Fundamentals of Spectral Analysis by Wavelet Transforms

## A Modern Review of Wavelet Transform in Its Spectral Analysis

*Francisco Bulnes*

#### **Abstract**

The spectral analysis, in much aspects as are the wavelet transform in its numerous versions and its relation with other transforms and special functions requires a special review, since the exploration in the frequency domain to the wavelet transform is more detailed and majorly more specific in different applications. For example, the wavelet transform of special function can be very useful to create and design special signal filters or, for example, to the interphase between reception-emission devices with sensorial parts of the human body. Also the quantum wavelet transform is very useful in the spectral study of traces of particles. Likewise, in this chapter, these aspects are considered as an inherent property of the wavelet transform in the spectral exploration of some phenomena. Finally, general results to the discrete case are given, which is analyzed to the wavelet transform and its spectra.

**Keywords:** discrete Fourier transform, discrete wavelet transform, fast Fourier transform, Gabor transform, short-time Fourier transform, spectra, quantum wavelet transform, wavelet transform

#### **1. Introduction**

Likewise, we consider a set of functions

$$\langle \Psi\_{1k}, \Psi\_{2k}, \dots, \Psi\_{nk}, \dots \rangle \in L^2(\mathbb{R}), \tag{1}$$

which define a Hilbert basis of square integrable functions [1]. Likewise, for each *j*, *k*∈ *Z*, the *ψ*jk represents dyadics and dilations of *ψ*, given by the functions:

$$
\mu\_{\rm jk}(\mathbf{x}) = \mathfrak{Z}^j \mathfrak{w} \left( \mathfrak{Z}^j \mathbf{x} - k \right), \tag{2}
$$

<sup>∀</sup>*j*, *<sup>k</sup>*<sup>∈</sup> *<sup>Z</sup>:* Likewise, for a function *<sup>ξ</sup>*ð Þ *<sup>x</sup>* <sup>∈</sup>*L*<sup>2</sup> ð Þ *R* , and using the orthonormal functions family, by completeness we have:

$$\xi(\mathbf{x}) = \sum\_{-\infty}^{+\infty} c\_{\mathbf{jk}} \nu\_{\mathbf{jk}}(\mathbf{x}),\tag{3}$$

Then the convergence of the series will be understood to be convergence in norm. Likewise, a representation of *ξ*ð Þ *x* is known as a wavelet series with wavelet

coefficients *c*jk*:* This implies that an orthonormal wavelet is self-dual. Then the wavelet integral transform is the integral transform [2] given for<sup>1</sup> :

$$\mathcal{W}\_{\psi}\xi = \frac{1}{\sqrt{|a|}} \int\_{-\infty}^{+\infty} \bar{\varphi}\left(\frac{\varkappa - b}{a}\right) \xi(\varkappa) d\mathbf{x},\tag{4}$$

Here *<sup>a</sup>* <sup>¼</sup> <sup>2</sup>�*<sup>j</sup>* is the binary dilation or dyadic dilation, and *<sup>b</sup>* <sup>¼</sup> *<sup>k</sup>*2�*<sup>j</sup>* is the binary or dyadic position. Then the wavelet transform can be modified depending on the response treatments that are given. For example in the images compression through impulse function *x n*ð Þ¼ *δ*ð Þ *n* � *ni* , for a discrete signal where impulse response can be used to evaluate the image compression-reconstruction system, the wavelet transform has been modified.

As has been said in different signal treatments, one of fundamental problems in electronics is obtaining a sufficiently clean signal in the different processes of communication, perception, and management of the signals in different continuous and discrete domains. For example, in signal processing in accelerometers for gait analysis, where actually is necessary to implement a good programming in real time for drones or other devices of vehicles, even human body parts with accelerations process in fault detections for design of low power pacemakers and also in ultrawideband in wireless [3]; the cleaning of signal is fundamental.

The wavelet transforms as transformation should allow only changes in time extension, but not shape. This could be affected by choosing suitable basis functions that allow for this. For example, in numerical analysis, we can consider the scale factor *cn* <sup>¼</sup> *<sup>c</sup><sup>n</sup>* 0, with the discrete frequency *<sup>η</sup><sup>m</sup>* <sup>¼</sup> mLc*<sup>n</sup>* 0, having the wavelets (considering the discrete formula with the basis wavelet):

$$\Psi(k,n,m,) = \frac{1}{\sqrt{c\_0^n}} \Psi\left[\frac{k - \text{mc}\_0^n}{c\_0^n} L\right] = \frac{1}{\sqrt{c\_0^n}} \Psi\left[\left(\frac{k}{c\_0^n} - m\right) L\right],\tag{5}$$

where such discrete wavelets can be used through the discrete wavelet transform version:

$$W\_D(n,m) = \frac{1}{\sqrt{c\_0^n}} \sum\_{k=0}^{K-1} w(k) \Psi\left[\left(\frac{k}{c\_0^n} - m\right)L\right],\tag{6}$$

whose continuous (or analogic) is the standard wavelet transform:

$$\mathcal{W}(c,\eta) = \frac{1}{\sqrt{c}} \int\_{-\infty}^{+\infty} w(t) \mathcal{Y}\left(\frac{t-\eta}{c}\right) \mathrm{d}t,\tag{7}$$

where *c*, is a scaling factor, and *η*, represents time shift factor. For the case (7) the Fourier transformation of signal *w k*ð Þ, is computed with the FFT. An adequate selection of a discrete scaling factor *cn*will be necessary*:* Changes in the time extension are expected to conform to the corresponding analysis frequency of the basis function, based on the uncertainty principle of signal processing.

