Use of Daubechies Wavelets in the Representation of Analytical Functions

*Paulo César Linhares da Silva*

#### **Abstract**

This chapter aims to use Daubechies' wavelets as basis functions to generate analytical functions, thus being able to rewrite the Taylor series using these wavelets. This makes it possible to analyze functions with a high degree of complexity, in problems that require a high degree of precision in their solution. Wavelet analysis can be applied to practical problems that require a high degree of precision, for example, in the study and analysis of electromagnetic propagation in optical fibers, solutions of differential equations involving engineering problems, in the transmission of WiFi signals, in the treatment and analysis of biomedical images, detection of oil sources through the study of seismic signals.

**Keywords:** wavelets, Daubechies, analytical functions, basis functions, Taylor series

#### **1. Introduction**

Wavelets [1] were born from the need to generate functions, especially those that present singularities, high gradients, discontinuities both in the time domain and in the frequency domain. Wavelets enable the high-resolution analysis of functions with these characteristics. An example of a problem that occurs when generating functions with a Fourier base is the Gibbs phenomenon. Such a phenomenon occurs because there is no way to represent functions that present discontinuities, even adding more elements in the base that will generate the function. A characteristic of wavelets is that they do not produce such an effect.

Wavelets are widely used in the solution of numerical problems in several areas of knowledge such as image compression, Numerical Harmonic Analysis [2], financial analysis, oil detection, differential Equations [3, 4], biomedical signals, analysis of electromagnetic integral Equations [5], optical fibers [6], among others. Many of these applications use the specific properties of wavelets, such as coefficients that are determined numerically, multi-resolution analysis to decompose a signal, integrals, and derivatives obtained numerically, energy concentrated in its compact and base with orthogonal elements.

#### **2. Short introduction to wavelet theory**

For the development of topics presented in this chapter, the reader must have as a prerequisite knowledge of functional analysis, linear algebra, measure theory and integration, differential and integral calculus. It is important to note that the wavelet basis is for the wavelet transform as well as the trigonometric basis is for the Fourier transform. Generally, the term wavelet is also used as a wavelet transform. The following subsections present these initial prerequisites to the reader.

#### **2.1 Preliminaries on Hilbert spaces**

In this subsection, some mathematical concepts necessary for a better formal understanding of the wavelet tool are defined. The definitions, contained in this section, are due to the author [2].

Definition 2.1 The space H is said to be a Hilbert space, if an inner product <, >, associated with a standard k ¼ ffiffiffiffiffiffiffiffiffiffiffiffi < , > p has been defined in it. And a set of vectors f g *vi* , for *i* ∈ an orthonormal system is said if the internal product <*vn*, *vm* > ¼ *δmn*, for *m*, *n* ∈ .

Definition 2.2 A set of vectors f g *vn* is orthonormal, if and only if, for every finite set of complex numbers *xn*, there is ∥ P *<sup>n</sup>anxn*∥<sup>2</sup> <sup>¼</sup> <sup>P</sup>j j *an* <sup>2</sup> , for *n* ∈ .

Definition 2.3 In Hilbert's H space, a set of vectors f g *vn* is said to be a Riez system, if there are constants 0 ≤*c*≤*C*< ∞ such that for any finite set of complex numbers *xn* if you have:

$$\mathcal{L}\sum |a\_n|^2 \le \|\sum\_n a\_n \mathbf{x}\_n\|^2 \le \mathcal{C} \sum\_n |a\_n|^2 \tag{1}$$

**2.2 Definition of wavelet**

*ψ <sup>j</sup>*,*k*ð Þ *x* n o

term mathematically wavelet.

is an orthogonal basis on *L*<sup>2</sup>

efficient than the basic Fourier functions.

into a base of wavelet functions.

wavelet respectively.

equal to zero.

given by (6):

**335**

This subsection aims to define wavelet [2], the main mathematical tool used in the development of this chapter. However, it is necessary to define the expansion

Definition 2.7 *Given a*>0*, the expansion operator, Da, defined over a f x*ð Þ *function*

Thus, using the expansion and translation operations defined above, a family of

ð Þ .

n o

ð Þ . Where *j* and *k* are the resolution and translation of

*<sup>j</sup>*,*k*∈ <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> f g *jTkψ*ð Þ *<sup>x</sup> <sup>j</sup>*,*k*∈ (4)

*j*,*k*∈

*dt* (5)

*f t*ð Þ*ψ<sup>j</sup>*,*<sup>k</sup>*ð Þ*t dt* (6)

, to define the

n o

*j*,*k* ∈

1 <sup>2</sup>*f x*ð Þ*.* Definition 2.8 *Given b*∈*R, the translation operator, Tb, defined over a function f x*ð Þ*,*

and translation of mathematical operations beforehand.

*Use of Daubechies Wavelets in the Representation of Analytical Functions*

functions *<sup>ψ</sup> <sup>j</sup>*,*k*ð Þ *<sup>x</sup>* was built: *<sup>L</sup>*<sup>2</sup> ! , base orthogonal to *<sup>L</sup>*<sup>2</sup>

<sup>2</sup> *ψ* 2 *<sup>j</sup>*

The Definition 2.9, uses the family of functions *ψ <sup>j</sup>*,*<sup>k</sup>*ð Þ *x*

The definition 2.10 is another way used to define a wavelet.

*x* � *k* n o � �

Definition 2.9 A function *ψ*ð Þ *x* is called wavelet if the collection *ψ <sup>j</sup>*,*<sup>k</sup>*ð Þ *x*

By varying the values of *j* and/or *k*, it is possible to analyze with greater precision, for example, the behavior of functions that present abrupt changes in values and discontinuity. This type of analysis makes the wavelet a tool as or more

Definition 2.10 A wavelet<sup>2</sup> is a short duration wave, which has an average value

Due to the definition 2.10, wavelets resemble Fourier sine and cosine basis functions. Analogously to what is done in the Fourier transform, which has sine and cosine functions as base functions, in wavelet analysis, a function is decomposed

The Fourier transform *F*ð Þ *ω* expression of a *f t*ð Þ function is given by (5):

ðþ<sup>∞</sup> �∞ *f t*ð Þ*e* �*iωt*

The expression (5) means that the Fourier transform is the sum of every *f t*ð Þ sign multiplied by a complex exponential, which can be separated into cosine and

Similarly, the expression of the wavelet transform *Wj*,*<sup>k</sup>*ð Þ*f* of a function *f t*ð Þ, is

ðþ<sup>∞</sup> �∞

Similarly, the expression of the wavelet transform (6) is the internal product of

*F*ð Þ¼ *ω*

sinusoidal components in the real and complex parts, respectively.

*Wj*,*<sup>k</sup>*ð Þ¼ *f*

<sup>2</sup> Anglophone term to designate a small wave, in the sense of having a fast duration.

the signal to be transformed by a wavelet function.

*in L*<sup>1</sup> *or L*<sup>2</sup> *over , is given by, Da f x*ð Þ¼ *<sup>a</sup>*

*DOI: http://dx.doi.org/10.5772/intechopen.93885*

*in L*<sup>1</sup> *or L*<sup>2</sup> *over , is given by, Tb f x*ð Þ¼ *f xb* ð Þ*.*

*<sup>j</sup>*,*<sup>k</sup>* <sup>∈</sup> <sup>¼</sup> <sup>2</sup> *<sup>j</sup>*

Definition 2.4 The space *L*<sup>2</sup> ð Þ is said to be an integrable square function space, that is,

$$L^2(\mathbb{R}) = \left\{ f : \mathbb{R} \mapsto \mathbb{C} : \int\_{\mathbb{R}} |f|(\mathfrak{x})|^2 d\mathfrak{x} < \infty \right\} \tag{2}$$

For *f*, *g* ∈*L*<sup>2</sup> ð Þ , define the inner product <sup>&</sup>lt;*f*, *<sup>g</sup>* <sup>&</sup>gt; <sup>¼</sup> <sup>Ð</sup> *f x*ð Þ*g x*ð Þ*dx*. On what, *g x*ð Þ is the complex conjugate of the function *g x*ð Þ.

In particular <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup> <sup>¼</sup> <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup> j j *f x*ð Þ <sup>2</sup> *dx* � �<sup>1</sup> 2 , and *f* is said to be an integrable square.

Definition 2.5 Let *f* : ↦ be a function. The support of *f*, denoted by *suppf*, is the closing of the set f g *x*∈ : *f x*ð Þ 6¼ 0 . A function *f* is said to have compact support if the *suppf* set is compact.<sup>1</sup>

Definition 2.6 We say that a function *f* is generated by the basis functions *f* <sup>1</sup>, …*f <sup>n</sup>* � �, if coefficients exist f g *<sup>c</sup>*1, …,*cn* such that:

$$f = \sum\_{i=1}^{n} f\_i c\_i \tag{3}$$

The concepts presented here about orthogonality and support of a *f* function, are fundamental to formalize the definition of wavelet. The following subsection presents the formal mathematical concept of wavelet.

<sup>1</sup> A set is said to be compact if it is limited and closed.

