þ*ξ ρ*

expressions in (48) and (50) thus Eq. (67) reads

Sð Þ¼ *x*, *ξ*

Sð Þ¼ *x*, *ξ*

polynomials and Bessel functions ([38], p. 116):

*P n*ð Þ , *ξ* ≔ *ψn*j*ϑξ*

*Wavelet Theory*

*for all ξ*∈ .

where the series

(68) becomes

**274**

�� � � � � 2

**5.2 Coherent state transform**

$$\mathcal{J}\_n(\xi) = \sqrt{\frac{\pi}{2\xi}} J\_{n + \frac{1}{2}}(\xi), \ n = 0, 1, 2, \dots \quad , \tag{72}$$

is the spherical Bessel function [20]. This ends the proof. □

The careful reader has certainly recognized in (70) the expression of nonlinear coherent states [38].

Let us note that, in view of the formula ([36], p. 667):

$$\sum\_{n=0}^{\infty} \binom{n+\frac{1}{2}}{n+\frac{1}{2}} I\_{n+\frac{1}{2}}(\eta) I\_{n+\frac{1}{2}}(\xi) = \frac{\sqrt{\eta \xi}}{\pi(\eta - \xi)} \sin \left(\eta - \xi\right),\tag{73}$$

the reproducing kernel arising from GCS (47) can be written as

$$K(\eta,\xi) := \left\langle \theta\_{\eta} | \theta\_{\xi} \right\rangle \tag{74}$$

$$=\pi\sum\_{n=0}^{\infty}\binom{n+\frac{1}{2}}{2}\frac{J\_{n+\frac{1}{2}}(\eta)J\_{n+\frac{1}{2}}(\xi)}{\sqrt{\eta}}=\frac{\sin\left(\eta-\xi\right)}{\eta-\xi},\tag{75}$$

denotes the Dyson's sine kernel, which is the reproducing kernel of the Paley-Wiener Hilbert space *PW*1. Then, the family ½ � *<sup>π</sup>*ð Þ *<sup>n</sup>* <sup>þ</sup> <sup>1</sup>*=*<sup>2</sup> *<sup>=</sup><sup>ξ</sup>* <sup>1</sup>*=*<sup>2</sup> *Jn*þ<sup>1</sup> 2 ð Þ*ξ* n o; *<sup>n</sup>* <sup>∈</sup> 0, forms an orthonormal basis of *PW*<sup>1</sup> [39].

Once we have a closed form of GCS, we can look for the associated CST, this transform should map the space <sup>H</sup> <sup>¼</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup> *dx* � � spanned by eigenstates f g *<sup>ψ</sup><sup>n</sup>* in (27) onto *PW*<sup>1</sup> <sup>⊂</sup>*L*<sup>2</sup> ð Þ , *dμ* as.

**Proposition 4.** *For <sup>φ</sup>*<sup>∈</sup> *<sup>L</sup>*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup> *dx* � �*, the CST is the unitary map*

$$\mathcal{W}\left(L^{\frac{2}{2}}\left[\left[-1,1\right],2^{-1}d\kappa\right) = PW\_1,\tag{76}$$

*defined by means of* (65) *as*

$$\mathcal{W}[\rho](\xi) = \left(\mathcal{N}(\xi)\right)^{1/2} \langle \rho | \xi \rangle\_{\mathcal{M}'} = \int\_{-1}^{1} e^{-i\mathbf{x}\cdot\xi} \overline{\rho(\mathbf{x})} \frac{d\mathbf{x}}{2},\tag{77}$$

*for all ξ*∈ .

**Corollary 2.** *The following integral*

$$\frac{\left(-i\right)^{n}}{\sqrt{\xi}}I\_{n+\frac{1}{2}}(\xi) = \frac{1}{\sqrt{2\pi}}\int\_{-1}^{1}P\_{n}(\varkappa)e^{-i\xi\varkappa}d\varkappa, \quad \xi \in \mathbb{R}.\tag{78}$$

*holds*.

**Proof.** From (75), the image of the basis vector f g *ψ<sup>n</sup>* under the transform W should exactly be

$$\mathcal{W}[\psi\_n](\xi) = (-i)^n \sqrt{\frac{\pi(2n+1)}{2\xi}} J\_{n+\frac{1}{2}}(\xi) . \tag{79}$$

Now, by writing (75) as

$$
\mathcal{W}[\boldsymbol{\mu}\_n](\xi) = \int\_{-1}^1 e^{-i\boldsymbol{x}\xi} \boldsymbol{\nu}\_n(\boldsymbol{x}) \frac{d\boldsymbol{x}}{2},
$$

and replacing *ψ<sup>n</sup>* by their values given in (27), we obtain

$$\mathcal{W}[\psi\_n](\xi) = \frac{\sqrt{2n+1}}{2} \int\_{-1}^{1} e^{-i\kappa \xi} P\_n(\varkappa) d\varkappa,$$

the integral 78 ð Þ can be evaluated by the help of the formula ([40], p. 456):

$$\int\_{-1}^{1} P\_n(\varkappa) e^{i\xi \varkappa} d\varkappa = i^n \sqrt{\frac{2\pi}{\xi}} J\_{n+\frac{1}{2}}(\xi),\tag{80}$$

**Exercise 1.** *Show that the vectors*

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

[22, 23]. For 1≤*p* ≤ ∞ and *μ*> 0, denote

Now, let us consider the function

where *Jμ*�<sup>1</sup>

defined by

**277**

2

Parseval formula also holds

*Lp*

*ϑξ* <sup>¼</sup> <sup>N</sup> ð Þ*<sup>ξ</sup>* �1*=*2X<sup>∞</sup>

*forms a set of GCS and gives the associated GCS transform*.

**6. Application 2: continuous Bessel wavelet transform**

*<sup>σ</sup>* ð Þ <sup>þ</sup> <sup>≔</sup> *<sup>ψ</sup> such as* k k *<sup>ψ</sup> <sup>p</sup>*

*dσ*ð Þ¼ *x*

*j x*ð Þ¼ <sup>2</sup>*<sup>μ</sup>*�<sup>1</sup>

*x* 2 � �*<sup>l</sup>*X<sup>∞</sup>

ð<sup>∞</sup> 0

and <sup>∥</sup>*ϕ*^∥∞,*<sup>σ</sup>* ≤ ∥*ϕ*∥1,*<sup>σ</sup>:* If *<sup>ϕ</sup>*, *<sup>ϕ</sup>*^ <sup>∈</sup>*L*1,*<sup>σ</sup>*ð Þ 0, <sup>∞</sup> , then by inversion, we have

ð<sup>∞</sup> 0

From ([45], p. 127) if *ϕ*ð Þ *x* and Φð Þ *x* are in *L*1,*<sup>σ</sup>*ð Þ 0, ∞ , then the following

*ϕ*ð Þ¼ *x*

*<sup>ϕ</sup>*^ð Þ*<sup>t</sup>* <sup>Φ</sup>^ ð Þ*<sup>t</sup> <sup>d</sup>σ*ðÞ¼ *<sup>t</sup>*

*Jl*ð Þ¼ *x*

*<sup>ϕ</sup>*^ð Þ *<sup>x</sup>* <sup>≔</sup>

ð<sup>∞</sup> 0

*n*¼0

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

q

The continuous wavelet transform (CWT) is used to decompose a signal into wavelets. In mathematics, the CWT is a formal tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. There are several ways to introduce the Bessel wavelet

*<sup>p</sup>*,*<sup>σ</sup>* ¼

*x*<sup>2</sup>*<sup>μ</sup>*

1 2 � �*<sup>x</sup>*

1 2 �*μJμ*�<sup>1</sup> 2

ð Þ �<sup>1</sup> *<sup>k</sup> k*!Γð Þ *k* þ *l* þ 1

For *μ* ¼ 1, the function *j x*ð Þ¼ O0ð Þ *x* coincides with equation 2ð Þ discussed in the introduction. For each function *ϕ*∈ *L*1,*<sup>σ</sup>*ð Þ 0, ∞ , the Hankel transform of order *μ* is

We know that from ([44], p. 316) that *<sup>ϕ</sup>*^ð Þ *<sup>x</sup>* is bounded and continuous on 0, ½ Þ <sup>∞</sup>

ð<sup>∞</sup> 0

*x* 2 � �<sup>2</sup>*<sup>k</sup>*

*j xt* ð Þ*ϕ*ð Þ*t dσ*ð Þ*t* , 0≤*x*< ∞*:* (88)

*j xt* ð Þ*ϕ*^ð Þ*<sup>t</sup> <sup>d</sup>σ*ð Þ*<sup>t</sup> :* (89)

*ϕ*ð Þ *x* Φð Þ *x dσ*ð Þ *x :* (90)

<sup>2</sup><sup>Γ</sup> *<sup>μ</sup>* <sup>þ</sup> <sup>3</sup> 2 � �

and ∥*ψ*∥∞,*<sup>σ</sup>* ¼ *ess*0<*x*<sup>&</sup>lt; <sup>∞</sup> sup∣*ψ*ð Þ *x* ∣< ∞ and *dσ*ð Þ *x* is the measure defined as

2*<sup>μ</sup>*þ<sup>1</sup>

<sup>2</sup>Γ *μ* þ

*k*¼0

ð Þ *x* is the Bessel function of order *l* ≔ *μ* � 1*=*2 given by

ð<sup>∞</sup> 0

� �

j j *<sup>ψ</sup>*ð Þ *<sup>x</sup> <sup>p</sup>*

*dσ*ð Þ *x* < ∞

*dx:* (85)

ð Þ *x* , (86)

*:* (87)

ffiffiffiffiffi <sup>2</sup>*<sup>π</sup>* <sup>p</sup> ð Þ �*i<sup>ξ</sup> <sup>n</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 22*<sup>n</sup>*þ<sup>1</sup>