<sup>1</sup> To recover the original signal *w t*ð Þ, the first inverse continuous wavelet transform can be used: <sup>ξ</sup>ðÞ¼ <sup>t</sup> <sup>χ</sup>�<sup>1</sup> ψ Ð <sup>þ</sup><sup>∞</sup> �∞ Ð <sup>þ</sup><sup>∞</sup> �<sup>∞</sup> <sup>W</sup>ψξð Þ a, b <sup>1</sup> ffiffiffiffi j j <sup>a</sup> <sup>p</sup> <sup>ψ</sup><sup>~</sup> <sup>t</sup>�<sup>b</sup> a � �db da a2 *:*

For example, the wavelet transform of Shannon function can be very useful to creation of windows *<sup>Ψ</sup>*Shað Þ *<sup>ω</sup>* , through functions Shað Þ*<sup>t</sup>* , and with gate functions *Π*ð Þ *x* , useful in the signal analysis by ideal band-pass filters that define a decomposition known as Shannon wavelets. Also, for example, the complex-valued Morlet wavelet is closely related to human perception, both hearing and the vision [4].

Likewise, the transition for "classical" wavelet transform (with some modifications accepted) to quantum wavelet transform can be approached by factoring the classical operators for the transformation into direct sums, direct products, and dot products of unitary matrices. Likewise, the permutation matrices play a vital role [5].

## **2. From the signal resolution problems until biological-sensorial perception**

A fundamental property of the wavelet transform and the signal resolution problem can be discussed and explored simultaneously in time and frequency domains starting from the wavelet spectra:

$$W(a) = \frac{1}{\sqrt{|a|}} \int\_{-\infty}^{+\infty} w\left(\frac{t-b}{a}\right) e^{-jat} dt = \sqrt{|a|} W(ao) e^{-jat},\tag{8}$$

where *W*ð Þ *ω* is the Fourier transform of the basic wavelet *w t*ð Þ*:* If the wavelets are normalized in terms of amplitude, the Fourier transforms of the wavelets with different scales will have the same amplitude that is suitable for implementation of the continuous wavelet transform using the frequency domain filtering. This property is fundamental in the samples of frequency pulses of signal spectra, since it shows that a dilatation *t=a a*ð Þ >1 of a function in the time domain produces a contraction *aω*, of its Fourier transform, which are "spectral wavelets" corresponding. Likewise, the term *t=a* has a metrology of frequency, which is equivalent to *ω:* However, in the technical convention, this term is known as scale, since the term "frequency" is reserved for the Fourier transform. Then the design of signal filters in frequency obey to the correlation between the signal and the wavelets, in the time domain that can be written as the inverse Fourier transform of the product of the conjugate Fourier transforms of the wavelets and the Fourier transform of the signal:

$$\mathcal{W}\_{\xi}(a,b) = \frac{\sqrt{|a|}}{2\pi} \int\_{-\infty}^{+\infty} \Xi(a)\mathcal{W}(ja a)e^{-j\alpha t} d\alpha \tag{9}$$

The Fourier transforms of the wavelets are referred to as the wavelet transform filters. The impulse response of the wavelet transform filter ffiffiffiffiffi j j *<sup>a</sup>* <sup>p</sup> *W a*ð Þ *<sup>ω</sup>* is the scaled wavelet <sup>1</sup> ffiffiffiffi j j *<sup>a</sup>* <sup>p</sup> *<sup>w</sup> <sup>t</sup> a* � �*:* Therefore, the wavelet transform is a collection of wavelet transform filters with different scales, *a:* Then we can relate the short-time Fourier transform (STFT) [6] with the idea of the wavelets to determine the sinusoidal frequency and phase content of local sections of a signal considering as changes over time. If we introduce the Gaussian function, which can be regarded as a window function, then the STFT is the Gabor transform. Here the Gabor atoms or functions used to build from translations and modulations of generating function a family of functions are constructed and characterized.

Likewise, we can have direct applications of the STFT, to samples in real time of the complex processes, which require a speed compute of data through direct relation between machine and real-time domains in the measured and perception of the phenomena. Likewise, the STFT is performed on a computer using the fast Fourier transform (FFT), so both variables are discrete and quantized.

Secondly the Morlet transform is a Gabor transform consisting of a wavelet composed of a complex exponential (carrier) multiplied by a Gaussian window (envelope). This wavelet is closely related to human perception, both hearing [2] and vision. Then the functions related with these bio-sensorial perceptions use Shað Þ*t* functions as special Gabor functions to discriminate steps of signal spectra in the perception and create of a signal response audible or visible required to the eye organ, the iris of eye, in the case of the vision and to audition, we have the audiphones that amplify the sounds to equilibrate the lack of the eardrum or other parts of middle and inner ear to perceive the sounds.