*Use of Daubechies Wavelets in the Representation of Analytical Functions DOI: http://dx.doi.org/10.5772/intechopen.93885*

#### **2.2 Definition of wavelet**

and integration, differential and integral calculus. It is important to note that the wavelet basis is for the wavelet transform as well as the trigonometric basis is for the Fourier transform. Generally, the term wavelet is also used as a wavelet transform. The following subsections present these initial prerequisites

In this subsection, some mathematical concepts necessary for a better formal understanding of the wavelet tool are defined. The definitions, contained in this

Definition 2.1 The space H is said to be a Hilbert space, if an inner product <, >,

Definition 2.2 A set of vectors f g *vn* is orthonormal, if and only if, for every finite

*<sup>n</sup>anxn*∥<sup>2</sup> <sup>¼</sup> <sup>P</sup>j j *an* <sup>2</sup>

X *n*

ð Þ is said to be an integrable square function space,

*dx*< ∞

*anxx*∥<sup>2</sup> ≤*C*

ð 

*dx* � �<sup>1</sup>

Definition 2.5 Let *f* : ↦ be a function. The support of *f*, denoted by *suppf*, is

2

� �

<sup>j</sup>*f x*j j ð Þ <sup>2</sup>

f g *vi* , for *i* ∈ an orthonormal system is said if the internal product <*vn*, *vm* > ¼ *δmn*,

P

*n*

<sup>X</sup>j j *an* <sup>2</sup> ≤ ∥<sup>X</sup>

ð Þ¼ *f* : ↦ :

ð Þ , define the inner product <sup>&</sup>lt;*f*, *<sup>g</sup>* <sup>&</sup>gt; <sup>¼</sup> <sup>Ð</sup>

j j *f x*ð Þ <sup>2</sup>

the closing of the set f g *x*∈ : *f x*ð Þ 6¼ 0 . A function *f* is said to have compact

Definition 2.6 We say that a function *f* is generated by the basis functions

*<sup>f</sup>* <sup>¼</sup> <sup>X</sup>*<sup>n</sup> i*¼1 *fi*

The concepts presented here about orthogonality and support of a *f* function, are fundamental to formalize the definition of wavelet. The following subsection

Definition 2.3 In Hilbert's H space, a set of vectors f g *vn* is said to be a Riez system, if there are constants 0 ≤*c*≤*C*< ∞ such that for any finite set of complex

< , > p has been defined in it. And a set of vectors

, for *n* ∈ .

j j *an* <sup>2</sup> (1)

*f x*ð Þ*g x*ð Þ*dx*. On what,

, and *f* is said to be an integrable

*ci* (3)

(2)

to the reader.

*Wavelet Theory*

for *m*, *n* ∈ .

that is,

square.

*f* <sup>1</sup>, …*f <sup>n</sup>*

**334**

For *f*, *g* ∈*L*<sup>2</sup>

**2.1 Preliminaries on Hilbert spaces**

section, are due to the author [2].

associated with a standard k ¼ ffiffiffiffiffiffiffiffiffiffiffiffi

set of complex numbers *xn*, there is ∥

Definition 2.4 The space *L*<sup>2</sup>

In particular <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup> <sup>¼</sup> <sup>∥</sup> *<sup>f</sup>* <sup>∥</sup><sup>2</sup> <sup>¼</sup> <sup>Ð</sup>

support if the *suppf* set is compact.<sup>1</sup>

*c*

*L*2

*g x*ð Þ is the complex conjugate of the function *g x*ð Þ.

� �, if coefficients exist f g *<sup>c</sup>*1, …,*cn* such that:

presents the formal mathematical concept of wavelet.

<sup>1</sup> A set is said to be compact if it is limited and closed.

numbers *xn* if you have:

This subsection aims to define wavelet [2], the main mathematical tool used in the development of this chapter. However, it is necessary to define the expansion and translation of mathematical operations beforehand.

Definition 2.7 *Given a*>0*, the expansion operator, Da, defined over a f x*ð Þ *function in L*<sup>1</sup> *or L*<sup>2</sup> *over , is given by, Da f x*ð Þ¼ *<sup>a</sup>* 1 <sup>2</sup>*f x*ð Þ*.*

Definition 2.8 *Given b*∈*R, the translation operator, Tb, defined over a function f x*ð Þ*, in L*<sup>1</sup> *or L*<sup>2</sup> *over , is given by, Tb f x*ð Þ¼ *f xb* ð Þ*.*

Thus, using the expansion and translation operations defined above, a family of functions *<sup>ψ</sup> <sup>j</sup>*,*k*ð Þ *<sup>x</sup>* was built: *<sup>L</sup>*<sup>2</sup> ! , base orthogonal to *<sup>L</sup>*<sup>2</sup> ð Þ .

$$\left\{\boldsymbol{\Psi}\,\_{j,k}(\mathbf{x})\right\}\_{j,k\in\mathbb{Z}} = \left\{\mathcal{D}^{\frac{j}{2}}\boldsymbol{\Psi}\left(\mathcal{D}^{j}\mathbf{x} - k\right)\right\}\_{j,k\in\mathbb{Z}} = \left\{D\_{2}\boldsymbol{T}\_{k}\boldsymbol{\Psi}\left(\mathbf{x}\right)\right\}\_{j,k\in\mathbb{Z}}\tag{4}$$

The Definition 2.9, uses the family of functions *ψ <sup>j</sup>*,*<sup>k</sup>*ð Þ *x* n o *j*,*k*∈ , to define the term mathematically wavelet.

Definition 2.9 A function *ψ*ð Þ *x* is called wavelet if the collection *ψ <sup>j</sup>*,*<sup>k</sup>*ð Þ *x* n o *j*,*k* ∈ is an orthogonal basis on *L*<sup>2</sup> ð Þ . Where *j* and *k* are the resolution and translation of wavelet respectively.

By varying the values of *j* and/or *k*, it is possible to analyze with greater precision, for example, the behavior of functions that present abrupt changes in values and discontinuity. This type of analysis makes the wavelet a tool as or more efficient than the basic Fourier functions.

The definition 2.10 is another way used to define a wavelet.

Definition 2.10 A wavelet<sup>2</sup> is a short duration wave, which has an average value equal to zero.

Due to the definition 2.10, wavelets resemble Fourier sine and cosine basis functions. Analogously to what is done in the Fourier transform, which has sine and cosine functions as base functions, in wavelet analysis, a function is decomposed into a base of wavelet functions.

The Fourier transform *F*ð Þ *ω* expression of a *f t*ð Þ function is given by (5):

$$F(o) = \int\_{-\infty}^{+\infty} f(t)e^{-iat}dt\tag{5}$$

The expression (5) means that the Fourier transform is the sum of every *f t*ð Þ sign multiplied by a complex exponential, which can be separated into cosine and sinusoidal components in the real and complex parts, respectively.

Similarly, the expression of the wavelet transform *Wj*,*<sup>k</sup>*ð Þ*f* of a function *f t*ð Þ, is given by (6):

$$\mathcal{W}\_{j,k}(f) = \int\_{-\infty}^{+\infty} f(t)\nu\_{j,k}(t)dt\tag{6}$$

Similarly, the expression of the wavelet transform (6) is the internal product of the signal to be transformed by a wavelet function.

<sup>2</sup> Anglophone term to designate a small wave, in the sense of having a fast duration.

In the following subsection, among the most varied types of wavelets, the Daubechies wavelets are highlighted, which are the basis for the development of this chapter.

#### **2.3 Daubechies wavelet properties**

At 1988, a family of compact support wavelets [7] is built by Ingrid Daubechies. This family of wavelets has highly well-located elements. Each member wavelet is governed by a set of *N* integer coefficients and *k* ¼ f g 0*:*1, …, *N* � 1 coefficients through scale relations (7) and (8). The *ak* and *a*1�*<sup>k</sup>* coefficients, which appear in the (7) and (8), are called filter coefficients and verify the following relations:

$$\phi(\mathbf{x}) = \sum\_{k=0}^{N-1} a\_k \phi(2\mathbf{x} - k) \tag{7}$$

To determine the filter coefficients *ak* and *a*<sup>1</sup>�*<sup>k</sup>*, which appear in the (7) and (8),

*Daubechies wavelet ψ. Source: This figure was generated by the author using the python programming language.*

*ak* ¼ 2 (9)

*akak*�*<sup>m</sup>* ¼ *δ*0,*<sup>m</sup>* (10)

*a*<sup>1</sup>�*kak*�2*<sup>m</sup>* ¼ 0 (11)

*<sup>n</sup>*! ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup> *<sup>n</sup>* (13)

*cmϕm*ð Þ *x :* (14)

<sup>2</sup> � 1, (12)

*N*

X *N*�1

*k*¼0

X *N*�1

*k*¼0

ð Þ �<sup>1</sup> *<sup>k</sup>*

**3. Generating an analytical function of the type** *x<sup>k</sup>* **using wavelets**

*kmak* <sup>¼</sup> 0, *<sup>m</sup>* <sup>¼</sup> <sup>0</sup>*:*1, …,

Analytical functions are those that can be locally around a point *x*<sup>0</sup> expanded in a

In general according to the author [8], any *f x*ð Þ function can be represented in

*ckϕ*ð Þ¼ *<sup>x</sup>* � *<sup>m</sup>* <sup>X</sup>

þ∞

*m*¼�∞

X *N*�1

*Use of Daubechies Wavelets in the Representation of Analytical Functions*

*DOI: http://dx.doi.org/10.5772/intechopen.93885*

*k*¼0

X *N*�1

ð Þ �<sup>1</sup> *<sup>k</sup>*

*k*¼0

where *δ<sup>k</sup>*,*<sup>m</sup>* is the Kronecker Delta function.

Taylor series, according to the following expression.

*f x*ð Þ¼ <sup>X</sup> þ∞

*m*¼�∞

terms of a wavelet base, as follows:

**337**

*f x*ð Þ¼ <sup>X</sup> þ∞

*n*¼0

*f* ð Þ *<sup>n</sup>* ð Þ *<sup>x</sup>*<sup>0</sup>

we use the relations (9)–(12) below.

**Figure 2.**

$$\psi(\mathbf{x}) = \sum\_{k=2-N}^{1} \left(-\mathbf{1}\right)^{k} a\_{1-k} \phi(2\mathbf{x} - k) \tag{8}$$

In the **Figures 1** and **2** below, we have the graphical representation of the Daubechies wavelet functions *ϕ* and *ψ* of kind 4.

The functions *ϕ* in (7) and *ψ* in (8) are called the scale function *ϕ* and wavelet function *ψ*, respectively. The fundamental support of the scale function<sup>3</sup> is the interval 0, ½ � *N* � 1 as the fundamental support of wavelet function *ψ*ð Þ *x* is the interval 1 � *<sup>N</sup>* <sup>2</sup> , *<sup>N</sup>* 2 � �. In the case of *<sup>N</sup>* <sup>¼</sup> 4, we have the graphs of the **Figures 1** and **<sup>2</sup>**.

**Figure 1.**

*Daubechies wavelets ϕ. Source: This figure was generated by the author using the python programming language.*

<sup>3</sup> We emphasize that the scale function has energy concentrated in its support that is determined by the genus of the wavelet, that is, *supp*ð Þ¼ *ϕ* ½ � 0, *N* � 1 , and that the total energy of the scale function is unitary, that is, Ð <sup>þ</sup><sup>∞</sup> �<sup>∞</sup> *<sup>ϕ</sup>dx* <sup>¼</sup> 1.