ð Þ 2*n* þ 1

<sup>Γ</sup> *<sup>n</sup>* <sup>þ</sup> <sup>1</sup> 2 � � *ψn:* (84)

this ends the proof. □

Note that, in view of ([28], p. 29), by considering *hn*ð Þ*ξ* ≔ *ρ* �1*=*<sup>2</sup> *<sup>n</sup>* <sup>Φ</sup>*n*ð Þ*<sup>ξ</sup>* and GCS K ð Þ *ξ*, *x* ≔ *x*j*ϑξ* � �, the basis element *<sup>ψ</sup><sup>n</sup>* <sup>∈</sup>*L*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup> *dx* � � has the integral representation

$$\Psi \varphi\_n(\mathbf{x}) = \int\_{-\infty}^{\infty} h\_n(\xi) \overline{\mathcal{X}'(\xi, \mathbf{x})} d\mu(\xi) \tag{81}$$

where the function Φ*n*ð Þ*ξ* and the positive sequences *ρ<sup>n</sup>* are given in (48) and (50) respectively, the measure *dμ ξ*ð Þ is given in (55), then the Legendre polynomial has the following integral representation

$$P\_n(\mathbf{x}) = \frac{(-i)^n}{\pi} \int\_{-\infty}^{\infty} \mathcal{J}\_n(t) e^{i\mathbf{x}\xi} d\xi,\tag{82}$$

where the functionJ*n*ð Þ*:* is given in (72), which is recognized as the Fourier transform of the spherical Bessel function (72) (see [40], p. 267):

$$\int\_{-\infty}^{\infty} e^{i\pi t} \, \mathcal{J}\_n(t) dt = \begin{cases} \pi i^n P\_n(\infty), & -1 < \infty < 1 \\\\ \frac{1}{2} \pi (\pm i)^n, & \varkappa = \pm 1, \\\\ 0, & \pm \varkappa > 1 \end{cases} \tag{83}$$

where *Pn*ð Þ*:* the Legendre's polynomial [40]. **Remark 1.** *Also note that:*


*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

**Exercise 1.** *Show that the vectors*

Now, by writing (75) as

*Wavelet Theory*

this ends the proof. □

has the following integral representation

ð<sup>∞</sup> �∞ *e*

**Remark 1.** *Also note that:*

*differential operator, on L*<sup>2</sup>

*(see [43])*.

**276**

• *For x*, *<sup>ξ</sup>*<sup>∈</sup> *, the function φξ*ð Þ¼ *<sup>x</sup> eix<sup>ξ</sup>*

where *Pn*ð Þ*:* the Legendre's polynomial [40].

*of the Heisenberg group in three dimensions, in L*<sup>2</sup>

K ð Þ *ξ*, *x* ≔ *x*j*ϑξ*

representation

W *ψ<sup>n</sup>* ½ �ð Þ¼ *ξ*

and replacing *ψ<sup>n</sup>* by their values given in (27), we obtain

*Pn*ð Þ *x e*

Note that, in view of ([28], p. 29), by considering *hn*ð Þ*ξ* ≔ *ρ*

� �, the basis element *<sup>ψ</sup><sup>n</sup>* <sup>∈</sup>*L*<sup>2</sup> ½ � �1, 1 , 2�<sup>1</sup>

*Pn*ð Þ¼ *<sup>x</sup>* ð Þ �*<sup>i</sup> <sup>n</sup>*

transform of the spherical Bessel function (72) (see [40], p. 267):

*ixt*J*n*ð Þ*<sup>t</sup> dt* <sup>¼</sup>

*π*

ð<sup>∞</sup> �∞

where the function Φ*n*ð Þ*ξ* and the positive sequences *ρ<sup>n</sup>* are given in (48) and (50) respectively, the measure *dμ ξ*ð Þ is given in (55), then the Legendre polynomial

> ð<sup>∞</sup> �∞

where the functionJ*n*ð Þ*:* is given in (72), which is recognized as the Fourier

*πi*

8 >>><

>>>:

*introduced in signal theory where the property ψξ* <sup>¼</sup> *<sup>T</sup>*^ð Þ*<sup>ξ</sup> <sup>ψ</sup>, with <sup>ψ</sup>* <sup>∈</sup>*L*<sup>2</sup>

1 2 *<sup>π</sup>*ð Þ �*<sup>i</sup> <sup>n</sup>*

• *The usefulness expansion of GCS was made very clear in a paper authored by Ismail and Zhang, where it was used to solve the eigenvalue problem for the left inverse of the*

*<sup>T</sup>*^ð Þ*<sup>ξ</sup> the unitary transformation, is obtained by using the standard representation*

*-spaces with ultraspherical weights [41, 42]*.

J*n*ð Þ*t e*

*ixξ*

*nPn*ð Þ *<sup>x</sup>* , �1<sup>&</sup>lt; *<sup>x</sup>*<<sup>1</sup>

0, �*x*>1

, *x* ¼ �1,

*, is known as the Gabor's coherent states*

ð Þ *, for more information*

*ψn*ð Þ¼ *x*

W *ψ<sup>n</sup>* ½ �ð Þ¼ *ξ*

ð1 �1 ð1 �1 *e* �*ixξ ψn*ð Þ *x*

ffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> <sup>p</sup> 2

the integral 78 ð Þ can be evaluated by the help of the formula ([40], p. 456):

*<sup>i</sup><sup>ξ</sup>xdx* <sup>¼</sup> *<sup>i</sup>*

ð1 �1 *e* �*ixξ*

*n* ffiffiffiffiffi 2*π ξ*

s

*Jn*þ<sup>1</sup> 2

*dx* 2 ,

*Pn*ð Þ *x dx*,

*dx* � � has the integral

*hn*ð Þ*ξ* K ð Þ *ξ*, *x dμ ξ*ð Þ (81)

ð Þ*ξ* , (80)

*dξ*, (82)

(83)

ð Þ *, and*

�1*=*<sup>2</sup> *<sup>n</sup>* <sup>Φ</sup>*n*ð Þ*<sup>ξ</sup>* and GCS

$$\theta\_{\xi} = \mathcal{J}'(\xi)^{-1/2} \sum\_{n=0}^{\infty} \frac{\sqrt{2\pi}(-i\xi)^n}{\sqrt{2^{2n+1}(2n+1)}\Gamma(n+\frac{1}{2})} \mathbb{1}\_n. \tag{84}$$

*forms a set of GCS and gives the associated GCS transform*.

#### **6. Application 2: continuous Bessel wavelet transform**

The continuous wavelet transform (CWT) is used to decompose a signal into wavelets. In mathematics, the CWT is a formal tool that provides an overcomplete representation of a signal by letting the translation and scale parameter of the wavelets vary continuously. There are several ways to introduce the Bessel wavelet [22, 23]. For 1≤*p* ≤ ∞ and *μ*> 0, denote

$$L^p\_\sigma(\mathbb{R}\_+) \coloneqq \left\{ \boldsymbol{\Psi} \text{ } \text{ such } \text{ as } \left\| \boldsymbol{\Psi} \boldsymbol{\nu} \right\|\_{p,\sigma}^p = \int\_0^\infty |\boldsymbol{\varphi}(\boldsymbol{\varkappa})|^p d\sigma(\boldsymbol{\varkappa}) < \infty \right\}.$$

and ∥*ψ*∥∞,*<sup>σ</sup>* ¼ *ess*0<*x*<sup>&</sup>lt; <sup>∞</sup> sup∣*ψ*ð Þ *x* ∣< ∞ and *dσ*ð Þ *x* is the measure defined as

$$d\sigma(\mathbf{x}) = \frac{\varkappa^{2\mu}}{\mathfrak{L}^{\mu + \frac{1}{2}}\Gamma\left(\mu + \frac{3}{2}\right)} d\mathbf{x}.\tag{85}$$

Now, let us consider the function

$$j(\mathbf{x}) = 2^{\mu - \frac{1}{2}} \Gamma \left(\mu + \frac{1}{2}\right) \varkappa^{\frac{1}{2} - \mu} J\_{\mu - \frac{1}{2}}(\mathbf{x}),\tag{86}$$

where *Jμ*�<sup>1</sup> 2 ð Þ *x* is the Bessel function of order *l* ≔ *μ* � 1*=*2 given by

$$J\_l(\mathbf{x}) = \left(\frac{\mathbf{x}}{2}\right)^l \sum\_{k=0}^{\infty} \frac{(-1)^k}{k!\Gamma(k+l+1)} \left(\frac{\mathbf{x}}{2}\right)^{2k}.\tag{87}$$

For *μ* ¼ 1, the function *j x*ð Þ¼ O0ð Þ *x* coincides with equation 2ð Þ discussed in the introduction. For each function *ϕ*∈ *L*1,*<sup>σ</sup>*ð Þ 0, ∞ , the Hankel transform of order *μ* is defined by

$$\hat{\phi}(\infty) \coloneqq \int\_0^\infty j(\infty t) \phi(t) d\sigma(t), \quad 0 \le \infty < \infty. \tag{88}$$

We know that from ([44], p. 316) that *<sup>ϕ</sup>*^ð Þ *<sup>x</sup>* is bounded and continuous on 0, ½ Þ <sup>∞</sup> and <sup>∥</sup>*ϕ*^∥∞,*<sup>σ</sup>* ≤ ∥*ϕ*∥1,*<sup>σ</sup>:* If *<sup>ϕ</sup>*, *<sup>ϕ</sup>*^ <sup>∈</sup>*L*1,*<sup>σ</sup>*ð Þ 0, <sup>∞</sup> , then by inversion, we have

$$
\phi(\mathbf{x}) = \int\_0^\infty j(\mathbf{x}t)\hat{\phi}(t)d\sigma(t). \tag{89}
$$

From ([45], p. 127) if *ϕ*ð Þ *x* and Φð Þ *x* are in *L*1,*<sup>σ</sup>*ð Þ 0, ∞ , then the following Parseval formula also holds

$$\int\_0^\infty \hat{\phi}(t)\hat{\Phi}(t)d\sigma(t) = \int\_0^\infty \phi(\mathbf{x})\Phi(\mathbf{x})d\sigma(\mathbf{x}).\tag{90}$$

Denoting therefore by

$$\mathcal{B}(\mathbf{x}, \mathbf{y}, \mathbf{z}) = \int\_0^\infty j(\mathbf{x}t)\dot{\mathbf{j}}(\mathbf{y}t)\dot{\mathbf{j}}(\mathbf{z}t)d\sigma(t). \tag{91}$$

ð<sup>∞</sup> 0

*Proof***.** For the function *f* ∈*L*<sup>2</sup>

<sup>¼</sup> <sup>1</sup> *a*<sup>2</sup>*μ*þ<sup>1</sup>

> <sup>D</sup> *<sup>b</sup> a* , *t a* , *z* � �

1 *a*<sup>2</sup>*μ*þ<sup>1</sup>

<sup>¼</sup> <sup>1</sup> *a*<sup>2</sup>*μ*þ<sup>1</sup>

ð 3 þ

> ð 2 þ

<sup>¼</sup> <sup>1</sup> *a*<sup>2</sup>*μ*þ<sup>1</sup>

In terms of the Parseval formula (90), we obtain

¼ ð<sup>∞</sup> 0

Now multiplying by *<sup>a</sup>*�2*μ*�<sup>1</sup>*dσ*ð Þ *<sup>a</sup>* and integrating, we get

ð þ

¼ ð<sup>∞</sup> 0

ð þ

**279**

ð þ ¼ ð þ

whenever

For all *μ*>0.