#### **3. Some important results in discrete signal analysis**

Let *S<sup>N</sup>* be the complex sphere of dimension *N*, and let

$$\dots, e^{-2\Omega^{\circ}}, e^{-\Omega^{\circ}}, \mathbf{1}, e^{\Omega^{\circ}}, e^{-2\Omega^{\circ}}, \dots, \tag{10}$$

be a linear basis of signals space in *L*<sup>2</sup> ½ � *K* that generates the subspace *W*, such that ∀*w* ∈ *W* is

$$w = \sum\_{n=1}^{N} c\_n e^{-n\mathcal{Q}\dagger},\tag{11}$$

We define the space of nilpotent classes on *E*½ � *<sup>K</sup>* , (being *E*½ � *<sup>K</sup>* ¼ *E*<sup>1</sup> ⊕ … ⊕ *En*, [7] the total discrete signal space) with the component:

$$N(E\_{[K]}) = \{ F \in D'(G^0/K) | F = \mathbf{0} \},\tag{12}$$

**Proposition 3.1** [8, 9]. If *z*∈ *N E*½ � *<sup>K</sup>* � �, and if *β* ∈ *H<sup>i</sup> n*0, *L*<sup>2</sup> ½ � *<sup>K</sup>* � �, (then)

$$z\beta = e^{-n\Omega \circ} x[m] = p(z)\varkappa[n],\tag{13}$$

where *x n*½ � is a Gabor discrete function<sup>2</sup> . *Proof*. Here

$$n\_0 \simeq n, \quad n\_0 \simeq p\_{I^\*}$$

<sup>2</sup> A discrete version of Gabor representation is

x tðÞ¼ <sup>P</sup><sup>∞</sup> n¼�∞ P∞ m¼∞ Cnmgnmð Þt ,

with *<sup>g</sup>*nmðÞ¼ *<sup>t</sup> s t* � *<sup>m</sup>η*<sup>0</sup> ð Þ*e*�*nΩ*jt*:*Similar to the DFT (discrete Fourier transformation) we have: x kð Þ¼ <sup>P</sup><sup>∞</sup> n¼�∞ P<sup>∞</sup> <sup>m</sup>¼<sup>∞</sup>Cnmgnmð Þ <sup>k</sup> ,

where the Gabor basis functions are *<sup>g</sup>*nmð Þ¼ *<sup>k</sup> s k* � *<sup>m</sup>η*<sup>0</sup> ð Þ*e*�*nΩ*jk*:*

where *pI* is the unitary sphere *p*∩*gI*, where *gI*, ¼ ½ � *g*, *g :* We demonstrate on the dimension *i*, of the cohomological space *H<sup>i</sup> gI*, *L*<sup>2</sup> ½ � *<sup>K</sup> :* If *<sup>i</sup>* <sup>¼</sup> *<sup>n</sup>* <sup>¼</sup> dim*n*0, then

$$H^i(\mathbb{g}\_I, L^2[K]) = n\_0 \ast \otimes L^2/n\_0 L^2,\tag{14}$$

Therefore, *z* acts by *I* ⊗ *p z*ð Þ*:* Then *p z*ð Þ acts for ð Þ *C* ⊗ *π p z*ð Þ*:*(Then)

$$(I \otimes p(\mathbf{z})) \beta = p\_C(\mathbf{z}) \beta,\tag{15}$$

As a special note, we have as a particular example an LTI system *L e*�*nΩ<sup>j</sup>* <sup>¼</sup> *<sup>H</sup>*ð Þ *<sup>Ω</sup> <sup>e</sup>*�*nΩ<sup>j</sup>* , where *H*ð Þ *Ω* is a projection of the system function.

Likewise, the result to *i* ¼ *r* þ 1≤ *n*, then demonstrate for *i* ¼ *r:* Let *C* be the periodic complex. Let *α*∈ *H* ∗ *Xi* ð Þ , *QbZa* , such that *α*ð Þ *g* ⊗ *υ gυ:* Then *α* is the homeomorphism

$$a: \operatorname{Hom}\_{\mathbb{K}}(p\_I, \mathbb{C}) \to \operatorname{Hom}\_{\mathbb{K}}(p\_I, L^2[\mathbb{K}]),\tag{16}$$

Let *X* ¼ ker*α*, where specifically

$$X = \left\{ a \in \text{Hom}\_K(\mathbb{C}, L^2[K]) | a(\mathbf{g} \otimes \nu) = \mathbf{0}, \quad \forall \mathbf{g} \in U(\mathbf{g}), \nu \in L^2[k] \right\},\tag{17}$$

Then we have

$$\mathbf{0} \to \mathbf{X}\_i \to \mathbf{C} \to L^2 \to \mathbf{0},\tag{18}$$

Now *U g*ð Þ, is a *U pI* �complex free under left translations. Therefore

$$H^i(p\_I, \mathcal{C}) = 0,\tag{19}$$

$$\forall \mathcal{C} = E\_{[K]} \otimes Q\_b Z\_a \text{, (then)}$$

$$H^i(p\_I, E\_{[K]} \otimes Q\_b Z\_a) = 0,\tag{20}$$

∀*j*<*n*, and *b* � *a*mod*j:* Then the long exact sequence of discrete cohomology for this case of periodic complexes and *N E*½ � *<sup>K</sup>* �complex is:

$$0 \to H^i\left(p\_I, L^2[K]\right) \to H^{i+1}\left(p\_I, X\right) \to H^{i+1}\left(p\_I, E\_{[K]} \otimes Q\_b Z\_a\right) \to 0,\tag{21}$$

where such injection implies the result.