*Use of Daubechies Wavelets in the Representation of Analytical Functions DOI: http://dx.doi.org/10.5772/intechopen.93885*

**Figure 2.** *Daubechies wavelet ψ. Source: This figure was generated by the author using the python programming language.*

To determine the filter coefficients *ak* and *a*<sup>1</sup>�*<sup>k</sup>*, which appear in the (7) and (8), we use the relations (9)–(12) below.

$$\sum\_{k=0}^{N-1} a\_k = 2 \tag{9}$$

$$\sum\_{k=0}^{N-1} a\_k a\_{k-m} = \delta\_{0,m} \tag{10}$$

$$\sum\_{k=0}^{N-1} (-1)^k a\_{1-k} a\_{k-2m} = 0 \tag{11}$$

$$\sum\_{k=0}^{N-1} (-1)^k k^m a\_k = 0, \quad m = 0.1, \ldots, \frac{N}{2} - 1,\tag{12}$$

where *δ<sup>k</sup>*,*<sup>m</sup>* is the Kronecker Delta function.

#### **3. Generating an analytical function of the type** *x<sup>k</sup>* **using wavelets**

Analytical functions are those that can be locally around a point *x*<sup>0</sup> expanded in a Taylor series, according to the following expression.

$$f(\mathbf{x}) = \sum\_{n=0}^{+\infty} \frac{f^{(n)}(\mathbf{x}\_0)}{n!} (\mathbf{x} - \mathbf{x}\_0)^n \tag{13}$$

In general according to the author [8], any *f x*ð Þ function can be represented in terms of a wavelet base, as follows:

$$f(\mathbf{x}) = \sum\_{m = -\infty}^{+\infty} c\_k \phi(\mathbf{x} - m) = \sum\_{m = -\infty}^{+\infty} c\_m \phi\_m(\mathbf{x}). \tag{14}$$

In the following subsection, among the most varied types of wavelets, the Daubechies wavelets are highlighted, which are the basis for the development of

At 1988, a family of compact support wavelets [7] is built by Ingrid Daubechies. This family of wavelets has highly well-located elements. Each member wavelet is governed by a set of *N* integer coefficients and *k* ¼ f g 0*:*1, …, *N* � 1 coefficients through scale relations (7) and (8). The *ak* and *a*1�*<sup>k</sup>* coefficients, which appear in the

*akϕ*ð Þ 2*x* � *k* (7)

*a*<sup>1</sup>�*<sup>k</sup>ϕ*ð Þ 2*x* � *k* (8)

(7) and (8), are called filter coefficients and verify the following relations:

*<sup>ϕ</sup>*ð Þ¼ *<sup>x</sup>* <sup>X</sup> *N*�1

1

*k*¼2�*N*

*<sup>ψ</sup>*ð Þ¼ *<sup>x</sup>* <sup>X</sup>

Daubechies wavelet functions *ϕ* and *ψ* of kind 4.

*k*¼0

ð Þ �<sup>1</sup> *<sup>k</sup>*

In the **Figures 1** and **2** below, we have the graphical representation of the

The functions *ϕ* in (7) and *ψ* in (8) are called the scale function *ϕ* and wavelet function *ψ*, respectively. The fundamental support of the scale function<sup>3</sup> is the interval 0, ½ � *N* � 1 as the fundamental support of wavelet function *ψ*ð Þ *x* is the

<sup>3</sup> We emphasize that the scale function has energy concentrated in its support that is determined by the genus of the wavelet, that is, *supp*ð Þ¼ *ϕ* ½ � 0, *N* � 1 , and that the total energy of the scale function is

*Daubechies wavelets ϕ. Source: This figure was generated by the author using the python programming*

� �. In the case of *<sup>N</sup>* <sup>¼</sup> 4, we have the graphs of the **Figures 1** and **<sup>2</sup>**.

this chapter.

*Wavelet Theory*

interval 1 � *<sup>N</sup>*

unitary, that is, Ð <sup>þ</sup><sup>∞</sup>

**Figure 1.**

*language.*

**336**

�<sup>∞</sup> *<sup>ϕ</sup>dx* <sup>¼</sup> 1.

<sup>2</sup> , *<sup>N</sup>* 2

**2.3 Daubechies wavelet properties**

The *ck* coefficients are called moments of the scale functions. In particular, for *f x*ð Þ¼ *xk*, we have the expression (15), below:

$$\boldsymbol{\infty}^{k} = \sum\_{m=-\infty}^{+\infty} \frac{M\_{m}^{k}}{\mathfrak{d}^{\boldsymbol{k}}} \phi \left( 2^{\boldsymbol{j}} \boldsymbol{\infty} - \boldsymbol{m} \right), \tag{15}$$

Substituting the Eq. (21) in (20), we have:

*DOI: http://dx.doi.org/10.5772/intechopen.93885*

of the wavelet respectively.

ical expression is used

**4. Moment generating function**

For *m* ¼ *k* ¼ 0, in (23), we have:

*M<sup>k</sup>* <sup>0</sup> ¼

Making the substitution *<sup>z</sup>* <sup>¼</sup> <sup>2</sup>*x*, *dz*

Using the substitution *<sup>x</sup>* � *<sup>m</sup>* <sup>¼</sup> *<sup>t</sup>*, *dx*

*M<sup>k</sup> <sup>m</sup>* ¼

**339**

*p x*ð Þ¼ <sup>X</sup>*<sup>n</sup>*

*Use of Daubechies Wavelets in the Representation of Analytical Functions*

*k*¼0

*ak* <sup>2</sup>*jk* <sup>X</sup> þ∞

*m*¼�∞

In the next subsection, the calculation of the moment generating function, which appears in the expression 21 as a coefficient of *xk*, is shown in detail.

where *k* is the degree of the polynomial *j* and *m* are the resolution and translation

The calculation of the moment generating function according to the author [11]

which refers to the moment of the wavelet scale *ϕ* in relation to the monomial *xk*.

ðþ<sup>∞</sup> �∞

*dx* <sup>¼</sup> 2, *dx* <sup>¼</sup> *dz*

X *N*�1

*asM<sup>k</sup>*

*k*

*r*¼0

*s*¼0

*<sup>ϕ</sup>*ð Þ *<sup>x</sup>* � *<sup>m</sup> dx* <sup>¼</sup> <sup>X</sup>

*xk*X *N*�1

*s*¼0

<sup>2</sup> , we have:

*dt* ¼ 1, *dx* ¼ *dt*, in (23), we have:

*k r* � �*mk*�*<sup>r</sup>*

is of fundamental importance to approximate the functions by wavelets. The deduction of the moment-generating function now begins. For this, the mathemat-

> ðþ<sup>∞</sup> �∞

> > ðþ<sup>∞</sup> �∞

*M<sup>k</sup> <sup>m</sup>* ¼

> *M*<sup>0</sup> <sup>0</sup> ¼

*ϕ*ð Þ *x dx* ¼

*s*¼0 *as* ðþ<sup>∞</sup> �∞ *xk*

*s*¼0 *as* ðþ<sup>∞</sup> �∞ *xk*

*Mk* <sup>0</sup> <sup>¼</sup> <sup>1</sup> 2*<sup>k</sup>*þ<sup>1</sup>

Note that the variable *s*, in the Eq. (26), also represents a translation.

Substituting *m* ¼ 0 in the Eq. (23), we have:

ðþ<sup>∞</sup> �∞ *xk*

> *M<sup>k</sup>* <sup>0</sup> <sup>¼</sup> <sup>X</sup> *N*�1

*M<sup>k</sup>* <sup>0</sup> <sup>¼</sup> <sup>X</sup> *N*�1

ðþ<sup>∞</sup> �∞ *xk* *ϕ* 2 *<sup>j</sup>*

*<sup>x</sup>* � *<sup>m</sup>* � �*M<sup>k</sup>*

*<sup>m</sup>* (22)

*xkϕ*ð Þ *<sup>x</sup>* � *<sup>m</sup> dx* (23)

*ϕ*ð Þ *x dx* ¼ 1*:* (24)

*asϕ*ð Þ 2*x* � *s dx* (25)

*ϕ*ð Þ 2*x* � *s dx:* (26)

*ϕ*ð Þ 2*x* � *s dx* (27)

*M<sup>r</sup>*

*<sup>s</sup>* (28)

<sup>0</sup> (29)

Since *Mk <sup>m</sup>* the moment of the wavelet scales concerning the *xk* monomial, where *k* is the degree of the polynomial, *m* and *j* are the translation and resolution of the *ϕ* wavelet. The justification for the construction of the equation is found in the work of [8–10], in which the author concludes that the *c<sup>j</sup> <sup>m</sup>* coefficients for approximating a monomial of the *x<sup>k</sup>* form, using a Daubechies wavelet base *ϕ*, looks like this:

$$
\sigma\_m^j = \frac{M\_m^k}{2^{jk}} \tag{16}
$$

The justification used in the approximation (15) of a polynomial function of type *f x*ð Þ¼ *xk* derives from the number of null moments,

$$\int\_{-\infty}^{+\infty} \mathbf{x}^k \boldsymbol{\nu}(\mathbf{x}) d\mathbf{x} = \mathbf{0}, \quad k = 0.1, \ldots, \frac{N}{2} - \mathbf{1} \tag{17}$$

According to the Eq. (17), the *N* Daubechies Wavelet has *<sup>N</sup>* <sup>2</sup> vanish moments, being possible to represent a polynomial of degree at most *<sup>N</sup>* <sup>2</sup> � 1, using the *ϕ*ð Þ *x* scale function. The polynomial approximation using the scale function is formalized in the following definition.