Now observe that

Hence whe have that

<sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ¼ *<sup>b</sup>*, *<sup>a</sup>*

Eq. (38) as

ð<sup>∞</sup> 0

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

C *<sup>ψ</sup>* ¼

ð<sup>∞</sup> 0 *t* �2*μ*�1

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

<sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ¼ *<sup>b</sup>*, *<sup>a</sup>*

¼ ð<sup>∞</sup> 0 *<sup>j</sup> bu a* � �*<sup>j</sup> tu*

*f t*ð Þ*ψ*ð Þ*<sup>z</sup> <sup>j</sup> bu*

^*f u a*

> ð þ ^*f u a*

*a* � �*<sup>j</sup> tu*

� �*ψ*ð Þ*<sup>z</sup> <sup>j</sup> bu*

ð<sup>∞</sup> 0

ð<sup>∞</sup> 0

<sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ *<sup>b</sup>*, *<sup>a</sup>* <sup>B</sup>*<sup>ψ</sup> <sup>g</sup>* � �ð Þ *<sup>b</sup>*, *<sup>a</sup> <sup>d</sup>σ*ð Þ *<sup>a</sup> <sup>d</sup>σ*ð Þ¼ *<sup>b</sup>* <sup>C</sup> *<sup>ψ</sup>* h i *<sup>f</sup>*, *<sup>g</sup>* (101)

*<sup>σ</sup>*ð Þ <sup>þ</sup> , let us write the Bessel wavelet by using

*dσ*ð Þ*t* < ∞*:* (102)

*f t*ð Þ*ψa*,*b*ð Þ*t dσ*ð Þ*t* (103)

� �*dσ*ð Þ*<sup>z</sup> <sup>d</sup>σ*ð Þ*<sup>t</sup> :* (104)

� �*j zu* ð Þ*dσ*ð Þ *<sup>u</sup> :* (105)

� �*j zu* ð Þ*dσ*ð Þ *<sup>u</sup> <sup>d</sup>σ*ð Þ*<sup>z</sup> <sup>d</sup>σ*ð Þ*<sup>t</sup>* (106)

� �*j zu* ð Þ*dσ*ð Þ *<sup>u</sup> <sup>d</sup>σ*ð Þ*<sup>z</sup>* (107)

^*f v*ð Þ*ψ*^ð Þ *av j bv* ð Þ*dσ*ð Þ*<sup>v</sup>* (109)

<sup>¼</sup> ^*f v*ð Þ*ψ*^ð Þ *av* � �b ð Þ *<sup>b</sup> :* (110)

^*f v*ð Þ*ψ*^ð Þ *av* � �b ð Þ *<sup>b</sup>* ^*g v*ð Þ*ψ*^ð Þ *av* � �b ð Þ *<sup>b</sup> <sup>d</sup>σ*ð Þ *<sup>u</sup>* (111)

^*f u*ð Þ*ψ*^ð Þ *au* ^*g u*ð Þ*ψ*^ð Þ *au <sup>d</sup>σ*ð Þ *<sup>u</sup>* (112)

*dσ*ð Þ *a dσ*ð Þ *b* (113)

� �*dσ*ð Þ *<sup>u</sup>* (108)

j j *<sup>ψ</sup>*^ð Þ*<sup>t</sup>* <sup>2</sup>

ð<sup>∞</sup> 0

> *a* , *t a* , *z*

> > *a*

*a*

*a*

� �*ψ*^ð Þ *<sup>u</sup> <sup>j</sup> bu*

<sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ *<sup>b</sup>*, *<sup>a</sup>* <sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ *<sup>b</sup>*, *<sup>a</sup> <sup>d</sup>σ*ð Þ *<sup>b</sup>*

<sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ *<sup>b</sup>*, *<sup>a</sup>* <sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ *<sup>b</sup>*, *<sup>a</sup> <sup>a</sup>*�2*μ*�<sup>1</sup>

*a*

*f t*ð Þ*ψ*ð Þ*<sup>z</sup>* <sup>D</sup> *<sup>b</sup>*

For a 1-variable function *ψ* ∈*L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup> , we define the Hankel translation operator

$$\pi\_{\mathcal{I}}\mu(\mathbf{x}) \coloneqq \boldsymbol{\mu}(\mathbf{x}, \boldsymbol{y}) = \int\_{0}^{\infty} \mathcal{Q}(\mathbf{x}, \boldsymbol{y}, \boldsymbol{z}) \boldsymbol{\mu}(\mathbf{z}) d\sigma(\mathbf{z}), \quad \forall \mathbf{x} > \mathbf{0}, \ \mathbf{y} < \infty. \tag{92}$$

Trime'che ([46], p. 177) has shown that the integral is convergent for almost all *y* and for each fixed *x*, and

$$\|\|\varphi(\mathbf{x}, \cdot.)\|\_{2,\sigma} \le \|\|\varphi\|\|\_{2,\sigma}.\tag{93}$$

The map *y* ↦ *τyψ* is continuous from 0, ½ Þ ∞ into 0, ð Þ ∞ . For a 2-variables the function *ψ*, we define a dilatation operator

$$D\_d \varphi(\mathbf{x}, \boldsymbol{\mathcal{y}}) = a^{-2\mu - 1} \varphi\left(\frac{\mathbf{x}}{a}, \frac{\boldsymbol{\mathcal{y}}}{a}\right). \tag{94}$$

From the inversion formula in (89), we have

$$\int\_0^\infty j(zt)\mathcal{B}(\infty,y,z)d\sigma(z) = j(\ge t)j(yt), \quad \forall 0 < \infty, y < \infty, \quad 0 \le t < \infty, \quad$$

for *t* ¼ 0 and *μ* � 1*=*2 ¼ 0, we arrive at

$$\int\_0^\infty \mathcal{Q}(x, y, z) d\sigma(z) = 1. \tag{95}$$

The Bessel Wavelet copy *ψ<sup>a</sup>*,*<sup>b</sup>* are defined from the Bessel wavelet mother *ψ* ∈ *L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup> by

$$
\lambda \boldsymbol{\mu}\_{a,b}(\boldsymbol{\kappa}) := D\_a \boldsymbol{\pi}\_b \boldsymbol{\mu}(\boldsymbol{\kappa}) = D\_a \boldsymbol{\mu}(b, \boldsymbol{\kappa}) \tag{96}
$$

$$=a^{-2\mu-1}\int\_0^\infty \mathcal{B}\left(\frac{b}{a},\frac{\varkappa}{a};z\right)\psi(z)d\sigma(z), \quad \forall\ a>0,\ b\in\mathbb{R},\tag{97}$$

the integral being convergent by virtue of (92). As in the classical wavelet theory on , let us define the continuous Bessel Wavelet transform (CBWT) of a function *f* ∈*L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup> , at the scale *a* and the position *b* by

$$\mathcal{AB}(b,a) \coloneqq \left(\mathcal{AB}\_{\emptyset}f\right)(b,a) = \left\langle f(t), \psi\_{b,a}(t) \right\rangle \tag{98}$$

$$=\int\_{0}^{\infty} f(t)\overline{\nu\_{a,b}(t)}d\sigma(t)\tag{99}$$

$$=a^{-2\mu-1}\int\_{0}^{\infty}\left[\int\_{0}^{\infty}f(t)\overline{\nu(z)}\,\mathcal{O}\left(\frac{b}{a},\frac{t}{a},z\right)d\sigma(z)d\sigma(t).\tag{100}$$

The continuity of the Bessel wavelet follows from the boundedness property of the Hankel translation ([46], (104), p. 177). The following result is due to [22]:

Theorem 1.3 Let *ψ* ∈*L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup> and *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup>*L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup> . Then *Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

$$\int\_0^\infty \int\_0^\infty (\mathcal{A}\_\Psi f)(b, a) \overline{(\mathcal{A}\_\Psi g)(b, a)} d\sigma(a) d\sigma(b) = \mathcal{C}\_\Psi \langle f, g \rangle \tag{101}$$

whenever

Denoting therefore by

*Wavelet Theory*

For a 1-variable function *ψ* ∈*L*<sup>2</sup>

*τyψ*ð Þ *x* ≔ *ψ*ð Þ¼ *x*, *y*

function *ψ*, we define a dilatation operator

From the inversion formula in (89), we have

for *t* ¼ 0 and *μ* � 1*=*2 ¼ 0, we arrive at

<sup>¼</sup> *<sup>a</sup>*�2*μ*�<sup>1</sup>

ð<sup>∞</sup> 0 <sup>D</sup> *<sup>b</sup> a* , *x a* ; *z* � �

*<sup>σ</sup>*ð Þ <sup>þ</sup> , at the scale *a* and the position *b* by

<sup>¼</sup> *<sup>a</sup>*�2*μ*�<sup>1</sup>

Theorem 1.3 Let *ψ* ∈*L*<sup>2</sup>

¼ ð<sup>∞</sup> 0

ð<sup>∞</sup> 0

*<sup>σ</sup>*ð Þ <sup>þ</sup> and *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup>*L*<sup>2</sup>

*f t*ð Þ*ψ*ð Þ*<sup>z</sup>* <sup>D</sup> *<sup>b</sup>*

The continuity of the Bessel wavelet follows from the boundedness property of the Hankel translation ([46], (104), p. 177). The following result is due to [22]:

*a* , *t a* , *z* � �

*<sup>σ</sup>*ð Þ <sup>þ</sup> . Then

ð<sup>∞</sup> 0

and for each fixed *x*, and

ð<sup>∞</sup> 0

*ψ* ∈ *L*<sup>2</sup>

*f* ∈*L*<sup>2</sup>

**278**

*<sup>σ</sup>*ð Þ <sup>þ</sup> by

Dð Þ¼ *x*, *y*, *z*

ð<sup>∞</sup> 0

ð<sup>∞</sup> 0

Trime'che ([46], p. 177) has shown that the integral is convergent for almost all *y*

The map *y* ↦ *τyψ* is continuous from 0, ½ Þ ∞ into 0, ð Þ ∞ . For a 2-variables the

*j zt* ð ÞDð Þ *x*, *y*, *z dσ*ð Þ¼ *z j xt* ð Þ*j yt* ð Þ, ∀0<*x*, *y*< ∞, 0≤ *t*< ∞,