A study realized in signal and systems analysis on a linear system can be approximated in the time-frequency domain due to the composition of an analysis filter-bank, a transfer matrix (sub-band model) and a synthesis filter-bank, which is a method known as sub-band technique.

In the varying case, time-frequency representations of LTV systems have connection with the Gabor expansion of signals through the corresponding integral equation. Then we will have an integral equation with Gabor function. For example, a work realized in that sense is the creation of 3D Gabor frame based in spatial spectral integral equation designed to solve the scattering from dielectric objects embedded in a multilayer medium. Likewise, this is based on the Gabor frame, as a new set of basic functions (belonging to a basis) [10] together with a set of equidistant Dirac-delta test functions.

**Proposition 3.2.** Exists an isomorphism given for the DFT that maps the proper nilpotent classes of the system controlled under transformations of *pI :* Then DS-TFT is the FFT.

Let DFT be the isomorphism of the discrete signals:

$$E\_{[K]} \longrightarrow \tilde{E}\_{[K]},\tag{22}$$

where the explicit rule for any ∀*υ*∈*L*<sup>2</sup> ½ � *K* is

$$\text{DFT}(\nu) = (\mathbf{1}/\mathbf{N})\text{DFT}(\nu) = \text{DWT}(\nu),\tag{23}$$

Then in each component of the space *E*½ � *<sup>K</sup>* , (*E*½ � *<sup>K</sup>* ¼ *E*<sup>1</sup> ⊕ … ⊕ *En:*) we have:

$$F^{(k)} = \mathbf{0},\tag{24}$$

where the DS-TFT satisfies in short-time interval. In each component, we have:

$$U(p\_I)\operatorname{Hom}\_K\left(F, L^2[K]\right) = \chi\_\Lambda p\_{\xi^\*}\tag{25}$$

which exists FFT� ð Þ*υ* , such that

$$\text{DWT}(\nu) = \text{FFT}(\nu),\tag{26}$$

More details of the demonstration can be consulted in [11].

#### **4. Conclusions**

In this introductory chapter, the various and several advantages of the wavelet transform and its properties on the signal and system analysis have been shown, considering different specialized window functions and the wavelet function basis. Likewise, wavelet analysis is known for its successful approach to solving the problem of signal analysis in both the time domain and frequency domain. Also, the analysis of the nonstationary signal generated by physical phenomena has a great challenge for various conversion techniques. In several studies, it has been shown that the transformation techniques such as Fourier transform and short Fourier transform fail to analyze nonstationary signals. But wavelet transform methods may be able to efficiently analyze both stable and unstable signals. All this the author develops with precision and accuracy. In the Gabor transform, the resolution analysis considers the uncertainty principles on nilpotent Lie groups and their corresponding algebras, which were established in the propositions given through spectral analysis given in the classes *H<sup>i</sup> pI*, *L*<sup>2</sup> ½ � *<sup>K</sup>* , *<sup>H</sup><sup>i</sup>*þ<sup>1</sup> *pI*, *<sup>X</sup>* , and *<sup>H</sup><sup>i</sup>*þ<sup>1</sup> *pI* , *E*½ � *<sup>K</sup>* ⊗ *QbZa* . A scheme with neural network as components of a dynamical system can be proposed to demonstrate that using neural networks and linear filters in cascade and/or feedback configurations, a rich class of models of signaling and systematization in wide perspective and prospective can be constructed, considering the different filters designed by the different wavelet transform versions in short-time resolution or conventional resolution improving the canonical Fourier transform resolution. The multi-resolution analysis or multi-scale approximation can design a method considering practically the relevant discrete wavelet transforms (DWT), which can be considered as a fundamental set of special functions to realize approximations to solutions of different processes in time and the justification for the algorithm of the fast wavelet transform (FWT), for the calculating methods develop started with good wavelet bases.

## **Acknowledgements**

I thank Eng. Rene Rivera-Roldán, Director of Electronics Engineering Program, for the support of hours for the investigation to carry out this work.

## **Nomenclature**


## **Author details**

Francisco Bulnes Research Department in Mathematics and Engineering, IINAMEI, TESCHA, Mexico

\*Address all correspondence to: francisco.bulnes@tesch.edu.mx

© 2022 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**

[1] Reed M, Simon B. Methods of Modern Mathematical Physics. 1st ed. San Diego, California, USA: Academic Press Inc.; 1970

[2] Meyer Y. Wavelets and Operators. Cambridge, UK: Cambridge University Press; 1992

[3] Martin E. Novel method for stride length estimation with body area network accelerometers. 2011 IEEE Topical Conference on Biomedical Wireless Technologies, Networks, and Sensing Systems. 2011;**1**:79-82. DOI: 10.1109/BIOWIRELESS.2011.5724356

[4] Bruns A. Fourier-, Hilbert- and wavelet-based signal analysis: Are they really different approaches? Journal of Neuroscience Methods. 2004;**137**(2): 321-332

[5] Sharma J, Kumar A. Uncertainty Principles on Nilpotent Lie Groups, Journal of Representation Theory, arXiv: 1901.01676v1, [Math. R. T]. USA; 2019