Definition 3.1 A wavelet has *p* vanish moments (18), if and only if, the wavelet scale function *ϕ* can generate polynomials of degree up to *p* � 1 [Eq. (19)]. That is, the scale function alone can be used to represent these polynomials. The fact that it has more null moments means that the scale function can represent more complex functions.

$$\int\_{-\infty}^{+\infty} x^m \psi(x) dx = 0; \quad m = 0, 1, \dots, \frac{N}{2} - 1 \tag{18}$$

$$f(\mathbf{x}) = p\_1 + p\_2 \mathbf{x} + \dots + p\_{k-1} \mathbf{x}^k, \quad k \le \frac{N}{2} - 1 \tag{19}$$

In general, a Daubechies wavelet of kind *N*, properly translated and adjusted to the appropriate resolution level, generates a polynomial of degree *k*, with the relation between *N* and *k* given by *N* ¼ 2*k* þ 2. For example, to generate a polynomial of degree 1 a wavelet of Daubechies of kind 4 is necessary.

To generate a polynomial with *n* þ 1 terms, in the function of Daubechies wavelets of genres 4, 6, 8, …, *N* � 1, we use the momentum equation and the polynomial expansion as a function of wavelets.

$$p(\mathbf{x}) = \sum\_{k=0}^{n} a\_k \mathbf{x}^k,\tag{20}$$

where *xk*, takes the form

$$\mathbf{x}^{k} = \sum\_{m = -\infty}^{+\infty} \frac{M\_{m}^{k}}{2^{jk}} \phi \left( 2^{j} \mathbf{x} - m \right) \tag{21}$$

*Use of Daubechies Wavelets in the Representation of Analytical Functions DOI: http://dx.doi.org/10.5772/intechopen.93885*

Substituting the Eq. (21) in (20), we have:

The *ck* coefficients are called moments of the scale functions. In particular, for

*<sup>m</sup>* the moment of the wavelet scales concerning the *xk* monomial, where *k*

*<sup>x</sup>* � *<sup>m</sup>* � �, (15)

*<sup>m</sup>* coefficients for approximating a

<sup>2</sup>*jk* (16)

<sup>2</sup> � <sup>1</sup> (17)

<sup>2</sup> vanish moments,

<sup>2</sup> � 1, using the *ϕ*ð Þ *x*

<sup>2</sup> � <sup>1</sup> (18)

<sup>2</sup> � <sup>1</sup> (19)

, (20)

*<sup>x</sup>* � *<sup>m</sup>* � � (21)

*N*

*N*

*N*

, *k*≤

*M<sup>k</sup> m* <sup>2</sup>*jk <sup>ϕ</sup>* <sup>2</sup> *<sup>j</sup>*

is the degree of the polynomial, *m* and *j* are the translation and resolution of the *ϕ* wavelet. The justification for the construction of the equation is found in the work of

monomial of the *x<sup>k</sup>* form, using a Daubechies wavelet base *ϕ*, looks like this:

*cj <sup>m</sup>* <sup>¼</sup> *<sup>M</sup><sup>k</sup> m*

The justification used in the approximation (15) of a polynomial function of type

*ψ*ð Þ *x dx* ¼ 0, *k* ¼ 0*:*1, …,

scale function. The polynomial approximation using the scale function is formalized

Definition 3.1 A wavelet has *p* vanish moments (18), if and only if, the wavelet scale function *ϕ* can generate polynomials of degree up to *p* � 1 [Eq. (19)]. That is, the scale function alone can be used to represent these polynomials. The fact that it has more null moments means that the scale function can represent more

*xmψ*ð Þ *<sup>x</sup> dx* <sup>¼</sup> 0; *<sup>m</sup>* <sup>¼</sup> 0, 1, …,

the appropriate resolution level, generates a polynomial of degree *k*, with the relation between *N* and *k* given by *N* ¼ 2*k* þ 2. For example, to generate a polynomial of degree 1 a wavelet of Daubechies of kind 4 is necessary.

To generate a polynomial with *n* þ 1 terms, in the function of Daubechies wavelets of genres 4, 6, 8, …, *N* � 1, we use the momentum equation and the

*p x*ð Þ¼ <sup>X</sup>*<sup>n</sup>*

*<sup>x</sup><sup>k</sup>* <sup>¼</sup> <sup>X</sup> þ∞

*m*¼�∞

*k*¼0

*M<sup>k</sup> m* <sup>2</sup>*jk <sup>ϕ</sup>* <sup>2</sup> *<sup>j</sup>*

*akx<sup>k</sup>*

In general, a Daubechies wavelet of kind *N*, properly translated and adjusted to

*f x*ð Þ¼ *<sup>p</sup>*<sup>1</sup> <sup>þ</sup> *<sup>p</sup>*2*<sup>x</sup>* <sup>þ</sup> … <sup>þ</sup> *pk*�<sup>1</sup>*xk*

*f x*ð Þ¼ *xk*, we have the expression (15), below:

[8–10], in which the author concludes that the *c<sup>j</sup>*

*f x*ð Þ¼ *xk* derives from the number of null moments,

According to the Eq. (17), the *N* Daubechies Wavelet has *<sup>N</sup>*

being possible to represent a polynomial of degree at most *<sup>N</sup>*

ðþ<sup>∞</sup> �∞ *xk*

ðþ<sup>∞</sup> �∞

polynomial expansion as a function of wavelets.

where *xk*, takes the form

**338**

in the following definition.

complex functions.

Since *Mk*

*Wavelet Theory*

*xk* <sup>¼</sup> <sup>X</sup> þ∞

*m*¼�∞

$$p(\mathbf{x}) = \sum\_{k=0}^{n} \frac{a\_k}{2^k} \sum\_{m=-\infty}^{+\infty} \phi\left(2^j \mathbf{x} - m\right) \mathcal{M}\_m^k \tag{22}$$

where *k* is the degree of the polynomial *j* and *m* are the resolution and translation of the wavelet respectively.

In the next subsection, the calculation of the moment generating function, which appears in the expression 21 as a coefficient of *xk*, is shown in detail.

#### **4. Moment generating function**

The calculation of the moment generating function according to the author [11] is of fundamental importance to approximate the functions by wavelets. The deduction of the moment-generating function now begins. For this, the mathematical expression is used

$$\mathcal{M}\_m^k = \int\_{-\infty}^{+\infty} \varkappa^k \phi(\varkappa - m) d\varkappa \tag{23}$$

which refers to the moment of the wavelet scale *ϕ* in relation to the monomial *xk*. For *m* ¼ *k* ¼ 0, in (23), we have:

$$M\_0^0 = \int\_{-\infty}^{+\infty} \phi(\mathbf{x}) d\mathbf{x} = \mathbf{1}.\tag{24}$$

Substituting *m* ¼ 0 in the Eq. (23), we have:

$$\mathcal{M}\_0^k = \int\_{-\infty}^{+\infty} \varkappa^k \phi(\varkappa) d\varkappa = \int\_{-\infty}^{+\infty} \varkappa^k \sum\_{\varkappa=0}^{N-1} a\_\varkappa \phi(2\varkappa - s) d\varkappa \tag{25}$$

$$M\_0^k = \sum\_{s=0}^{N-1} a\_s \int\_{-\infty}^{+\infty} \varkappa^k \phi(2\varkappa - s) d\varkappa. \tag{26}$$

Note that the variable *s*, in the Eq. (26), also represents a translation. Making the substitution *<sup>z</sup>* <sup>¼</sup> <sup>2</sup>*x*, *dz dx* <sup>¼</sup> 2, *dx* <sup>¼</sup> *dz* <sup>2</sup> , we have:

$$M\_0^k = \sum\_{s=0}^{N-1} a\_s \int\_{-\infty}^{+\infty} \varkappa^k \phi(2\varkappa - s) d\varkappa \tag{27}$$

$$\mathcal{M}\_0^k = \frac{1}{2^{k+1}} \sum\_{s=0}^{N-1} a\_s \mathcal{M}\_s^k \tag{28}$$

Using the substitution *<sup>x</sup>* � *<sup>m</sup>* <sup>¼</sup> *<sup>t</sup>*, *dx dt* ¼ 1, *dx* ¼ *dt*, in (23), we have:

$$M\_m^k = \int\_{-\infty}^{+\infty} x^k \phi(\infty - m) dx = \sum\_{r=0}^k \binom{k}{r} m^{k-r} M\_0^r \tag{29}$$

Now consider the equations:

$$\mathcal{M}\_0^k = \frac{1}{2^{k+1}} \sum\_{s=0}^{N-1} a\_s \mathcal{M}\_s^k \tag{30}$$

Similar to what was done with the calculation of the moments for the function *ϕ*,

*N*

<sup>2</sup> � <sup>1</sup>*:* (38)

*M*<sup>1</sup> 0

<sup>2</sup> *<sup>j</sup> <sup>ϕ</sup>*ð Þ *<sup>x</sup>* (39)

(40)

there is also the calculation of the moments for the function *ψ*. This is given by

*Use of Daubechies Wavelets in the Representation of Analytical Functions*

*<sup>x</sup>mψ*ð Þ *<sup>x</sup> dx* <sup>¼</sup> 0; *<sup>m</sup>* <sup>¼</sup> 0, 1, …,

Example 4.1 In this example, the Daubechies wavelet of kind 4 is used to generate the analytical polynomial function *f x*ð Þ¼ *x*. According to the definition 3.1, the scale function of Daubechies of genus *N* ¼ 4, generates a line (polynomial of degree 1). To represent a 1 monomial with a 4 Daubechies wavelet in the 0, 1 ½ �

range, the translations *ϕ*ð Þ *x* , *ϕ*ð Þ *x* þ 1 , *ϕ*ð Þ *x* þ 2 , whose supports are

respectively, that form a base to generate the function *f x*ð Þ¼ *x*.

Eqs. (9)–(12), which gives rise to the following non-linear system.