The Bessel Wavelet copy *ψ<sup>a</sup>*,*<sup>b</sup>* are defined from the Bessel wavelet mother

the integral being convergent by virtue of (92). As in the classical wavelet theory on , let us define the continuous Bessel Wavelet transform (CBWT) of a function

*ψ x a* , *y a* � �

*Daψ*ð Þ¼ *<sup>x</sup>*, *<sup>y</sup> <sup>a</sup>*�2*μ*�<sup>1</sup>

ð<sup>∞</sup> 0

*j xt* ð Þ*j yt* ð Þ*j zt* ð Þ*dσ*ð Þ*t :* (91)

*<sup>σ</sup>*ð Þ <sup>þ</sup> , we define the Hankel translation operator

Dð Þ *x*, *y*, *z ψ*ð Þ*z dσ*ð Þ*z* , ∀*x*>0, *y*< ∞*:* (92)

∥*ψ*ð Þ *x*, *:* ∥2,*<sup>σ</sup>* ≤ ∥*ψ*∥2,*<sup>σ</sup>:* (93)

Dð Þ *x*, *y*, *z dσ*ð Þ¼ *z* 1*:* (95)

*ψ*ð Þ*z dσ*ð Þ*z* , ∀ *a*>0, *b*∈ , (97)

*f t*ð Þ*ψ<sup>a</sup>*,*<sup>b</sup>*ð Þ*t dσ*ð Þ*t* (99)

*dσ*ð Þ*z dσ*ð Þ*t :* (100)

*ψ<sup>a</sup>*,*<sup>b</sup>*ð Þ *x* :¼ *Daτbψ*ð Þ¼ *x Daψ*ð Þ *b*, *x* (96)

<sup>B</sup>ð Þ *<sup>b</sup>*, *<sup>a</sup>* <sup>≔</sup> <sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ¼ *<sup>b</sup>*, *<sup>a</sup> f t*ð Þ, *<sup>ψ</sup><sup>b</sup>*,*<sup>a</sup>*ð Þ*<sup>t</sup>* � � (98)

*:* (94)

$$\mathcal{H}\_{\Psi} = \int\_0^\infty t^{-2\mu - 1} |\hat{\psi}(t)|^2 d\sigma(t) < \infty. \tag{102}$$

For all *μ*>0.

*Proof***.** For the function *f* ∈*L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup> , let us write the Bessel wavelet by using Eq. (38) as

$$\left(\mathcal{B}\!\_{\psi}f\right)(b,a) = \int\_{0}^{\infty} f(t)\varphi\_{a,b}(t)d\sigma(t)\tag{103}$$

$$=\frac{1}{a^{2\mu+1}}\int\_0^\infty \int\_0^\infty f(t)\overline{\mu}(z)\mathcal{B}\left(\frac{b}{a},\frac{t}{a},z\right)d\sigma(z)d\sigma(t). \tag{104}$$

Now observe that

$$\mathcal{B}\left(\frac{b}{a},\frac{t}{a},x\right) = \int\_0^\infty j\left(\frac{bu}{a}\right) j\left(\frac{tu}{a}\right) j(xu) d\sigma(u). \tag{105}$$

Hence whe have that

$$\left(\left(\mathcal{B}\_{\mathbb{W}}f\right)(b,a) = \frac{1}{a^{2\mu+1}}\right]\_{\mathbb{R}\_+^{\mathbb{Y}}} f(t)\psi(z)j\left(\frac{bu}{a}\right)j\binom{tu}{a}j(zu)d\sigma(u)d\sigma(z)d\sigma(t) \tag{106}$$

$$=\frac{1}{a^{2\mu+1}}\int\_{\mathbb{R}^{2}\_{+}}\hat{f}\left(\frac{u}{a}\right)\psi(z)\hat{j}\left(\frac{bu}{a}\right)\hat{j}(zu)d\sigma(u)d\sigma(z)\tag{107}$$

$$=\frac{1}{a^{2\mu+1}}\int\_{\mathbb{R}\_+} \hat{f}\left(\frac{u}{a}\right)\hat{\wp}(u)\hat{j}\left(\frac{bu}{a}\right)d\sigma(u) \tag{108}$$

$$=\int\_{\mathbb{R}\_+} \hat{f}(v)\hat{\wp}(av)\dot{\jmath}(bv)d\sigma(v) \tag{109}$$

$$= \left(\hat{f}\left(\boldsymbol{v}\right)\hat{\boldsymbol{\mu}}\left(\boldsymbol{a}\boldsymbol{v}\right)\right)^{\sim}\left(\boldsymbol{b}\right).\tag{110}$$

In terms of the Parseval formula (90), we obtain

$$\int\_{\mathbb{R}\_+} \left( \mathcal{A} \mathfrak{g} f \right)(b, a) \overline{\left( \mathcal{A} \mathfrak{g} f \right)}(b, a) d\sigma(b)$$

$$= \int\_0^\infty \left( \hat{f}(v) \hat{\wp}(av) \right)^\sim(b) \overline{\left( \hat{\wp}(v) \overline{\hat{\wp}(av)} \right)^\sim}(b) d\sigma(u) \tag{111}$$

$$=\int\_0^\infty \hat{f}(u)\overline{\hat{\wp}(au)}\overline{\hat{\wp}(u)\overline{\hat{\wp}(au)}}d\sigma(u)\tag{112}$$

Now multiplying by *<sup>a</sup>*�2*μ*�<sup>1</sup>*dσ*ð Þ *<sup>a</sup>* and integrating, we get

$$\int\_{\mathbb{R}\_+} \int\_{\mathbb{R}\_+} \left( \mathcal{A} \mathfrak{G} f \right)(b, a) \overline{\left( \mathcal{A} \mathfrak{G} f \right)}(b, a) a^{-2\mu - 1} d\sigma(a) d\sigma(b) \tag{113}$$

$$= \int \int\_0^\infty \hat{f}(u)\overline{\hat{\wp}(au)}\overline{\hat{\wp}(u)}\overline{\hat{\wp}(au)}\frac{d\sigma(a)}{a^{2\mu+1}}d\sigma(u) \tag{114}$$

$$=\int\_{\mathbb{R}}\hat{f}(u)\overline{\hat{g}(u)}\left(\int\_{\mathbb{R}}\left|\hat{\Psi}(au)\right|^{2}\frac{d\sigma(a)}{a^{2\mu+1}}\right)d\sigma(u) = C\_{\mathbb{V}}\int\_{\mathbb{R}}\hat{f}(u)\overline{\hat{g}(u)}d\sigma(u)\tag{115}$$

$$
\mathfrak{g} = \mathfrak{G}\_{\mathbb{A}} \langle f, \mathfrak{g} \rangle. \tag{116}
$$

Where the integral

Then (125) reads

O*a*,*w*<sup>0</sup> ð Þ¼ *u*

B*<sup>ψ</sup>*

<sup>¼</sup> ð Þ *qr <sup>ν</sup> πp<sup>μ</sup>*þ2*<sup>ν</sup>*

of *<sup>q</sup>*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>* ð Þ <sup>2</sup> � <sup>2</sup>*qr* cos *<sup>ϕ</sup>* <sup>1</sup>*=*<sup>2</sup>

**281**

where

1 *n*! *y<sup>n</sup>*�1*=*<sup>2</sup> *e* �*py* <sup>¼</sup> O*a*,*w*<sup>0</sup> ð Þ¼ *u*

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �5*=*<sup>2</sup> <sup>¼</sup> *<sup>w</sup>*<sup>2</sup>

*<sup>x</sup>*<sup>1</sup>*=*<sup>2</sup> *<sup>p</sup>*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> � ��<sup>1</sup>

For parameters *n* ¼ 2 and *p* ¼ *w*0, we find that

In terms of the above result, the CBWT read as

M*<sup>a</sup>*,*w*<sup>0</sup> ð Þ¼ *z*

Γð Þ *μ* þ 2*ν* 2*ν* þ 1

M*<sup>a</sup>*,*w*<sup>0</sup> ð Þ¼ *z*

where *<sup>ζ</sup>* <sup>¼</sup> *<sup>a</sup>*�<sup>1</sup> ð Þ *<sup>b</sup>* <sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> � <sup>2</sup>*a*�<sup>1</sup>*bz* cos *<sup>ϕ</sup>*

2*w*<sup>2</sup> <sup>0</sup> � *t* 2

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

ð<sup>∞</sup> 0 *w*2 <sup>0</sup> þ *t* <sup>2</sup> � ��3*=*<sup>2</sup>

ð<sup>∞</sup> 0

> 2*w*<sup>2</sup> <sup>0</sup> � *t* 2

( ) !

2 *w*<sup>2</sup> <sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup>

2 *w*<sup>2</sup>

ð<sup>∞</sup> 0

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

In terms of the Legendre polynomial *P*2ð Þ*t* , the function

2*w*<sup>2</sup> <sup>0</sup> � *t* 2

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �5*=*<sup>2</sup> *<sup>J</sup>*<sup>0</sup>

*t a u*

*P*<sup>2</sup> *w*<sup>0</sup> *w*<sup>2</sup>

<sup>0</sup> þ *t*

*t a u*

1*=*2

� �*:* (129)

ψð Þ*z* M*<sup>a</sup>*,*w*<sup>0</sup> ð Þ*z dσ*ð Þ*z* (130)

� �*J*0ð Þ *zu du:* (131)

*dt* (132)

*ϕdϕ*

*p*2

� �*d<sup>ϕ</sup>* (133)

� �*dσ*ð Þ*<sup>t</sup> :* (125)

<sup>2</sup> � ��1*=*<sup>2</sup> h i*:* (126)

� �*dσ*ð Þ*<sup>t</sup> :* (127)

*J*0ð Þ *xy dx:* (128)

2 *w*<sup>2</sup>

<sup>0</sup> þ *t* <sup>2</sup> � ��3*=*<sup>2</sup>

*P*<sup>2</sup> *w*<sup>0</sup> *w*<sup>2</sup>

The above equation can be evaluated by means of the formula ([47], p. 13):

1 4

ð Þ¼ *<sup>b</sup>*, *<sup>a</sup> <sup>a</sup>*�<sup>2</sup>

2 *<sup>n</sup>*�<sup>1</sup> 2

O*<sup>a</sup>*,*w*<sup>0</sup> ð Þ¼ *u*

ð<sup>∞</sup> 0 8�<sup>1</sup> *u*2 *e* �*w*<sup>0</sup> *<sup>a</sup> uJ*<sup>0</sup> *b a u*

ð<sup>∞</sup> 0 *e*

ð*π* <sup>2</sup>*F*<sup>1</sup>

<sup>0</sup> þ *t* <sup>2</sup> � ��1*=*<sup>2</sup> h i*J*<sup>0</sup>

*u* exp �*w*<sup>0</sup>

ð<sup>∞</sup> 0

To evaluated (131) we make appeal to the Lipschitz-Hankel integrals ([48], p. 389):

with conditions ℜð Þ *p* � *iq* � *ir* >0 and ℜð Þ *μ* þ 2*ν* >0, while *ζ* is written in place

3 2

.