[6] Allen JB. Short time spectral analysis, synthesis, and modification by discrete Fourier transform. IEEE Transactions on Acoustics, Speech, and Signal Processing. 1977;**ASSP-25**(3):235-238

[7] Akansu AN, Haddad RA. Multiresolution Signal Decomposition: Transforms, Subbands, and Wavelets. Boston, MA: Academic Press; 1992

[8] Bulnes F. Controlabilidad Digital Total sobre una Cohomología Discreta con Coeficientes en L2[K]. In: Proceedings of the Appliedmath III, International Conference in Applied Mathematics (APPLIEDMATH '03); 9–12 October 2007. Mexico City: IPN, UNAM, CINVESTAV, UNAM, Tec de Monterrey; 2007. pp. 71-77

[9] Bulnes F. Teoría de los (g, K)- Módulos. 1st ed. UNAM: Instituto de Matemáticas; 2000

[10] Dilz RJ, van Beurden MC. Fast operations for a Gabor-frame-based integral equation with equidistant sampling. IEEE Antennas and Wireless Propagation Letters. 2018;**17**(1):82-85 [8115279]

[11] Mallat S. A Wavelet Tour of Signal Processing. 2nd ed. San Diego, CA: Academic; 1999

## **Chapter 4**

## Bases of Wavelets and Multiresolution in Analysis on Wiener Space

*Claude Martias*

### **Abstract**

The multiresolution analysis is applied into the space of square integrable Wiener functionals for extending well-known constructions of orthonormal wavelets in L<sup>2</sup> (**R)** to this space denoted by L2 (μ), μ being the Wiener measure, as for instance Mallat's construction or furthermore Goodman–Lee and Tang construction. We also extend the Calderon–Zygmund decomposition theorem into the L1 (μ) framework. Even if L<sup>1</sup> spaces do not have unconditional bases, wavelets still outperform Fourier analysis in some sense. We illustrate this by introducing periodized Wiener wavelets.

**Keywords:** Wiener functionals, Wiener space, wavelet, multiresolution analysis

#### **1. Introduction**

The wavelet transform for Wiener functionals has been considered by the author and applied to diffusion processes and to the solutions to backward stochastic equation in [1]. This application is a purely mathematical one; others, having more practical aspect could be considered as an extension in networks domain (see [2]) for instance or in detection of change and chronological series analysis [3, 4]. Extension of the well-known concept in finite dimension of wavelet transform to analysis on Wiener space is one among many possible others, which could be useful to infinite dimensional analysis. The chapter devotes to this extension, more precisely, extension of multiresolution and bases of wavelets. We start by recalling notion of multiresolution analysis, exclusively on the space of square integrable Wiener functional. We follow the Mallat's construction [5] and notice that an extension to wavelets generated by a finite set of Wiener functionals can easily be done if we follow arguments of Goodman–Lee–Tang [6]. We give an example of multiresolution approximation generated by cardinal Hermite B-splines as in [6]. We complete our work by a study of unconditional bases for *LP*ð Þ *<sup>μ</sup>* , 1<sup>&</sup>lt; *<sup>p</sup>*<sup>&</sup>lt; <sup>∞</sup>, *<sup>μ</sup>* being the Wiener measure. We start it by first proving an extension of the wellknown Calderon–Zygmund decomposition theorem. As L<sup>1</sup> -spaces do not have unconditional bases, we introduce a notion of "periodized Wiener wavelets" and show that wavelets still perform Fourier–Wiener analysis in some sense, as in finite dimension [7].

#### **1.1 Multiresolution analysis in square Integrable Wiener Functionals**

A multiresolution analysis in *L*<sup>2</sup> ð Þ *μ* , the space of μ-square integrable Wiener functionals, μ denoting the Wiener measure, consists of a sequence approximation spaces **V***<sup>j</sup>* � � *<sup>j</sup>* <sup>∈</sup> <sup>ℤ</sup>, **<sup>V</sup>***<sup>j</sup>* <sup>⊂</sup> *<sup>L</sup>*<sup>2</sup> ð Þ *μ* ; these subspaces are assumed to be closed and satisfy the following:

$$\mathbf{V}\_{j} \subset \mathbf{V}\_{j-1}, \text{for all } j \in \mathbb{Z}, \tag{1}$$

$$\text{the closed space generated by } \overline{\bigcup\_{j \in \mathbb{Z}} \mathbf{V}\_j} \text{ is } L^2(\mu), \tag{2}$$

$$\bigcap\_{j \in \mathbb{Z}} \mathbf{V}\_j = \{\mathbf{0}\}. \tag{3}$$

If we denote by Q *<sup>j</sup>* the orthogonal projection operator onto **V**<sup>j</sup> , then (2) ensures that lim *<sup>j</sup>*!<sup>∞</sup> Q *j <sup>φ</sup>* <sup>¼</sup> *<sup>φ</sup>* for all *<sup>φ</sup>* <sup>∈</sup>*L*<sup>2</sup> ð Þ *μ* . There exist many ladders of spaces satisfying (1)–(3) that nothing to have with multiresolution; the multiresolution aspect is a consequence of the additional requirement

$$
\rho \in \mathbf{V}\_{\hat{\jmath}} \Leftrightarrow \rho(\mathcal{Q}^{\hat{\jmath}}.) \in \mathbf{V}\_0. \tag{4}
$$