<sup>0</sup> <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

�2

*<sup>x</sup>* � *<sup>m</sup>* � � <sup>¼</sup> *<sup>M</sup>*<sup>1</sup>

The following is an example of the calculation of the moments for the case of

<sup>2</sup> *<sup>j</sup> <sup>ϕ</sup>*ð Þþ *<sup>x</sup>* <sup>þ</sup> <sup>2</sup> *<sup>M</sup>*<sup>1</sup>

The support of the linear combination (39), represented in **Figure 3**, is obtained

by the intersection of the supports of the translations of the function *ϕ*ð Þ *x* . This intersection results in the interval *I* ¼ ½ � 0*:*1 . This fact defines well the function to be integrated in the *I* range. In **Figure 3**, the number of translations of the function

**Figure 4** shows the graph of translated functions *ϕ*ð Þ *x* , *ϕ*ð Þ *x* þ 1 and *ϕ*ð Þ *x* þ 2

*a*<sup>0</sup> þ *a*<sup>1</sup> þ *a*<sup>2</sup> þ *a*<sup>3</sup> ¼ 2

*a*0*a*<sup>2</sup> þ *a*1*a*<sup>3</sup> ¼ 0

�*a*<sup>1</sup> þ 2*a*<sup>2</sup> � 3*a*<sup>3</sup> ¼ 0

*Translations required to represent the analytical function f x*ð Þ¼ *x using Daubechies wavelets of kind* 4*. Source:*

<sup>2</sup> <sup>þ</sup> *<sup>a</sup>*<sup>2</sup>

<sup>3</sup> ¼ 2

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

The calculation using the moment generating function depends on the Daubechies wavelet coefficients of kind 4. These coefficients are obtained by the

�1

<sup>2</sup> *<sup>j</sup> <sup>ϕ</sup>*ð Þþ *<sup>x</sup>* <sup>þ</sup> <sup>1</sup>

ðþ<sup>∞</sup> �∞

*DOI: http://dx.doi.org/10.5772/intechopen.93885*

Daubechies wavelets of a kind *N* ¼ 4.

½ � 0*:*3 , ½ � �1*:*2 , ½ � �2*:*1 , that is:

*M*<sup>1</sup> *m* <sup>2</sup>*jk <sup>ϕ</sup>* <sup>2</sup> *<sup>j</sup>*

*ϕ*ð Þ *x* to generate *f x*ð Þ¼ *x* is illustrated.

*a*2

8 >>>>>>><

>>>>>>>:

*<sup>x</sup>* <sup>¼</sup> <sup>X</sup> 0

**Figure 3.**

**341**

*Own authorship.*

*m*¼�2

integral (38)

$$\mathcal{M}\_s^k = \sum\_{r=0}^k \binom{k}{r} s^{k-r} \mathcal{M}\_0^r \tag{31}$$

Substituting *M<sup>k</sup> <sup>s</sup>* in *M<sup>k</sup>* 0, we have: (note that *m* ¼ *s*)

$$\mathbf{M}\_0^k = \frac{1}{2^{k+1}} \sum\_{s=0}^{N-1} a\_s \sum\_{r=0}^k \binom{k}{r} s^{k-r} \mathbf{M}\_0^r \tag{32}$$

Now separate the last term of the sum (32), ð Þ *r* ¼ *k* , to place the term on the left side of the equation:

$$\boldsymbol{M}\_{0}^{k} = \frac{1}{2^{k+1}} \sum\_{s=0}^{N-1} \boldsymbol{a}\_{s} \sum\_{r=0}^{k-1} \binom{k}{r} \boldsymbol{s}^{k-r} \boldsymbol{M}\_{0}^{r} + \frac{1}{2^{k+1}} \sum\_{s=0}^{N-1} \boldsymbol{a}\_{s} \binom{k}{r} \boldsymbol{s}^{k-k} \boldsymbol{M}\_{0}^{k} \tag{33}$$

Using the fact that *N* P�1 *s*¼0 *as* ¼ 2, we have:

$$M\_0^k = \frac{1}{2(2^k - 1)} \sum\_{r=0}^{k-1} \binom{k}{r} M\_0^r \sum\_{s=0}^{N-1} a\_s s^{k-r} \tag{34}$$

Thus, the equations are obtained:

$$M\_m^k = \sum\_{r=0}^k \binom{k}{r} m^{k-r} M\_0^r \tag{35}$$

$$\mathbf{M}\_0^k = \frac{1}{2\left(2^k - 1\right)} \sum\_{r=0}^{k-1} \binom{k}{r} \sum\_{s=0}^{N-1} a\_s s^{k-r} \mathbf{M}\_0^r \tag{36}$$

From (35), (34), and (24), we get the moment generating function *M<sup>k</sup> <sup>m</sup>* : W ! R, where W is wavelet space, *m* is the translation of the scale function and *k* is the degree of the polynomial to be approximated.

$$\mathcal{M}\_m^k = \begin{cases} \frac{1}{2(2^k - 1)} \sum\_{r=0}^{k-1} \binom{k}{r} \sum\_{s=0}^{N-1} a\_s s^{k-r} \mathcal{M}\_0^r, & \text{se} \quad m = 0; k \neq 0 \\\\ \sum\_{r=0}^k \binom{k}{r} m^{k-r} \mathcal{M}\_0^r, & \text{se} \quad m \neq 0; k \neq 0 \\\\ 1, & \text{se} \quad m = k = 0, \end{cases} \tag{37}$$

The analytical expression for *M<sup>k</sup> <sup>m</sup>* was developed during the author's research [11] and to validate the results found, a comparative study was made with other numerical results [12, 13] of the scientific literature.

*Use of Daubechies Wavelets in the Representation of Analytical Functions DOI: http://dx.doi.org/10.5772/intechopen.93885*

Similar to what was done with the calculation of the moments for the function *ϕ*, there is also the calculation of the moments for the function *ψ*. This is given by integral (38)

$$\int\_{-\infty}^{+\infty} \mathfrak{x}^{m} \varphi(\mathfrak{x}) d\mathfrak{x} = 0; \quad m = 0, 1, \ldots, \frac{N}{2} - 1. \tag{38}$$

The following is an example of the calculation of the moments for the case of Daubechies wavelets of a kind *N* ¼ 4.

Example 4.1 In this example, the Daubechies wavelet of kind 4 is used to generate the analytical polynomial function *f x*ð Þ¼ *x*. According to the definition 3.1, the scale function of Daubechies of genus *N* ¼ 4, generates a line (polynomial of degree 1). To represent a 1 monomial with a 4 Daubechies wavelet in the 0, 1 ½ � range, the translations *ϕ*ð Þ *x* , *ϕ*ð Þ *x* þ 1 , *ϕ*ð Þ *x* þ 2 , whose supports are ½ � 0*:*3 , ½ � �1*:*2 , ½ � �2*:*1 , that is:

$$\alpha = \sum\_{m=-2}^{0} \frac{M\_m^1}{2^{\circ k}} \phi(2^j x - m) = \frac{M\_{-2}^1}{2^j} \phi(x+2) + \frac{M\_{-1}^1}{2^j} \phi(x+1) + \frac{M\_0^1}{2^j} \phi(x) \tag{39}$$

The support of the linear combination (39), represented in **Figure 3**, is obtained by the intersection of the supports of the translations of the function *ϕ*ð Þ *x* . This intersection results in the interval *I* ¼ ½ � 0*:*1 . This fact defines well the function to be integrated in the *I* range. In **Figure 3**, the number of translations of the function *ϕ*ð Þ *x* to generate *f x*ð Þ¼ *x* is illustrated.

**Figure 4** shows the graph of translated functions *ϕ*ð Þ *x* , *ϕ*ð Þ *x* þ 1 and *ϕ*ð Þ *x* þ 2 respectively, that form a base to generate the function *f x*ð Þ¼ *x*.

The calculation using the moment generating function depends on the Daubechies wavelet coefficients of kind 4. These coefficients are obtained by the Eqs. (9)–(12), which gives rise to the following non-linear system.

**Figure 3.**

*Translations required to represent the analytical function f x*ð Þ¼ *x using Daubechies wavelets of kind* 4*. Source: Own authorship.*

Now consider the equations:

*<sup>s</sup>* in *M<sup>k</sup>*

X *N*�1

*s*¼0 *as* X *k*�1

> *N* P�1 *s*¼0

> > *M<sup>k</sup>*

*M<sup>k</sup>*

1 2 2*<sup>k</sup>* � <sup>1</sup> � �<sup>X</sup>

*k*

1 <sup>A</sup>*m<sup>k</sup>*�*<sup>r</sup>*

numerical results [12, 13] of the scientific literature.

0 @

The analytical expression for *M<sup>k</sup>*

*r*

P *k r*¼0

8

>>>>>>>>>>>><

>>>>>>>>>>>>:

Thus, the equations are obtained:

*M<sup>k</sup>* <sup>0</sup> <sup>¼</sup> <sup>1</sup> 2*k*þ<sup>1</sup>

*r*¼0

<sup>0</sup> <sup>¼</sup> <sup>1</sup>

*M<sup>k</sup> <sup>m</sup>* <sup>¼</sup> <sup>X</sup> *k*

<sup>0</sup> <sup>¼</sup> <sup>1</sup>

and *k* is the degree of the polynomial to be approximated.