*μ*�1

� � sin <sup>2</sup>*<sup>ν</sup>*

, where <sup>2</sup>*F*<sup>1</sup> denotes the hypergeometric function. For

, 2; 1; � *aw*�<sup>1</sup>

<sup>2</sup> ; *<sup>ν</sup>* <sup>þ</sup> 1; � *<sup>ζ</sup>*<sup>2</sup>

<sup>0</sup> *<sup>ζ</sup>* � �<sup>2</sup>

*μ* þ 2*ν* þ 1

�*ptJν*ð Þ *qt <sup>J</sup>ν*ð Þ *rt <sup>t</sup>*

*μ* þ 2*ν* <sup>2</sup> ,

parameters *p* ¼ *w*0*=a*, *q* ¼ b*=*a,*r* ¼ *z*, *μ* ¼ 3 and *n* ¼ 0, we arrive at

*a*3 4*πw*<sup>3</sup> 0

h i<sup>1</sup>*=*<sup>2</sup>

ð*π* <sup>2</sup>*F*<sup>1</sup>

*Pn p p*<sup>2</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> � ��1*=*<sup>2</sup> h ið Þ *xy*

*u a*

The *admissible condition* (102) requires that *ψ*^ð Þ¼ 0 0. If *ψ*^ is continuous then from (88) it follows that

$$\int\_0^\infty \varphi(\varkappa) d\sigma(\varkappa) = 0.\tag{117}$$

#### **6.1 Example**

Let us consider the function

$$f(t) = \frac{2w\_0^2 - t^2}{2\left(w\_0^2 + t^2\right)^{5/2}}, \ w\_0 > 0, \ t \in \mathbb{R}\_+. \tag{118}$$

In the case *μ* ¼ 1*=*2, the measure (85) takes the form

$$d\sigma(t) = \frac{t}{2}dt\tag{119}$$

and the function (86) reduces to

$$j(t) = J\_0(t),\tag{120}$$

where *J*0ð Þ *x* the Bessel's function of the first kind. Also note that

$$\int\_{0}^{\infty} \frac{\left(2w\_{0}^{2} - t^{2}\right)^{2}}{2\left(w\_{0}^{2} + t^{2}\right)^{5}} d\sigma(t) < \infty. \tag{121}$$

The Bessel wavelet transform of *f t*ð Þ is given by

$$\left\{ \mathcal{J}\_{\psi} \left( \frac{2w\_0^2 - t^2}{2 \left(w\_0^2 + t^2\right)^{5/2}} \right) \right\} (b, a) = a^{-2} \int\_0^\infty \frac{2w\_0^2 - t^2}{2 \left(w\_0^2 + t^2\right)^{5/2}} \psi\left(\frac{b}{a}, \frac{t}{a}\right) d\sigma(t) \tag{122}$$

$$=\left.a^{-2}\right\vert\_{0}^{\infty}\psi(z)\left(\int\_{0}^{\infty}\frac{2w\_{0}^{2}-t^{2}}{2\left(w\_{0}^{2}+t^{2}\right)^{5/2}}\mathcal{B}\left(\frac{b}{a},\frac{t}{a},z\right)d\sigma(t)\right)d\sigma(z)\tag{123}$$

Using the representation

$$\mathcal{O}\left(\frac{b}{a},\frac{t}{a},x\right) = \int\_0^\infty f\_0\left(\frac{b}{a}u\right)f\_0\left(\frac{t}{a}u\right)f\_0(xu)d\sigma(u) \tag{124}$$

then (122) becomes

$$a^{-2} \int\_0^\infty \Psi(z) \left( \int\_0^\infty J\_0\left(\frac{b}{a}u\right) J\_0(zu) \mathfrak{D}\_{a,w\_0}(u) d\sigma(u) \right) d\sigma(z) \,\,\_2\pi$$

**280**

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

Where the integral

¼ ð ð<sup>∞</sup> 0

ð 

j j ψ^ð Þ *au*

*f t*ðÞ¼ <sup>2</sup>*w*<sup>2</sup>

In the case *μ* ¼ 1*=*2, the measure (85) takes the form

2 *w*<sup>2</sup>

� �

^*f u*ð Þ^*g u*ð Þ

Let us consider the function

and the function (86) reduces to

¼ ð 

*Wavelet Theory*

**6.1 Example**

B*<sup>ψ</sup>*

2*w*<sup>2</sup> <sup>0</sup> � *t* 2

( ) !

ð<sup>∞</sup> 0 *ψ*ð Þ*z*

> <sup>D</sup> *<sup>b</sup> a* , *t a* , *z* � �

*a*�<sup>2</sup> ð<sup>∞</sup> 0 ψð Þ*z*

Using the representation

2 *w*<sup>2</sup> <sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup>

<sup>¼</sup> *<sup>a</sup>*�<sup>2</sup>

then (122) becomes

**280**

from (88) it follows that

^*f u*ð Þ*ψ*^ð Þ *au* ^*g u*ð Þ*ψ*^ð Þ *au*

The *admissible condition* (102) requires that *ψ*^ð Þ¼ 0 0. If *ψ*^ is continuous then

<sup>2</sup> *dσ*ð Þ *a a*2*μ*þ<sup>1</sup>

ð<sup>∞</sup> 0

> <sup>0</sup> � *t* 2

> > *dσ*ðÞ¼ *t*

where *J*0ð Þ *x* the Bessel's function of the first kind. Also note that

2*w*<sup>2</sup> <sup>0</sup> � *t* <sup>2</sup> � �<sup>2</sup>

2 *w*<sup>2</sup>

ð Þ¼ *<sup>b</sup>*, *<sup>a</sup> <sup>a</sup>*�<sup>2</sup>

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup> <sup>D</sup> *<sup>b</sup>*

2*w*<sup>2</sup> <sup>0</sup> � *t* 2

> ¼ ð<sup>∞</sup> 0 *J*0 *b a u* � � *J*0 *t a u* � �

ð<sup>∞</sup> 0 *J*0 *b a u* � �

2 *w*<sup>2</sup>

ð<sup>∞</sup> 0

!

2*w*<sup>2</sup> <sup>0</sup> � *t* 2

*a* , *t a* , *z* � �

*J*0ð Þ *zu* O*<sup>a</sup>*,*w*<sup>0</sup> ð Þ *u dσ*ð Þ *u*

� �

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup> *<sup>ψ</sup>*

2 *w*<sup>2</sup>

ð<sup>∞</sup> 0

The Bessel wavelet transform of *f t*ð Þ is given by

ð<sup>∞</sup> 0

*t* 2 *dσ*ð Þ *a*

ð 

¼ C *<sup>ψ</sup>* h i *f*, *g :* (116)

*ψ*ð Þ *x dσ*ð Þ¼ *x* 0*:* (117)

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup> , *<sup>w</sup>*<sup>0</sup> <sup>&</sup>gt;0, *<sup>t</sup>*<sup>∈</sup> þ*:* (118)

*dt* (119)

*j t*ðÞ¼ *J*0ð Þ*t* , (120)

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup> *<sup>d</sup>σ*ð Þ*<sup>t</sup>* <sup>&</sup>lt; <sup>∞</sup>*:* (121)

*dσ*ð Þ*t*

*b a* , *t a* � �

*dσ*ð Þ*t* (122)

*dσ*ð Þ*z* (123)

*J*0ð Þ *zu dσ*ð Þ *u* (124)

*dσ*ð Þ*z*

*dσ*ð Þ¼ *u C<sup>ψ</sup>*

*<sup>a</sup>*2*μ*þ<sup>1</sup> *<sup>d</sup>σ*ð Þ *<sup>u</sup>* (114)

^*f u*ð Þ^*g u*ð Þ*dσ*ð Þ *<sup>u</sup>* (115)

$$\mathfrak{D}\_{a,w\_0}(u) = \int\_0^\infty \frac{2w\_0^2 - t^2}{2\left(w\_0^2 + t^2\right)^{5/2}} J\_0\left(\frac{t}{a}u\right) d\sigma(t). \tag{125}$$

In terms of the Legendre polynomial *P*2ð Þ*t* , the function

$$\frac{2w\_0^2 - t^2}{2\left(w\_0^2 + t^2\right)^{5/2}} = \left(w\_0^2 + t^2\right)^{-3/2} P\_2\left[w\_0\left(w\_0^2 + t^2\right)^{-1/2}\right].\tag{126}$$

Then (125) reads

$$\mathcal{D}\_{a,w\_0}(u) = \int\_0^\infty \left(w\_0^2 + t^2\right)^{-3/2} P\_2\left[w\_0\left(w\_0^2 + t^2\right)^{-1/2}\right] J\_0\left(\frac{t}{a}u\right) d\sigma(t). \tag{127}$$

The above equation can be evaluated by means of the formula ([47], p. 13):

$$\frac{1}{n!}p^{n-1/2}e^{-py} = \int\_0^\infty x^{1/2} \left(p^2 + x^2\right)^{-\frac{1}{2p-\frac{4}{7}}} P\_n\left[p\left(p^2 + x^2\right)^{-1/2}\right] (xy)^{1/2} f\_0(xy) dx. \tag{128}$$

For parameters *n* ¼ 2 and *p* ¼ *w*0, we find that

$$\mathfrak{D}\_{a,w\_0}(u) = \frac{1}{4} u \exp\left(-w\_0 \frac{u}{a}\right). \tag{129}$$

In terms of the above result, the CBWT read as

$$\left\{ \mathcal{J} \mathbb{1}\_{\psi} \left( \frac{2w\_0^2 - t^2}{2 \left(w\_0^2 + t^2\right)^{5/2}} \right) \right\} (b, a) = a^{-2} \int\_0^\infty \mathbb{1}(z) \mathfrak{M}\_{a, w\_0}(z) d\sigma(z) \tag{130}$$

where

$$\mathfrak{M}\_{\mathfrak{u},w\_0}(z) = \int\_0^\infty \mathbf{8}^{-1} u^2 e^{-\frac{w\_0}{a}u} f\_0\left(\frac{b}{a}u\right) f\_0(zu) du. \tag{131}$$