That is, all the spaces are scaled versions of the central space **V**0. An example corresponding to the Haar multiresolution analysis in real analysis (see [8, 9]) is the following:

$$\mathbf{V}\_j := \left\{ \boldsymbol{\varrho} \in L^2(\mu) ; \forall k \in \mathbb{Z} : \boldsymbol{\varrho}\_{\left[ \mathcal{Q}^j \boldsymbol{e}\_{(k)}, \mathcal{Q}^{j+1} \boldsymbol{e}\_{(k+1)} \right]} = constant \right\},$$

where *e k*ð Þ, *k*∈ ℤ (or *ek*, *k*∈ ℤ, another notation) is a fixed orthonormal basis in the Cameron – Martin space **H** defined by

$$\mathbf{H} := \left\{ h : [0, 1] \to \mathbb{R}^d / h(t) = \int\_{[0, t]} h'(s) ds, \ \left( \|h\|\_{\mathbf{H}} \right)^2 := \int\_{[0, 1]} \left| h'(s) \right|^2 ds < \infty \right\},$$

and the above interval in definition of **V**<sup>j</sup> is in the sense of usual order relation of functions. This example exhibits another feature that we require from a multiresolution analysis: invariance of **V**<sup>0</sup> under " integer " translations,

$$
\rho \in \mathbf{V}\_0 \Rightarrow \rho(. - n e\_0) \in \mathbf{V}\_0, \forall n \in \mathbb{Z}. \tag{5}
$$

Because of (4) this implies that if *φ*∈ **V***j*, then *φ :*–2*<sup>j</sup> ne*<sup>0</sup> � � belongs to **V**<sup>j</sup> for all *n*∈ ℤ. Finally, we require also that there exists ϕ∈ **V**<sup>0</sup> so that

$$\left\{\phi\_{0,n}; n \in \mathbb{Z}\right\} \text{ is an orthonormal basis in } \mathbf{V}\_0,\tag{6}$$

where, for all *<sup>j</sup>*, *<sup>n</sup>*<sup>∈</sup> <sup>ℤ</sup>, <sup>ϕ</sup>*<sup>j</sup>*,*<sup>n</sup>*ð Þ *<sup>ω</sup>* <sup>≔</sup> <sup>2</sup>�*j=*<sup>2</sup> ϕ 2�*<sup>j</sup> ω*–*ne*<sup>0</sup> � �. Together, (4) and (6) imply that ϕ*<sup>j</sup>*,*<sup>n</sup>*; *n*∈ ℤ n o is an orthonormal basis for **<sup>V</sup>**<sup>j</sup> for all *<sup>j</sup>*<sup>∈</sup> <sup>ℤ</sup>. In the example given above, the possible choice for ϕ is the indicator Wiener functional for ½ � 0,*e*<sup>0</sup> ≔ f g *ω*∈**W**; 0 ≤*ω*ð Þ*s* ≤*e*0ð Þ*s* , ∀*s*∈½ � 0, 1 , that is, ϕð Þ¼ *ω* 1 if *ω*∈ ½ � 0,*e*<sup>0</sup> , interval in the lattice space (**W**,≤), and ϕð Þ¼ *ω* 0 otherwise. We call ϕ the scaling Wiener

*Bases of Wavelets and Multiresolution in Analysis on Wiener Space DOI: http://dx.doi.org/10.5772/intechopen.104713*

functional of the multiresolution analysis. Note that ϕ depends on the choice of e0. Hence, we will say that ϕ is the scaling Wiener functional in direction e0.

The basic tenet of multiresolution analysis is that whenever a collection of closed subspaces satisfy (1)–(6), then there exists an orthonormal Wiener wavelet basis *ψj*,*k*; *j*, *k*∈ ℤ n o of *<sup>L</sup>*<sup>2</sup> ð Þ *μ* , *<sup>ψ</sup>j*,*k*ð Þ *<sup>ω</sup>* <sup>≔</sup> <sup>2</sup>�*j=*<sup>2</sup> *ψ* 2�*<sup>j</sup> ω*–*ke*<sup>0</sup> � �, such that, for all *φ* ∈*L*<sup>2</sup> ð Þ *μ* ,

$$\prod\_{j=1} \rho = \prod\_j \rho + \sum\_{k \in \mathbb{Z}} \left< \rho, \psi\_{j,k} \right>\_{\mu} \nu\_{j,k},\tag{7}$$

where the bracket h i *:*, *: <sup>μ</sup>* denotes the scalar product in *<sup>L</sup>*<sup>2</sup> ð Þ *μ* . For every *j*∈ ℤ, define **W**<sup>j</sup> to be the complement orthogonal of **V**<sup>j</sup> in **V**<sup>j</sup> – 1. We have

$$\mathbf{V}\_{j-1} = \mathbf{V}\_j \oplus \mathbf{W}\_j \tag{8}$$

and

$$\mathbf{W}\_{\circ} \sqcup \mathbf{W}\_{\circ} \text{ if } j \neq j'. \tag{9}$$