*k*

1 A<sup>X</sup> *N*�1

0 @

*r*

*Mr*

*k*�1

*r*¼0

*k r* � � *s k*�*r M<sup>r</sup>* <sup>0</sup> þ

*as* ¼ 2, we have:

2 2*<sup>k</sup>* � <sup>1</sup> � �<sup>X</sup>

*k*�1

*k r* � � *M<sup>r</sup>* 0 X *N*�1

*r*¼0

*k r* � �

> *k r* � �X *N*�1

*mk*�*<sup>r</sup> M<sup>r</sup>*

> *s*¼0 *ass k*�*r M<sup>r</sup>*

0, se *m* 6¼ 0; *k* 6¼ 0

0, se *m* ¼ 0; *k* 6¼ 0

*<sup>m</sup>* was developed during the author's research

*r*¼0

*k*�1

*r*¼0

*<sup>m</sup>* : W ! R, where W is wavelet space, *m* is the translation of the scale function

1, se *m* ¼ *k* ¼ 0,

[11] and to validate the results found, a comparative study was made with other

2 2*<sup>k</sup>* � <sup>1</sup> � �<sup>X</sup>

From (35), (34), and (24), we get the moment generating function

*s*¼0 *ass k*�*r M<sup>r</sup>*

Substituting *M<sup>k</sup>*

*Wavelet Theory*

side of the equation:

*Mk* <sup>0</sup> <sup>¼</sup> <sup>1</sup> 2*<sup>k</sup>*þ<sup>1</sup>

Using the fact that

*M<sup>k</sup>*

**340**

*M<sup>k</sup> <sup>m</sup>* ¼ *Mk* <sup>0</sup> <sup>¼</sup> <sup>1</sup> 2*k*þ<sup>1</sup>

*M<sup>k</sup> <sup>s</sup>* <sup>¼</sup> <sup>X</sup> *k*

*r*¼0

0, we have: (note that *m* ¼ *s*)

X *N*�1

*s*¼0 *as* X *k*

X *N*�1

*asM<sup>k</sup>*

*k r* � � *s k*�*r M<sup>r</sup>*

*<sup>s</sup>* (30)

<sup>0</sup> (31)

<sup>0</sup> (32)

*<sup>k</sup>*�*<sup>r</sup>* (34)

<sup>0</sup> (35)

<sup>0</sup> (36)

(37)

<sup>0</sup> (33)

*s*¼0

*k r* � � *s k*�*r M<sup>r</sup>*

*r*¼0

Now separate the last term of the sum (32), ð Þ *r* ¼ *k* , to place the term on the left

1 2*<sup>k</sup>*þ<sup>1</sup> X *N*�1

*s*¼0 *as k r* � � *s k*�*k M<sup>k</sup>*

> *s*¼0 *ass*

#### **Figure 4.**

*Translations required to represent the analytical function f x*ð Þ¼ *x using Daubechies wavelets of kind* 4*. Source: This figure was generated by the author using the python programming language.*

The solution of this system is the irrational numbers *a*0, *a*1, *a*2, *a*3, given by:

$$\begin{cases} a\_0 = 0, 683012701892219 \\ a\_1 = 1.183012701892219 \\ a\_2 = 0, 316987298107781 \\ a\_3 = -0, 183012701892219 \end{cases} \tag{41}$$

So, the representation for the *x* polynomial (for a resolution *j* ¼ 0) is:

wavelets *ϕ*ð Þ *x* , *ϕ*ð Þ *x* þ 1 and *ϕ*ð Þ *x* þ 2 .

*DOI: http://dx.doi.org/10.5772/intechopen.93885*

Daubechies wavelets is given by

**Figure 5.**

**343**

*the python programming language.*

*x* ¼ 0, 634*ϕ*ð Þ� *x* 0, 366*ϕ*ð Þ� *x* þ 1 1*:*366*ϕ*ð Þ *x* þ 2 (46)

*x* � *m*<sup>1</sup>

*x* � *m*<sup>2</sup>

*x* � *m*<sup>3</sup>

� � (47)

� � (48)

� � (49)

In **Figure 5**, we have the graphical representation of the function obtained of the expression (46). Here the function *f x*ð Þ¼ *x* is generated by linear combination of

The representation for the expression (46) using the summation is given by,

The expression for writing polynomials of degrees *k* ¼ 2 and *k* ¼ 3 in terms of

*M*<sup>2</sup> *m*<sup>2</sup> <sup>2</sup>*jk <sup>ϕ</sup>* <sup>2</sup> *<sup>j</sup>*

*M*<sup>3</sup> *m*<sup>3</sup> <sup>2</sup>*jk <sup>ϕ</sup>* <sup>2</sup> *<sup>j</sup>*

See that to generate the polynomials (48), (49) is necessary to use Daubechies

*Function f x*ð Þ¼ *x using Daubechies wavelets of kind* 4*. Source: This figure was generated by the author using*

*M*<sup>1</sup> *m*<sup>1</sup> <sup>2</sup>*jk <sup>ϕ</sup>* <sup>2</sup> *<sup>j</sup>*

*<sup>x</sup>* <sup>¼</sup> <sup>X</sup> 0

*Use of Daubechies Wavelets in the Representation of Analytical Functions*

*<sup>x</sup>*<sup>2</sup> <sup>¼</sup> <sup>X</sup> 2

*<sup>x</sup>*<sup>3</sup> <sup>¼</sup> <sup>X</sup> 0

wavelets of kind 6 and 8, according with the definition 3.1.

*m*2¼�4

*m*3¼�6

*m*1¼�2

Using the moment generating function for the case where *m* ¼ 0, we have:

$$\begin{split} M\_0^1 &= \frac{1}{2} \sum\_{r=0}^0 \binom{1}{r} M\_0^r \sum\_{s=0}^3 a\_s s^{1-r} = \frac{1}{2} M\_0^0 (a\_1 + 2a\_2 + 3a\_3) = \frac{a\_1 + 2a\_2 + 3a\_3}{2} \\ &= 0,633974600 \end{split} \tag{42}$$

Proceeding with the calculations, we obtain:

$$M\_m^k = \sum\_{r=0}^1 \binom{1}{r} m^{1-r} M\_0^r = m + M\_0^1 = m + \frac{a\_1 + 2a\_2 + 3a\_3}{2} = m + 0,633974600 \tag{43}$$

Replacing the value of *m* by *m* ¼ �1, *m* ¼ �2 and *k* ¼ 1, we obtain:

$$M\_{-1}^1 = -0, \mathbf{3} \mathbf{6} \mathbf{6} \mathbf{0} \mathbf{2} \mathbf{5} \mathbf{4} \mathbf{0} \mathbf{0} \tag{44}$$

$$M\_{-2}^1 = -1,366025400\tag{45}$$

*Use of Daubechies Wavelets in the Representation of Analytical Functions DOI: http://dx.doi.org/10.5772/intechopen.93885*

So, the representation for the *x* polynomial (for a resolution *j* ¼ 0) is:

$$\text{l.x} = \text{0, } \mathfrak{G}\mathfrak{Z}4\phi(\mathfrak{x}) - \text{0, } \mathfrak{G}\mathfrak{G}\phi(\mathfrak{x} + \mathfrak{1}) - \text{1.} \mathfrak{G}\mathfrak{G}\phi(\mathfrak{x} + \mathfrak{2}) \tag{46}$$

In **Figure 5**, we have the graphical representation of the function obtained of the expression (46). Here the function *f x*ð Þ¼ *x* is generated by linear combination of wavelets *ϕ*ð Þ *x* , *ϕ*ð Þ *x* þ 1 and *ϕ*ð Þ *x* þ 2 .

The representation for the expression (46) using the summation is given by,

$$\infty = \sum\_{m\_1=-2}^{0} \frac{M\_{m\_1}^1}{\mathfrak{2}^{jk}} \phi \big( 2^j \mathfrak{x} - m\_1 \big) \tag{47}$$

The expression for writing polynomials of degrees *k* ¼ 2 and *k* ¼ 3 in terms of Daubechies wavelets is given by

$$\mathbf{x}^2 = \sum\_{m\_2=-4}^2 \frac{M\_{m\_2}^2}{2^{jk}} \phi \left(2^j \mathbf{x} - m\_2\right) \tag{48}$$

$$\mathbf{x}^3 = \sum\_{m\_3=-6}^{0} \frac{M\_{m\_3}^3}{2^{jk}} \phi \left(2^j \mathbf{x} - m\_3\right) \tag{49}$$

See that to generate the polynomials (48), (49) is necessary to use Daubechies wavelets of kind 6 and 8, according with the definition 3.1.

#### **Figure 5.**

The solution of this system is the irrational numbers *a*0, *a*1, *a*2, *a*3, given by:

*Translations required to represent the analytical function f x*ð Þ¼ *x using Daubechies wavelets of kind* 4*. Source:*

Using the moment generating function for the case where *m* ¼ 0, we have:

<sup>0</sup> ¼ *m* þ

Replacing the value of *m* by *m* ¼ �1, *m* ¼ �2 and *k* ¼ 1, we obtain:

¼ 0, 633974600 (42)

<sup>0</sup>ð*a*<sup>1</sup> <sup>þ</sup> <sup>2</sup>*a*<sup>2</sup> <sup>þ</sup> <sup>3</sup>*a*3Þ ¼ *<sup>a</sup>*<sup>1</sup> <sup>þ</sup> <sup>2</sup>*a*<sup>2</sup> <sup>þ</sup> <sup>3</sup>*a*<sup>3</sup>

�<sup>1</sup> ¼ �0, 366025400 (44)

�<sup>2</sup> ¼ �1, 366025400 (45)

*a*<sup>1</sup> þ 2*a*<sup>2</sup> þ 3*a*<sup>3</sup>

2

<sup>2</sup> <sup>¼</sup> *<sup>m</sup>* <sup>þ</sup> 0, 633974600

8 >>><

*This figure was generated by the author using the python programming language.*

>>>:

<sup>1</sup>�*<sup>r</sup>* <sup>¼</sup> <sup>1</sup> 2 *M*<sup>0</sup>

<sup>0</sup> <sup>¼</sup> *<sup>m</sup>* <sup>þ</sup> *<sup>M</sup>*<sup>1</sup>

*M*<sup>1</sup>

*M*<sup>1</sup>

*s*¼0 *ass*

Proceeding with the calculations, we obtain:

*M*<sup>1</sup> <sup>0</sup> <sup>¼</sup> <sup>1</sup> 2 X 0

**Figure 4.**

*Wavelet Theory*

*M<sup>k</sup> <sup>m</sup>* <sup>¼</sup> <sup>X</sup> 1

**342**

*r*¼0

*r*¼0

1 *r* � � *Mr* 0 X 3

1 *r* � �

*m*<sup>1</sup>�*<sup>r</sup> M<sup>r</sup>* *a*<sup>0</sup> ¼ 0, 683012701892219 *a*<sup>1</sup> ¼ 1*:*183012701892219 *a*<sup>2</sup> ¼ 0, 316987298107781 *a*<sup>3</sup> ¼ �0, 183012701892219

(41)

(43)

*Function f x*ð Þ¼ *x using Daubechies wavelets of kind* 4*. Source: This figure was generated by the author using the python programming language.*

#### **4.1 Taylor polynomial using Daubechies wavelets**

The Taylor polynomial or Taylor series is an expression that allows the calculation of the local value of a function *f* using your derivatives. For this, the function *f* must be of class *C* infinite (represented by *C*<sup>∞</sup>) which implies that the *f* is infinitely derivable in an interval containing a point *x*0. The expression for the Taylor polynomial for the function *f* is as follows,

$$f(\mathbf{x}) = \sum\_{k=0}^{+\infty} \frac{f^{(k)}(\mathbf{x}\_0)}{k!} (\mathbf{x} - \mathbf{x}\_0)^k \tag{50}$$

The expression (50) developed around *x*<sup>0</sup> ¼ 0 is:

$$f(\mathbf{x}) = \sum\_{k=0}^{+\infty} \frac{f^{(k)}(\mathbf{0})}{k!} (\mathbf{x})^k \tag{51}$$

sinh ð Þ¼ *<sup>x</sup>* <sup>X</sup>

*Comparison of the values obtained by the Taylor series and by Daubechies wavelets.*

**Function Value in** *x* ¼ **1, Taylor**

*DOI: http://dx.doi.org/10.5772/intechopen.93885*

*Calculation using Daubechies wavelets of kind 4.*

*Calculation using Taylor Series.*

**Series<sup>4</sup>**

*Use of Daubechies Wavelets in the Representation of Analytical Functions*

ln 1ð Þ¼ <sup>þ</sup> *<sup>x</sup>* <sup>X</sup>

wavelets of kind 4.