To evaluated (131) we make appeal to the Lipschitz-Hankel integrals ([48], p. 389):

$$\int\_0^\infty e^{-pt} f\_\nu(qt) f\_\nu(\tau t) t^{\nu - 1} dt \tag{132}$$

$$= \frac{(qr)^\nu}{\pi p^{\mu + 2\nu}} \frac{\Gamma(\mu + 2\nu)}{2\nu + 1} \Big|\_{zF\_1}^\pi \left(\frac{\mu + 2\nu}{2}, \frac{\mu + 2\nu + 1}{2}; \nu + 1; -\frac{\zeta^2}{p^2}\right) \sin^{2\nu}\phi d\phi$$

with conditions ℜð Þ *p* � *iq* � *ir* >0 and ℜð Þ *μ* þ 2*ν* >0, while *ζ* is written in place of *<sup>q</sup>*<sup>2</sup> <sup>þ</sup> *<sup>r</sup>* ð Þ <sup>2</sup> � <sup>2</sup>*qr* cos *<sup>ϕ</sup>* <sup>1</sup>*=*<sup>2</sup> , where <sup>2</sup>*F*<sup>1</sup> denotes the hypergeometric function. For parameters *p* ¼ *w*0*=a*, *q* ¼ b*=*a,*r* ¼ *z*, *μ* ¼ 3 and *n* ¼ 0, we arrive at

$$
\mathfrak{M}\_{\mathfrak{t},\mu\_0}(z) = \frac{a^3}{4\pi w\_0^3} \int\_{z^{\overline{\mathfrak{t}}}}^{\pi} \left(\frac{3}{2}, 2; 1; -\left(aw\_0^{-1}\zeta\right)^2\right) d\phi \tag{133}
$$

$$
\text{where } \zeta = \left[\left(a^{-1}b\right)^2 + z^2 - 2a^{-1}bz\cos\phi\right]^{1/2}.
$$

Next, by using the representation of the hypergeometric <sup>2</sup>*F*1-sum ([49], p. 404, Eq. 209) (**Figure 2**):

$${}\_{2}F\_{1}\left(\frac{3}{2},2;1;z\right)=\frac{1}{2}(2+z)(1-z)^{-5/2}.\tag{134}$$

is the mother wavelet. The Hankel transform of *f t*ð Þ is given by

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

<sup>C</sup> *<sup>f</sup>* <sup>¼</sup> <sup>1</sup> 2 ð<sup>∞</sup> 0

<sup>¼</sup> <sup>1</sup> 128*w*<sup>2</sup> 0

ð<sup>∞</sup> 0

*For which numbers n* ∈ , *the following function*

*<sup>f</sup> <sup>n</sup>*ðÞ¼ *<sup>t</sup> <sup>w</sup>*<sup>2</sup>

The Hankel transformation ^*f*ð Þ¼ <sup>0</sup> 0, so by the help of (140) we obtain

2 *<sup>n</sup>*�<sup>1</sup> 2

In this chapter we are interested in the construction of the generalized coherent state (GCS) and the theory of wavelets. As it is well know wavelets constructed on the basis of group representation theory have the same properties as coherent states. In other words, the wavelets can actually be thought of as the coherent state associated with these groups. Coherent state is very important because of three properties they have: coherence, overcompleteness, intrinsic geometrization. We have seen that it is possible to construct coherent states without taking into account the theory of group representation. Throughout this chapter we have used the Bessel function to construct the coherent state transform and Bessel continuous wavelets transform. We have prove that the kernel of the finite Fourier transform


Building coherent states in this chapter is always not easy because it is necessary to find coefficients which will make it possible to find vectors which will certainly satisfy certain conditions but the procedure based on Wavelet's theory makes it

It should be noted that the theory of classical wavelets finds several applications ranging from the analysis of geophysical and acoustic signals to quantum theory. This theory solves difficult problems in mathematics, physics and engineering, with several modern applications such as data compression, wave propagation, signal processing, computer graphics, pattern recognition, pattern processing. Wavelet

*Pn w*<sup>0</sup> *w*<sup>2</sup>

*t* 2*w*<sup>2</sup> <sup>0</sup> � *t* <sup>2</sup> � �

*w*2

<sup>0</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> � ��<sup>1</sup>

*Is the mother wavelet where Pn*ð Þ*: the Legendre's polynomial*.

1 <sup>4</sup> *ye*�*w*0*<sup>y</sup>*

^*f*ð Þ*<sup>ξ</sup>* � � �

� � � 2 , ∀ 0≤ *y*< ∞*:* (140)

*<sup>ξ</sup> <sup>d</sup><sup>ξ</sup>* (141)

, *w*<sup>0</sup> >0*:* (142)

<sup>0</sup> <sup>þ</sup> *<sup>x</sup>*<sup>2</sup> � ��1*=*<sup>2</sup> h i (144)

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup> *dt* <sup>¼</sup> <sup>0</sup>*:* (143)

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �5*=*<sup>2</sup> *<sup>J</sup>*0ð Þ *xy <sup>d</sup>σ*ðÞ¼ *<sup>t</sup>*

^*f y*ð Þ¼ <sup>ð</sup><sup>∞</sup>

**Exercise 2**

**7. Conclusions**

(FFT) of *L*<sup>2</sup>

easier.

**283**

makes it easier.

0

2*w*<sup>2</sup> <sup>0</sup> � *t* 2

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

and satisfy the admissible condition

2 *w*<sup>2</sup>

Then (131) takes the form

$$\mathfrak{M}\_{a,w\_0}(\mathbf{z}) = \frac{a^3}{8\pi w\_0^3} \int\_0^\pi \left( 2 - \left( w\_0^{-1} a \zeta \right)^2 \right) \left( \mathbf{1} + \left( w\_0^{-1} a \zeta \right)^2 \right)^{-5/2} d\zeta,\tag{135}$$

This leads to the following CBWT

$$\left\{ \mathcal{B}\_{\boldsymbol{\Psi}} \left( \frac{2w\_0^2 - t^2}{2 \left(w\_0^2 + t^2\right)^{5/2}} \right) \right\} (b, a) = \frac{a}{4\pi} \int\_0^\infty \boldsymbol{\psi}(\boldsymbol{z}) \int\_0^\pi \frac{2w\_0^2 - (a\boldsymbol{\zeta})^2}{2 \left(w\_0^2 + \left(a\boldsymbol{\zeta}\right)^2\right)^{5/2}} d\boldsymbol{\phi} d\sigma(\boldsymbol{z}). \tag{136}$$

We have given an example of a signal *f t*ð Þ<sup>∈</sup> *<sup>L</sup>*<sup>2</sup> *<sup>σ</sup>*ð Þ 0, ∞ such that the CBWT is written as

$$\left\{ \mathcal{J}\theta\_{\Psi}(\, \!\!/f(t)) \right\}(b, a) = \frac{a}{4\pi} \int\_{0}^{\pi} \int\_{0}^{\infty} \varphi(z) f(a\zeta) d\sigma(z) d\phi. \tag{137}$$

According to Theorem 1.3, let *ψ* ∈*L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup> and *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup>*L*<sup>2</sup> *<sup>σ</sup>*ð Þ <sup>þ</sup> , then

$$\int\_{0}^{\infty} \int\_{0}^{\infty} (\mathcal{B}\_{\text{y}} f)(b, a) \overline{(\mathcal{B}\_{\text{y}} \text{g})(b, a)} d\sigma(a) d\sigma(b) = \frac{1}{128 w\_{0}^{2}} \langle f, \text{g} \rangle. \tag{138}$$

Note that, for all *w*<sup>0</sup> >0, the given function

$$f(t) = \frac{2w\_0^2 - t^2}{2\left(w\_0^2 + t^2\right)^{5/2}}, \ t \in \mathbb{R}\_+,\tag{139}$$

**Figure 2.** *Plots of the mother wavelet f t*ð Þ *defined in* ð Þ 6*:*34 *versus t, for various values of the parameters w*0*.*

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

is the mother wavelet. The Hankel transform of *f t*ð Þ is given by

$$\hat{f}(\mathbf{y}) = \int\_0^\infty \frac{2w\_0^2 - t^2}{2\left(w\_0^2 + t^2\right)^{5/2}} J\_0(\mathbf{x}\mathbf{y}) d\sigma(t) = \frac{1}{4}\mathcal{Y} e^{-w\_0 \mathbf{y}}, \quad \forall \quad 0 \le \mathbf{y} < \infty. \tag{140}$$

and satisfy the admissible condition

$$\mathfrak{G}\_f = \frac{1}{2} \int\_0^\infty \frac{\left| \hat{f}(\xi) \right|^2}{\xi} d\xi \tag{141}$$

$$= \frac{1}{128w\_0^2},\ \ w\_0 > 0.\tag{142}$$

The Hankel transformation ^*f*ð Þ¼ <sup>0</sup> 0, so by the help of (140) we obtain

$$\int\_{0}^{\infty} \frac{t \left(2w\_{0}^{2} - t^{2}\right)}{\left(w\_{0}^{2} + t^{2}\right)^{5/2}} dt = 0. \tag{143}$$

#### **Exercise 2**

Next, by using the representation of the hypergeometric <sup>2</sup>*F*1-sum ([49], p. 404,

ð Þ <sup>2</sup> <sup>þ</sup> *<sup>z</sup>* ð Þ <sup>1</sup> � *<sup>z</sup>* �5*=*<sup>2</sup>

<sup>1</sup> <sup>þ</sup> *<sup>w</sup>*�<sup>1</sup> <sup>0</sup> *<sup>a</sup><sup>ζ</sup>* � �<sup>2</sup> � ��5*=*<sup>2</sup>

2*w*<sup>2</sup>

2 *w*<sup>2</sup>

*<sup>σ</sup>*ð Þ <sup>þ</sup> and *<sup>f</sup>*, *<sup>g</sup>* <sup>∈</sup>*L*<sup>2</sup>

<sup>0</sup> � ð Þ *<sup>a</sup><sup>ζ</sup>* <sup>2</sup>

<sup>0</sup> <sup>þ</sup> ð Þ *<sup>a</sup><sup>ζ</sup>* <sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup> *<sup>d</sup>ϕdσ*ð Þ*<sup>z</sup> :* (136)

*<sup>σ</sup>*ð Þ 0, ∞ such that the CBWT is

*ψ*ð Þ*z f a*ð Þ*ζ dσ*ð Þ*z dϕ:* (137)

h i *f*, *g :* (138)

*<sup>σ</sup>*ð Þ <sup>þ</sup> , then

128*w*<sup>2</sup> 0

<sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup> , *<sup>t</sup>*<sup>∈</sup> þ, (139)

*:* (134)

*dζ*, (135)

¼ 1 2

Eq. 209) (**Figure 2**):

*Wavelet Theory*

Then (131) takes the form

M*a*,*w*<sup>0</sup> ð Þ¼ *z*

2*w*<sup>2</sup> <sup>0</sup> � *t* 2

( ) !