It follows that, for j < j0,

$$\mathbf{V}\_{j} = \mathbf{V}\_{j0} \oplus \left( \oplus\_{k=0}^{j(0)-1} \mathbf{W}\_{j(0)-k} \right),\tag{10}$$

where all these subspaces are orthogonal. By virtue of (2) and (3), this implies

$$L^2(\mu) = \bigoplus\_{k \in \mathbb{Z}} \mathbf{W}\_j,\tag{11}$$

A decomposition of *L*<sup>2</sup> ð Þ *μ* into mutually orthogonal subspaces. Furthermore, the **W**<sup>j</sup> spaces inherit the scaling property (4) from the **V**<sup>j</sup> :

$$
\varphi \in \mathbf{W}\_{\hat{\jmath}} \Leftrightarrow \varphi(\mathcal{D}^{\hat{\jmath}}.) \in \mathbf{W}\_0. \tag{12}
$$

Formula (7) is equivalent to saying that, for fixed *j*, *ψ<sup>j</sup>*,*<sup>k</sup>*; *k*∈ ℤ n o constitutes an orthonormal basis for **W**j. Because of (11), (2) and (3), this then automatically implies that the whole collection *ψ<sup>j</sup>*,*<sup>k</sup>*; *j*, *k*∈ ℤ n o is an orthonormal basis for *<sup>L</sup>*<sup>2</sup> ð Þ *μ* . On the other hand, (12) ensures that if *ψ*0,*<sup>k</sup>*; *k*∈ ℤ � � is an orthonormal basis for **W**0, then *ψ<sup>j</sup>*,*<sup>k</sup>*; *k*∈ ℤ n o will likewise be an orthonormal basis for **<sup>W</sup>**j, for any *<sup>j</sup>*<sup>∈</sup> <sup>ℤ</sup>. Construction of *ψ* can be done as in the case of real analysis, using Fourier–Wiener transform (see [10] for an introduction to this notion) in the place of Fourier transform [11].

Our task thus reduces to finding *ψ* ∈**W**<sup>0</sup> such that the *ψ*ð Þ *:*–*ke*<sup>0</sup> constitute an orthonormal basis for **W**0. Let us first write out some interesting properties of ϕ and **W**0.

1.Since ϕ∈ **V**<sup>0</sup> ⊂ **V**�1, and the ϕ�1,*<sup>n</sup>* are an orthonormal basis in **V**�1, we have

$$
\Phi = \sum\_{n} c\_{n} \Phi\_{-1,n},
\tag{13}
$$

with

$$\left| \mathcal{L}\_n = \left\langle \phi, \left. \phi\_{-1,n} \right\rangle\_{\mu} \right\rangle\_{\mu} \text{ and } \sum\_{n} \left| \mathcal{c}\_n \right|^2 = \mathbf{1}. \tag{14}$$

we can rewrite (13) as either

$$\Phi(o) = 2^{1/2} \sum\_{n} c\_n \Phi(2o - ne\_0) \tag{15}$$

or

$$\hat{\Phi}(\xi) = 2^{-1/2} \sum\_{n} \exp\left[-\mathrm{in}\langle \xi, e\_0 \rangle\_{\mathbf{H}}/2\right] \hat{\Phi}(\xi/2),\tag{16}$$

where the convergence in either sum holds in *L*<sup>2</sup> ð Þ *<sup>μ</sup>* —sense, <sup>ϕ</sup>^ denoting the Fourier– Wiener transform. Formula (16) can be rewritten as

$$
\hat{\Phi}(\xi) = m\_0(\xi/2)\hat{\Phi}(\xi/2),
\tag{17}
$$

where

$$m\_0(\xi) = 2^{-1/2} \sum\_{\mathfrak{n}} c\_{\mathfrak{n}} \exp \left[ -\text{in} \langle \xi, e\_0 \rangle\_{\mathbf{H}} \right]. \tag{18}$$

Equality in (17) holds pointwise *μ*—almost everywhere.

2.The orthonormality of the ϕð Þ *:*–*ke*<sup>0</sup> leads to special properties for m0. We have

$$\begin{split} \delta\_{k,0} &= \int\_{\mathbf{H}} \left| \dot{\Phi}(\xi) \right|^{2} \exp\left[ \mathrm{ik}\langle \xi, e\_{0} \rangle\_{\mathbf{H}} \right] \mu(d\xi) \\ &= \sum\_{l} \int\_{\{\langle \xi, e(0) \rangle \in [2\pi l, 2\pi(l+1)]\}} \left| \dot{\Phi}(\xi) \right|^{2} \exp\left[ \mathrm{ik}\langle \xi, e\_{0} \rangle\_{\mathbf{H}} \right] \mu(d\xi) \\ &= \sum\_{l} \exp\left[ -2\pi l^{2} \right] \int\_{\{\langle h, e(0) \rangle \in [0, 2\pi]\}} \left| \dot{\Phi}(h + 2\pi l e\_{0}) \right|^{2} \exp\left[ \mathrm{ik}\langle h, e\_{0} \rangle\_{\mathbf{H}} \right] . \end{split}$$
 
$$\begin{split} \exp\left[ -2\pi l \int\_{\begin{subarray}{c} \mathbf{0} \\ \langle 0, 1 \rangle \end{subarray}} \underline{\boldsymbol{\varrho}}(\mathbf{s}) d\mathbf{l}(\mathbf{s}) \right] \mu(d\mathbf{h}) \\ &= \int\_{\begin{subarray}{c} \mathbf{0} \\ \langle \boldsymbol{\vartheta}, \mathbf{e}(0) \rangle \in [0, 2\pi] \end{subarray}} \left| \dot{\boldsymbol{\varrho}}(\mathbf{h} + 2\pi l e\_{0}) \right|^{2} \exp\left[ \mathrm{ik}\langle \boldsymbol{\vartheta}, e\_{0} \rangle\_{\mathbf{H}} \right] \mu(d\mathbf{h}), \end{split}$$