*4*

*5*

**Table 1.**

**5. Conclusions**

research.

**345**

**Acknowledgements**

þ∞

*s x*ð Þ¼ *ex* <sup>2</sup>*:*<sup>716666667</sup> <sup>2</sup>*:*<sup>716735469443329</sup> <sup>0</sup>*:*0025% *f x*ð Þ¼ cosh ð Þ *x* 1*:*543088161791753 1*:*543058311287478 0*:*0019% *g x*ð Þ¼ sinh ð Þ *x* 1*:*1750199840127897 1*:*1750591521108822 0*:*00376% *h x*ð Þ¼ ln 1ð Þ þ *x* 0*:*6456349203122008 0*:*6456349190214307 1, 9*:*10�7%

*x*<sup>2</sup>*n*þ<sup>1</sup>

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

*n*

*xn*

**Value in** *x* ¼ **1, Daubechies Wavelets<sup>5</sup>**

ð Þ <sup>2</sup>*<sup>n</sup>* <sup>þ</sup> <sup>1</sup> ! (57)

(58)

**Error %**

*n*¼0

þ∞

*n*¼1

In order to verify the potentiality of the application of Daubechies wavelets we will calculate the value of the functions in (53), (56), (57) and (58) evaluated at point x = 1. Considering only 7 terms in each summation. For obtain the results using Daubechies wavelets we apply the expression (55) in each summation (53), (56), (57) and (58). In the **Table 1** we have a comparison between the calculation of the values of the functions *s x*ð Þ¼ *<sup>e</sup><sup>x</sup>*, *f x*ð Þ¼ cosh ð Þ *<sup>x</sup>* , *g x*ð Þ¼ sinh ð Þ *<sup>x</sup>* and *h x*ð Þ¼ ln 1ð Þ þ *x* evaluated at point *x* ¼ 1, using the Taylor series and the Daubechies

**Table 1** appears here only as a way of showing the quality of the approximations using the Daubechies wavelets of kind 4. Obviously if we want more precise values, we must use Daubechies wavelets of the kind greater than 4. This will cause changes in the resolution and translation of each wavelet, but the result will be even better.

Daubechies wavelets are quite versatile mathematical tools. They can be used to analyze, generate, decompose a function, or even a signal that is represented by an analytical function. This type of application is widely used, for example, in electrical engineering in studies of magnetic fields and electric fields. The theory exposed in this chapter provides tools to carry out these studies. The use of the Taylor series as a way of approximating analytical functions is a very used technique in applied mathematics. Making use of the Taylor series with wavelets is another option to perform an approximation of analytical functions. In future work, we are

researching other wavelets, for example Deslauries-Dubuc interpolets, that have an even better approach quality. As Deslauriers-Dubuc interpolets and others in

The author would like to thank UFERSA for support during my doctoral studies.

Making use of the expression (21), we have:

$$f(\mathbf{x}) = \sum\_{k=0}^{+\infty} \sum\_{m=-\infty}^{+\infty} \frac{f^{(k)}(\mathbf{0})}{k!} \frac{M\_m^k}{2^k} \phi\left(2^j \mathbf{x} - m\right) \tag{52}$$

The expression (52) is another way of writing Taylor's polynomial using Daubechies Wavelets.

Example 4.2 Consider the analytical function *f x*ð Þ¼ *ex*, using Daubechies wavelet of kind a *N* ¼ 4 is possible to write this function *f* in terms of this wavelet. For this, Taylor's series development around the point *x*<sup>0</sup> ¼ 0 of this function is given by:

$$\mathcal{e}^{\mathfrak{x}} = \sum\_{n=0}^{+\infty} \frac{\mathfrak{x}^n}{n!} \tag{53}$$

Using only two summation terms in the expression (53), we have:

$$e^{\mathbf{x}} \approx \sum\_{n=0}^{1} \frac{\mathbf{x}^n}{n!} = \mathbf{1} + \mathbf{x} \tag{54}$$

Using the expression (46), we have:

$$
\epsilon^\* \approx \mathbf{1} + \mathbf{x} = \mathbf{1} + \mathbf{0}, \mathbf{63}\mathbf{4}\phi(\mathbf{x}) - \mathbf{0}, \mathbf{36}\mathbf{6}\phi(\mathbf{x} + \mathbf{1}) - \mathbf{1}. \mathbf{36}\mathbf{6}\phi(\mathbf{x} + \mathbf{2}) \tag{55}
$$

The expression (55) allows us to approximate the exponential function using a base of Daubechies wavelets. This type of approximation, although simple for this case, is very useful in the case of representation for functions other types.

In the following example, the expression (46) is used to approximate Taylor's series developments for the functions *s x*ð Þ¼ *<sup>e</sup><sup>x</sup>*, *f x*ð Þ¼ cosh ð Þ *<sup>x</sup>* , *g x*ð Þ¼ sinh ð Þ *<sup>x</sup>* and *h x*ð Þ¼ ln 1ð Þ þ *x* .

Example 4.3 For the functions *f x*ð Þ¼ cosh ð Þ *x* , *g x*ð Þ¼ cosð Þ *x* and *h x*ð Þ¼ sec ð Þ *x* . Taylor's series development of these functions around the point *x*<sup>0</sup> ¼ 0 is:

$$\cosh\left(\mathbf{x}\right) = \sum\_{n=0}^{+\infty} \frac{\mathbf{x}^{2n}}{(2n)!} \tag{56}$$


*Use of Daubechies Wavelets in the Representation of Analytical Functions DOI: http://dx.doi.org/10.5772/intechopen.93885*

#### **Table 1.**

**4.1 Taylor polynomial using Daubechies wavelets**

the Taylor polynomial for the function *f* is as follows,

The expression (50) developed around *x*<sup>0</sup> ¼ 0 is:

Making use of the expression (21), we have:

Daubechies Wavelets.

*Wavelet Theory*

given by:

*e*

*h x*ð Þ¼ ln 1ð Þ þ *x* .

**344**

*f x*ð Þ¼ <sup>X</sup> þ∞

*k*¼0

*f x*ð Þ¼ <sup>X</sup> þ∞

*k*¼0

*f x*ð Þ¼ <sup>X</sup> þ∞

> X þ∞

*m*¼�∞

*k*¼0

*f* ð Þ*<sup>k</sup>* ð Þ <sup>0</sup> *k*!

The expression (52) is another way of writing Taylor's polynomial using

Example 4.2 Consider the analytical function *f x*ð Þ¼ *ex*, using Daubechies wavelet of kind a *N* ¼ 4 is possible to write this function *f* in terms of this wavelet. For this, Taylor's series development around the point *x*<sup>0</sup> ¼ 0 of this function is

*n*¼0

*xn n*!

*<sup>x</sup>* <sup>≈</sup><sup>1</sup> <sup>þ</sup> *<sup>x</sup>* <sup>¼</sup> <sup>1</sup> <sup>þ</sup> 0, 634*ϕ*ð Þ� *<sup>x</sup>* 0, 366*ϕ*ð Þ� *<sup>x</sup>* <sup>þ</sup> <sup>1</sup> <sup>1</sup>*:*366*ϕ*ð Þ *<sup>x</sup>* <sup>þ</sup> <sup>2</sup> (55)

The expression (55) allows us to approximate the exponential function using a base of Daubechies wavelets. This type of approximation, although simple for this

In the following example, the expression (46) is used to approximate Taylor's series developments for the functions *s x*ð Þ¼ *<sup>e</sup><sup>x</sup>*, *f x*ð Þ¼ cosh ð Þ *<sup>x</sup>* , *g x*ð Þ¼ sinh ð Þ *<sup>x</sup>* and

Example 4.3 For the functions *f x*ð Þ¼ cosh ð Þ *x* , *g x*ð Þ¼ cosð Þ *x* and *h x*ð Þ¼ sec ð Þ *x* .

þ∞

*x*<sup>2</sup>*<sup>n</sup>*

*n*¼0

*xn*

*e <sup>x</sup>* <sup>¼</sup> <sup>X</sup> þ∞

Using only two summation terms in the expression (53), we have:

*n*¼0

case, is very useful in the case of representation for functions other types.