2 *w*<sup>2</sup> <sup>0</sup> <sup>þ</sup> *<sup>t</sup>*<sup>2</sup> � �<sup>5</sup>*=*<sup>2</sup>

ð<sup>∞</sup> 0

ð<sup>∞</sup> 0

B*<sup>ψ</sup>*

written as

**Figure 2.**

**282**

<sup>2</sup>*F*<sup>1</sup> 3 2 , 2; 1; *z* � �

*a*3 8*πw*<sup>3</sup> 0

This leads to the following CBWT

According to Theorem 1.3, let *ψ* ∈*L*<sup>2</sup>

Note that, for all *w*<sup>0</sup> >0, the given function

ð*π* 0

ð Þ¼ *b*, *a*

We have given an example of a signal *f t*ð Þ<sup>∈</sup> *<sup>L</sup>*<sup>2</sup>

<sup>B</sup>*<sup>ψ</sup>* ð Þ *f t*ð Þ � �ð Þ¼ *<sup>b</sup>*, *<sup>a</sup>*

<sup>2</sup> � *<sup>w</sup>*�<sup>1</sup> <sup>0</sup> *<sup>a</sup><sup>ζ</sup>* � �<sup>2</sup> � �

> *a* 4*π* ð<sup>∞</sup> 0 *ψ*ð Þ*z* ð*π* 0

> > *a* 4*π* ð*π* 0 ð<sup>∞</sup> 0

<sup>0</sup> � *t* 2

<sup>B</sup>*<sup>ψ</sup> <sup>f</sup>* � �ð Þ *<sup>b</sup>*, *<sup>a</sup>* <sup>B</sup>*<sup>ψ</sup> <sup>g</sup>* � �ð Þ *<sup>b</sup>*, *<sup>a</sup> <sup>d</sup>σ*ð Þ *<sup>a</sup> <sup>d</sup>σ*ð Þ¼ *<sup>b</sup>* <sup>1</sup>

*f t*ðÞ¼ <sup>2</sup>*w*<sup>2</sup>

2 *w*<sup>2</sup>

*Plots of the mother wavelet f t*ð Þ *defined in* ð Þ 6*:*34 *versus t, for various values of the parameters w*0*.*

*For which numbers n* ∈ , *the following function*

$$f\_n(t) = \left(w\_0^2 + \varkappa^2\right)^{-\frac{1}{2}n - \frac{1}{2}} P\_n\left[w\_0 \left(w\_0^2 + \varkappa^2\right)^{-1/2}\right] \tag{144}$$

*Is the mother wavelet where Pn*ð Þ*: the Legendre's polynomial*.

#### **7. Conclusions**

In this chapter we are interested in the construction of the generalized coherent state (GCS) and the theory of wavelets. As it is well know wavelets constructed on the basis of group representation theory have the same properties as coherent states. In other words, the wavelets can actually be thought of as the coherent state associated with these groups. Coherent state is very important because of three properties they have: coherence, overcompleteness, intrinsic geometrization. We have seen that it is possible to construct coherent states without taking into account the theory of group representation. Throughout this chapter we have used the Bessel function to construct the coherent state transform and Bessel continuous wavelets transform. We have prove that the kernel of the finite Fourier transform (FFT) of *L*<sup>2</sup> -functions supported on ½ � �1, 1 form a set of GCS. We therefore discussed another way of building a set of coherent states based on Wavelet's theory makes it easier.

Building coherent states in this chapter is always not easy because it is necessary to find coefficients which will make it possible to find vectors which will certainly satisfy certain conditions but the procedure based on Wavelet's theory makes it easier.

It should be noted that the theory of classical wavelets finds several applications ranging from the analysis of geophysical and acoustic signals to quantum theory. This theory solves difficult problems in mathematics, physics and engineering, with several modern applications such as data compression, wave propagation, signal processing, computer graphics, pattern recognition, pattern processing. Wavelet

analysis is a robust technique used for investigative methods in quantifying the timing of measurements in Hamiltonian systems.

**References**

2529

(1963) 2766

14, 664–666 (1926)

[1] Schrödinger, E.: Der stetige übergang von der Mikro- zur Makromechanik. Naturwissenschaften

[2] R. Glauber, The quantum theory of optical coherence, *Phys. Rev.* **130** (1963)

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

[12] H. Fakhri, B. Mojaveri and M. A. Gomshi Nobary, Landau Levels as a Limiting case of a model with the Morse-Like Magnetic field, *Rep. Math.*

[13] B. Mojaveri and A. Dehghani, New Generalized coherent states arising from generating functions: A novel approach,

[14] A.Dehghani, B.Mojaveri and M. Mahdian, New Even and ODD Coherent

Polynomials, *Rep. Math. Phys*, **75** (2015)

[16] V. V. Borzov, E. V. Damaskinsky, Coherent states for the Legendre oscillator, ZNS POMI, 285–52 (2002)

polynomials and generalized oscillator algebras, *Integral Transf. and Special*

[18] K. Thirulogasantar and N. Saad, Coherent states associated with the wavefunctions and the spectrum of the isotonic oscillator, *J. Phys. A, Math. Gen.*

[19] A. Horzela and F. H. Szafraniec, A measure-free approach to coherent states, *J. Phys. A: Math. Theor.* **45** (2012).

[20] G. Arfken, Mathematical methods for physicists, Academic Press Inc, 1985.

[21] Neal C. Gallagher, J. R., and Gary L. Wise, A representation for Band-Limited Functions, *Proceedings of the*

Continuous and discrete Bessel wavelet

*Rep. Math. Phys*, **75** (2015) 47

States Attached to the Hermite

[15] H. Fakhri and B. Mojaveri, Generalized Coherent States for the Spherical Harmonics *Int. J. Mod. Phys. A*,

[17] V. V. Borzov, Orthogonal

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267

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[22] R. S. Pathak, M. M. Dixit,

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representation theory. I. Postulayes of continuous representation, *J. Math.*

[5] J. R. Klauder, Generalized relation between quantum and classical

dynamics, *J. Math. Phys.* **4** (1963) 1058.

[6] J. R. Klauder and B-S. Shagertan, Coherent states, (world scientific,

[7] A. M. Perelomov, Generalized coherent states and their Application,

[8] Peter. W. Milonni and Michael Martin Nieto, Coherent states, In: Compendium of Quantum Physics, pp.106–108. Springer (2009)

[9] S. T. Ali, J-P. Antoine and J-P. Gazeau, Coherent states, Wavelets and their Gneralization, (Springer-Verlag,

[10] B. Mojaveri and S. Amiri Faseghandis, Generalized coherent states related to the associated Bessel functions and Morse potential, *Phys. Scr.*

**89** (2014) 085204 (8pp)

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**285**

## **Conflict of interest**

The authors declare no conflict of interest.

## **Author details**

Kayupe Kikodio Patrick Faculty of Sciences, Lubumbashi University, DRC

\*Address all correspondence to: kayupepatrick@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory DOI: http://dx.doi.org/10.5772/intechopen.94865*

#### **References**

analysis is a robust technique used for investigative methods in quantifying the

timing of measurements in Hamiltonian systems.

The authors declare no conflict of interest.

**Conflict of interest**

*Wavelet Theory*

**Author details**

**284**

Kayupe Kikodio Patrick

Faculty of Sciences, Lubumbashi University, DRC

provided the original work is properly cited.

\*Address all correspondence to: kayupepatrick@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

[1] Schrödinger, E.: Der stetige übergang von der Mikro- zur Makromechanik. Naturwissenschaften 14, 664–666 (1926)

[2] R. Glauber, The quantum theory of optical coherence, *Phys. Rev.* **130** (1963) 2529

[3] R. Glauber, Coherent and incoherent states of radiation field, *J. Phys. Rev.* **131** (1963) 2766

[4] J. R. Klauder, Continuous representation theory. I. Postulayes of continuous representation, *J. Math. Phys.* **4** (1963) 1055.

[5] J. R. Klauder, Generalized relation between quantum and classical dynamics, *J. Math. Phys.* **4** (1963) 1058.

[6] J. R. Klauder and B-S. Shagertan, Coherent states, (world scientific, Singapore, 1985)

[7] A. M. Perelomov, Generalized coherent states and their Application, (Spring-Verlag Berlin 1986)

[8] Peter. W. Milonni and Michael Martin Nieto, Coherent states, In: Compendium of Quantum Physics, pp.106–108. Springer (2009)

[9] S. T. Ali, J-P. Antoine and J-P. Gazeau, Coherent states, Wavelets and their Gneralization, (Springer-Verlag, New York, 2000)

[10] B. Mojaveri and S. Amiri Faseghandis, Generalized coherent states related to the associated Bessel functions and Morse potential, *Phys. Scr.* **89** (2014) 085204 (8pp)

[11] B. Mojaveri, Klauder-Perelomov and Gazeau-Klauder coherent states for an electron in the Morse-like magnetic field, *Eur. Phys. J. D,* **67** (2013) 105

[12] H. Fakhri, B. Mojaveri and M. A. Gomshi Nobary, Landau Levels as a Limiting case of a model with the Morse-Like Magnetic field, *Rep. Math. Phys*, **66** (2010) 299

[13] B. Mojaveri and A. Dehghani, New Generalized coherent states arising from generating functions: A novel approach, *Rep. Math. Phys*, **75** (2015) 47

[14] A.Dehghani, B.Mojaveri and M. Mahdian, New Even and ODD Coherent States Attached to the Hermite Polynomials, *Rep. Math. Phys*, **75** (2015) 267

[15] H. Fakhri and B. Mojaveri, Generalized Coherent States for the Spherical Harmonics *Int. J. Mod. Phys. A*, **25** (2010) 2165.

[16] V. V. Borzov, E. V. Damaskinsky, Coherent states for the Legendre oscillator, ZNS POMI, 285–52 (2002)

[17] V. V. Borzov, Orthogonal polynomials and generalized oscillator algebras, *Integral Transf. and Special Funct.* **12**,(2001).