Taylor's series development of these functions around the point *x*<sup>0</sup> ¼ 0 is:

cosh ð Þ¼ *<sup>x</sup>* <sup>X</sup>

*e <sup>x</sup>* ≈ X 1

Using the expression (46), we have:

*f* ð Þ*<sup>k</sup>* ð Þ <sup>0</sup> *<sup>k</sup>*! ð Þ *<sup>x</sup>*

*f* ð Þ*<sup>k</sup>* ð Þ *<sup>x</sup>*<sup>0</sup>

*<sup>k</sup>*! ð Þ *<sup>x</sup>* � *<sup>x</sup>*<sup>0</sup>

*M<sup>k</sup> m* <sup>2</sup>*jk <sup>ϕ</sup>* <sup>2</sup> *<sup>j</sup>* *<sup>k</sup>* (50)

*<sup>k</sup>* (51)

*<sup>x</sup>* � *<sup>m</sup>* � � (52)

*<sup>n</sup>*! (53)

¼ 1 þ *x* (54)

ð Þ <sup>2</sup>*<sup>n</sup>* ! (56)

The Taylor polynomial or Taylor series is an expression that allows the calculation of the local value of a function *f* using your derivatives. For this, the function *f* must be of class *C* infinite (represented by *C*<sup>∞</sup>) which implies that the *f* is infinitely derivable in an interval containing a point *x*0. The expression for

*Comparison of the values obtained by the Taylor series and by Daubechies wavelets.*

$$\sinh\left(\infty\right) = \sum\_{n=0}^{+\infty} \frac{\varkappa^{2n+1}}{(2n+1)!} \tag{57}$$

$$\ln\left(\mathbf{1} + \mathbf{x}\right) = \sum\_{n=1}^{+\infty} \frac{(-\mathbf{1})^{n+1} \mathbf{x}^n}{n} \tag{58}$$

In order to verify the potentiality of the application of Daubechies wavelets we will calculate the value of the functions in (53), (56), (57) and (58) evaluated at point x = 1. Considering only 7 terms in each summation. For obtain the results using Daubechies wavelets we apply the expression (55) in each summation (53), (56), (57) and (58). In the **Table 1** we have a comparison between the calculation of the values of the functions *s x*ð Þ¼ *<sup>e</sup><sup>x</sup>*, *f x*ð Þ¼ cosh ð Þ *<sup>x</sup>* , *g x*ð Þ¼ sinh ð Þ *<sup>x</sup>* and *h x*ð Þ¼ ln 1ð Þ þ *x* evaluated at point *x* ¼ 1, using the Taylor series and the Daubechies wavelets of kind 4.

**Table 1** appears here only as a way of showing the quality of the approximations using the Daubechies wavelets of kind 4. Obviously if we want more precise values, we must use Daubechies wavelets of the kind greater than 4. This will cause changes in the resolution and translation of each wavelet, but the result will be even better.

#### **5. Conclusions**

Daubechies wavelets are quite versatile mathematical tools. They can be used to analyze, generate, decompose a function, or even a signal that is represented by an analytical function. This type of application is widely used, for example, in electrical engineering in studies of magnetic fields and electric fields. The theory exposed in this chapter provides tools to carry out these studies. The use of the Taylor series as a way of approximating analytical functions is a very used technique in applied mathematics. Making use of the Taylor series with wavelets is another option to perform an approximation of analytical functions. In future work, we are researching other wavelets, for example Deslauries-Dubuc interpolets, that have an even better approach quality. As Deslauriers-Dubuc interpolets and others in research.

#### **Acknowledgements**

The author would like to thank UFERSA for support during my doctoral studies.

*Wavelet Theory*

#### **Author details**

Paulo César Linhares da Silva UFERSA, Mossoró-RN, Brazil

\*Address all correspondence to: linhares@ufersa.edu.br

© 2020 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.

**References**

user Boston, 2002.

Analysis,3, 1–9, 1996.

[1] Daubechies I. Recent results in wavelets applications. Journal of Electronic Imaging, 1998, 7, 719–724.

*DOI: http://dx.doi.org/10.5772/intechopen.93885*

*Use of Daubechies Wavelets in the Representation of Analytical Functions*

scaling functions. Thermal Engineering,

[11] Silva P, Silva J, Garcia A. Daubechies wavelets as basis functions for the vectorial beam propagation method, Journal of Electromagnetic Waves and Applications, 2019, 33:8, 1027-1041, DOI: 10.1080/09205071.2019.1587319

[12] Gopinath R and Burrus C. On the moments of the scaling function psi. Departament of Electrical and Computer Engineering-IEEE, 1992,

[13] Butzer P, Fischer A, Ruckforth K. Scaling functions and wavelets with vanishing moments. Computers Math.

Applic, 1994, 27, 33–39.

2016, 15, pp. 68–75.

963–966,

[2] Walnut D. An Introduction to Wavelet Analysis. Applied and Numerical Harmonic Analysis. Birkha

[3] Bertoluzza S. A wavelet collocation method for the numerical solution of partial differential equations. Applied

[4] Choudhury A. Wavelet method for numerical solution of parabolic equations. Journal of Computational Engineering, 2014, 2014, 1–12, 2014. h ttps://doi.org/10.1155/2014/346731

[5] Robert L, Weng C. A study of wavelets for the solution of

electromagnetic integral equations. IEEE Transactions on antennas and propagation, 1995, 43, 802–810.

2179-10742020v19i3825

PUC, Setembro 2009.

**347**

[6] Silva P, Melo R, Silva J. Optical Fiber Coupler Analysis Using Daubechies Wavelets. Journal of Microwaves, Optoelectronics and Electromagnetic Applications (JMOe), 2020, 19(3), AoP 294-300. https://doi.org/10.1590/

[7] Ingrid Daubechies. Ten lectures on wavelets. 1992, Society for Industrial and Applied Mathematics, USA.

[8] Burgos R. Análise de Estruturas Utilizando Wavelets de Daubechies e Interpolets de Deslauriers-Dubuc. PhD thesis, Pontifícia Universidade Católica,

[9] Burgos R. Finite elements based on deslauriers-dubuc wavelets for wave propagation problems, Applied Mathematics, 2016, 7, pp. 1490–1497.

[10] Burgos R. Solution of 1d and 2d poisson's equation by using wavelet

and Computational Harmonic

*Use of Daubechies Wavelets in the Representation of Analytical Functions DOI: http://dx.doi.org/10.5772/intechopen.93885*

### **References**

[1] Daubechies I. Recent results in wavelets applications. Journal of Electronic Imaging, 1998, 7, 719–724.

[2] Walnut D. An Introduction to Wavelet Analysis. Applied and Numerical Harmonic Analysis. Birkha user Boston, 2002.

[3] Bertoluzza S. A wavelet collocation method for the numerical solution of partial differential equations. Applied and Computational Harmonic Analysis,3, 1–9, 1996.

[4] Choudhury A. Wavelet method for numerical solution of parabolic equations. Journal of Computational Engineering, 2014, 2014, 1–12, 2014. h ttps://doi.org/10.1155/2014/346731

[5] Robert L, Weng C. A study of wavelets for the solution of electromagnetic integral equations. IEEE Transactions on antennas and propagation, 1995, 43, 802–810.

[6] Silva P, Melo R, Silva J. Optical Fiber Coupler Analysis Using Daubechies Wavelets. Journal of Microwaves, Optoelectronics and Electromagnetic Applications (JMOe), 2020, 19(3), AoP 294-300. https://doi.org/10.1590/ 2179-10742020v19i3825

[7] Ingrid Daubechies. Ten lectures on wavelets. 1992, Society for Industrial and Applied Mathematics, USA.

[8] Burgos R. Análise de Estruturas Utilizando Wavelets de Daubechies e Interpolets de Deslauriers-Dubuc. PhD thesis, Pontifícia Universidade Católica, PUC, Setembro 2009.

[9] Burgos R. Finite elements based on deslauriers-dubuc wavelets for wave propagation problems, Applied Mathematics, 2016, 7, pp. 1490–1497.

[10] Burgos R. Solution of 1d and 2d poisson's equation by using wavelet

scaling functions. Thermal Engineering, 2016, 15, pp. 68–75.

[11] Silva P, Silva J, Garcia A. Daubechies wavelets as basis functions for the vectorial beam propagation method, Journal of Electromagnetic Waves and Applications, 2019, 33:8, 1027-1041, DOI: 10.1080/09205071.2019.1587319

[12] Gopinath R and Burrus C. On the moments of the scaling function psi. Departament of Electrical and Computer Engineering-IEEE, 1992, 963–966,

[13] Butzer P, Fischer A, Ruckforth K. Scaling functions and wavelets with vanishing moments. Computers Math. Applic, 1994, 27, 33–39.

**Author details**

*Wavelet Theory*

**346**

Paulo César Linhares da Silva UFERSA, Mossoró-RN, Brazil

\*Address all correspondence to: linhares@ufersa.edu.br

provided the original work is properly cited.

© 2020 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,

**Chapter 16**

**Abstract**

**1. Introduction**

type is examined in [13].

**349**

Equations

mesh and the method parameter values used.

earthquake prediction and other algorithms.

Higher Order Haar Wavelet

Method for Solving Differential

*Jüri Majak, Mart Ratas, Kristo Karjust and Boris Shvartsman*

The study is focused on the development, adaption and evaluation of the higher order Haar wavelet method (HOHWM) for solving differential equations. Accuracy and computational complexity are two measurable key characteristics of any numerical method. The HOHWM introduced recently by authors as an improvement of the widely used Haar wavelet method (HWM) has shown excellent accuracy and convergence results in the case of all model problems studied. The practical value of the proposed HOHWM approach is that it allows reduction of the computational cost by several magnitudes as compared to HWM, depending on the

**Keywords:** higher order Haar wavelet method, convergence analysis, accuracy

Wavelets are most commonly used in signal processing applications to denoise the real signal, to cut a signal into different frequency components, to analyze the components with a resolution matched to its scale, also in image compression,

However, the current study is focused on the area where the use of wavelet methods shows a growth trend, i.e., in the solution of differential equations. Many different wavelets based methods have been introduced for solving differential and integro-differential equations. The Legendre wavelets are utilized to solve fractional differential equations in [1–4] and integro-differential equations in [5, 6]. In [7, 8], the Daubechies wavelet based approximation algorithms are derived to solve ordinary and partial differential equations. In [9], the Lucas wavelets are combined with Legendre–Gauss quadrature for solving fractional Fredholm–Volterra integrodifferential equations. The series solution of partial differential equations through separation of variables is developed by using the Fourier wavelets in [10]. The Riesz wavelets- based method for solving singular fractional integro-differential equations was developed in [11]. In the studies in [12], the Galerkin method was combined with the quadratic spline wavelets for solving Fredholm linear integral equations and second-order integro-differential equations. The Chebyshev wavelets method for partial differential equations with boundary conditions of the telegraph

estimates, improvement of widely used Haar wavelet method

#### **Chapter 16**