[18] K. Thirulogasantar and N. Saad, Coherent states associated with the wavefunctions and the spectrum of the isotonic oscillator, *J. Phys. A, Math. Gen.* **37**, 4567–4577 (2004).

[19] A. Horzela and F. H. Szafraniec, A measure-free approach to coherent states, *J. Phys. A: Math. Theor.* **45** (2012).

[20] G. Arfken, Mathematical methods for physicists, Academic Press Inc, 1985.

[21] Neal C. Gallagher, J. R., and Gary L. Wise, A representation for Band-Limited Functions, *Proceedings of the IEEE*, **63** 11, 1975.

[22] R. S. Pathak, M. M. Dixit, Continuous and discrete Bessel wavelet transforms, Journal of Computational and Applied Mathematics, 160 (2003), pp. 240–250.

[23] R. S. Pathak, S. K. Upadhyay and R. S. Pandey, The Bessel wavelet convolution product, Rend.Sem.Mat. Univ.Politec.Torino, 96(3) (2011), pp. 267–279

[24] A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, *SIAM J. Math. Anal.*, Vol. 15, No. 4 (1984), 723–736.

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[26] T.H. Koornwinder, The continuous wavelet transform. Vol. 1. Wavelets: An Elementary Treatment of Theory and Applications. Edited by T.H. Koornwinder, World Scientific, 1993, 27–48

[27] Y. Meyer, Wavelets and Operators, Cambridge University Press, Cambridge, 1992

[28] S. T. Ali, J. P. Antoine and J. P. Gazeau, Coherent states, Wavelets, and their generalization, Springer Science +Business Media New York 2014

[29] M. M. Pelosso, Classical spaces of holomorphic functions, Technical report, Università di Milano, 2011.

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[31] N. I. Akhieser, The classical moment problem and some related questions in analysis, Hafner Publ. Co, New York, 1965

[32] J-M. Sixdenierrs, K. A. Penson and A. I. Solomon, Mittag-Leffler coherent states, *J. Phys. A.* **32**, (1999)

[33] Ali, S.T., Reproducing kernels in coherent states, wavelets, and quantization, in Operator Theory, Springer, Basel (2015), p. 14; doi 10.1007/978-3-0348-0692-3 63–1

polynomials into Fourier systems, *Constr. Approx.* **28**, 219–235.

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*Math*.8 (1960–1961) 307–336.

New York, 1968

Amsterdam, 1997

Heildelberg Berlin 1972

Press, Second Edition 1944

[46] K.Trime<sup>0</sup>

**287**

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[43] D. Robert, La cohérence dans tous ses états. *SMF Gazette* **132** (2012)

*DOI: http://dx.doi.org/10.5772/intechopen.94865*

diminishing Hankel transforms, *J.Anal.*

and Hypergroups, Gordon and Breach,

[47] Fritz Oberhettinger, Tables of Bessel Transforms, Springer-Verlag New York

[48] G.N. Watson, A treatise of Bessel functions, Cambridge at the University

[49] A. P. Prudnikov, Yu. A. Brychkov, Integrals and series, More special Functions, volume 3, computing center of the USSR Academy of sciences Moscow, from Russian version

che, Generalized Wavelets

*Case Study: Coefficient Training in Paley-Wiener Space, FFT, and Wavelet Theory*

[34] Imen Rezgui and Anouar Ben Marbrouk, Some Generalized *q*-Bessel type Wavelets and Associated Transforms, *Anal. Theory Appl.*, **34** (2018), pp. 57–76

[35] Ali, S.T., Antoine, J.-P., Gazeau, J.- P.: Coherent States, Wavelets and their Generalizations. Springer, New York (2014)

[36] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and series. volume 2, Special functions, OPA (Overseas Publishers Association), Amsterdam 1986

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[39] J. R. Higgins, Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford, U.K.: Clarendon, 1996.

[40] F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, editors. NIST Handbook of Mathematical Functions. Cambridge University Press, 2010

[41] M. E. H. Ismail and R. Zhang, Diagonalization of certain integral operators. *Adv. Math.* **109** (1994), no. 1, 1–33

[42] L. D. Abreu, The reproducing kernel structure arising from a combination of continuous and discrete orthogonal

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polynomials into Fourier systems, *Constr. Approx.* **28**, 219–235.

transforms, Journal of Computational and Applied Mathematics, 160 (2003), [33] Ali, S.T., Reproducing kernels in coherent states, wavelets, and quantization, in Operator Theory, Springer, Basel (2015), p. 14; doi 10.1007/978-3-0348-0692-3 63–1

[34] Imen Rezgui and Anouar Ben Marbrouk, Some Generalized *q*-Bessel

[35] Ali, S.T., Antoine, J.-P., Gazeau, J.- P.: Coherent States, Wavelets and their Generalizations. Springer, New York

[36] A. P. Prudnikov, Y. A. Brychkov and O. I. Marichev, Integrals and series. volume 2, Special functions, OPA (Overseas Publishers Association),

[37] J. R. Higgins, Sampling theory in

[38] Mourad E. H. Ismail, Classical and Quantum Orthogonal Polynomials in one Variable, Encyclopedia of Mathematics and its Applications, Cambridge University Press 2005

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Foundations. Oxford, U.K.: Clarendon,

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[41] M. E. H. Ismail and R. Zhang, Diagonalization of certain integral operators. *Adv. Math.* **109** (1994), no. 1,

[42] L. D. Abreu, The reproducing kernel structure arising from a combination of continuous and discrete orthogonal

Fourier and Signal Analysis:

1996.

1–33

Fourier and signal analysis: Foundations, Oxford Science

type Wavelets and Associated Transforms, *Anal. Theory Appl.*, **34**

(2018), pp. 57–76

Amsterdam 1986

Publication, 1996.

(2014)

[23] R. S. Pathak, S. K. Upadhyay and R.

S. Pandey, The Bessel wavelet convolution product, Rend.Sem.Mat. Univ.Politec.Torino, 96(3) (2011),

[24] A. Grossmann and J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, *SIAM J. Math. Anal.*, Vol. 15, No.

[25] C.K. Chui, An Introduction to Wavelets, Academic Press, 1992

Applications. Edited by T.H.

Cambridge University Press,

Cambridge, 1992

1980 (in Russian).

1965

**286**

[26] T.H. Koornwinder, The continuous wavelet transform. Vol. 1. Wavelets: An Elementary Treatment of Theory and

Koornwinder, World Scientific, 1993,

[27] Y. Meyer, Wavelets and Operators,

[28] S. T. Ali, J. P. Antoine and J. P. Gazeau, Coherent states, Wavelets, and their generalization, Springer Science +Business Media New York 2014

[29] M. M. Pelosso, Classical spaces of holomorphic functions, Technical report, Università di Milano, 2011.

[30] M. S. Birman and M. Z. Solomyak, Spectral theory of self-adjoint operators in Hilbert space, Leningrad Univ. Press,

[31] N. I. Akhieser, The classical moment problem and some related questions in analysis, Hafner Publ. Co, New York,

[32] J-M. Sixdenierrs, K. A. Penson and A. I. Solomon, Mittag-Leffler coherent

states, *J. Phys. A.* **32**, (1999)

pp. 240–250.

*Wavelet Theory*

pp. 267–279

27–48

4 (1984), 723–736.

[43] D. Robert, La cohérence dans tous ses états. *SMF Gazette* **132** (2012)

[44] I.I.Hirschman Jr., Variation diminishing Hankel transforms, *J.Anal. Math*.8 (1960–1961) 307–336.

[45] A.H. Zemanian, Generalized Integral Transformations, Interscience, New York, 1968

[46] K.Trime<sup>0</sup> che, Generalized Wavelets and Hypergroups, Gordon and Breach, Amsterdam, 1997

[47] Fritz Oberhettinger, Tables of Bessel Transforms, Springer-Verlag New York Heildelberg Berlin 1972

[48] G.N. Watson, A treatise of Bessel functions, Cambridge at the University Press, Second Edition 1944

[49] A. P. Prudnikov, Yu. A. Brychkov, Integrals and series, More special Functions, volume 3, computing center of the USSR Academy of sciences Moscow, from Russian version

**Chapter 13**

*Xi Zhang*

**Abstract**

the proposed method.

orthogonality

**289**

**1. Introduction**

wavelet filter banks in this chapter.

can realize both of orthogonality and symmetry.

Allpass Filters

Wavelet Filter Banks Using

Allpass filter is a computationally efficient versatile signal processing building block. The interconnection of allpass filters has found numerous applications in digital filtering and wavelets. In this chapter, we discuss several classes of wavelet filter banks by using allpass filters. Firstly, we describe two classes of orthogonal wavelet filter banks composed of two real allpass filters or a complex allpass filter, and then consider design of orthogonal filter banks without or with symmetry, respectively. Next, we present two classes of filter banks by using allpass filters in lifting scheme. One class is causal stable biorthogonal wavelet filter bank and another class is orthogonal wavelet filter bank, all with approximately linear phase response. We also give several design examples to demonstrate the effectiveness of

**Keywords:** wavelet, filter bank, allpass filter, perfect reconstruction, symmetry,

The discrete wavelet transform (DWT), which is implemented by a two band perfect reconstruction (PR) filter bank, has been applied extensively to digital signal processing, image processing, medical and health care, economy and so on [1–4]. In many applications such as image processing, wavelets are required to be real since the signal is real-valued in general. We restrict ourselves to real-valued

In addition to orthogonality, one desirable property for wavelets is symmetry, which requires all filters in the filter bank to possess exactly linear phase, because the symmetric extension method is generally used to treat the boundaries of images [5, 6]. It is known in [1–4] that finite impulse response (FIR) filters (corresponding to the compactly supported wavelets) can easily realize exactly linear phase. However, it is widely appreciated that the only FIR solution that produces a real orthogonal symmetric wavelet basis is the Haar wavelet, which is not continuous and the corresponding filter is of order 1 only that is not enough for many practical applications. To obtain wavelet filter banks with higher degrees of freedom, infinite impulse response (IIR) filters have been used to construct wavelet filter banks with some of the desired properties [7–12]. Among the existing IIR wavelet filter banks, wavelet filter banks composed of allpass filters are attractive [7, 9, 10, 12], which

Allpass filter is a computationally efficient versatile signal processing building

block and quite useful in many applications [13]. Allpass filter possesses unit

#### **Chapter 13**