Implying

$$\sum\_{l} \left| \hat{\Phi}(h + 2\pi l e\_0) \right|^2 = (2\pi)^{-1} \mu - \text{a.e.}\tag{19}$$

Substituting (17) leads to

$$\sum\_{l} \left| m\_0(\xi + \pi l e\_0) \right|^2 \left| \hat{\Phi}(\xi + \pi l e\_0) \right|^2 = (2\pi)^{-1};$$

*Bases of Wavelets and Multiresolution in Analysis on Wiener Space DOI: http://dx.doi.org/10.5772/intechopen.104713*

Splitting the sum into even and odd l, using the periodicity of m0 and applying (19) gives

$$|m\_0(\xi)|^2 + |m\_0(\xi + \pi e\_0)|^2 = \mathbf{1}\,\mu - \mathbf{a.e.}\tag{20}$$

1.Let us now characterize **W**0. *φ*∈**W**<sup>0</sup> is equivalent to *φ*∈ **V**�<sup>1</sup> and *φ* is orthogonal to **V**0. Since *φ*∈ **V**�1, we have

 $\rho = \sum\_{n} \lambda\_n \Phi\_{-1,n}$ , with  $\lambda\_n = \left\langle \rho, \ \Phi\_{-1,n} \right\rangle\_{\mu}$ . This implies  $\rho$ 

$$\hat{\rho}(\xi) = 2^{-1/2} \sum\_{n} \lambda\_n \exp\left(-\text{in}\langle \xi, e\_0 \rangle\_{\mathbf{H}}/2\right) \hat{\Phi}\left(2^{-1}\xi\right) = m\_{\varphi}(2^{-1}\xi)\hat{\Phi}(2^{-1}\xi),\tag{21}$$

where

$$m\_{\boldsymbol{\uprho}}(\boldsymbol{\xi}) = \mathcal{Z}^{-1} \sum\_{n} \lambda\_{n} \exp\left(-\text{in}\langle \boldsymbol{\xi}, \boldsymbol{e}\_{0} \rangle\_{\mathbf{H}}\right);\tag{22}$$

*m<sup>φ</sup>* is a 2*πe*0-periodic Wiener functional; convergence in (22) holds pointwise *μ*—a. e.. The constraint " *φ* orthogonal to **V**<sup>0</sup> " implies that *φ* is orthogonal to Φ0,*<sup>k</sup>* for all k, that is,

$$\int\_{\mathbf{H}} (\hat{\rho}(h)) \left( \overline{\hat{\Phi}(h)} \right) \, \exp\left(ik \langle h, e\_0 \rangle\_{\mathbf{H}}\right) \mu(dh) = \mathbf{0}$$

where zc denotes the conjugate complex of complex z, or

$$\int\_{\{\langle h,e\_0\rangle \in [0,2\pi]\}} \exp\left(\mathrm{ik}\langle h,e\_0\rangle\_{\mathrm{H}}\right) \sum\_{l} \left(\hat{\rho}(h+2\pi l e\_0) \overline{\hat{\Phi}(h+2\pi l e\_0)}\right) \mu(dh) = \mathbf{0};$$

note that, out of the ordinary, we denoted by *h*∈ **H** the integration variable. Hence

$$\sum\_{l} \left( \hat{\rho}(h + 2\pi l e\_0) \overline{\hat{\phi}(h + 2\pi l e\_0)} \right) = 0,\tag{23}$$

where the series in (23) converges absolutely in *L*<sup>1</sup> ð Þ *μ* . Substituting (17) and (21), regrouping the sums for odd and even l (which we are allowed to do as we have an absolutely convergence), and using (19) leads to

$$m\_{\wp}(h)\left(\overline{m\_0(h)}\right) + m\_{\wp}(h + \pi e\_0)\left(\overline{m\_0(h + \pi e\_0)}\right) = \mathbf{0} \text{ } \mu-\text{a.e.}\tag{24}$$

Since *m*0ð Þ *h* � � and *<sup>m</sup>*0ð Þ *<sup>h</sup>* <sup>þ</sup> *<sup>π</sup>e*<sup>0</sup> � � cannot vanish together on a set of nonzero Wiener measure (because of (20)), this implies the existence of a 2*πe*0-periodic Wiener functional ϴ so that

$$m\_{\wp}(h) = \Theta(h)\Big(\overline{m\_0(h + \pi e\_0)}\Big)\,\mu - \text{a.e.}\tag{25}$$

and

$$\left(\Theta(h) + \Theta(h + \mathfrak{ne}\_0) = \mathbf{0}\,\,\mu - \mathbf{a}.\mathbf{e}.\tag{26}$$

This last equation can be rewrite as

$$
\Theta(h) = e(i \langle h, e\_0 \rangle\_{\mathbf{H}}) \nu(\mathcal{D}h),
\tag{27}
$$

where *ν* is a 2*πe*0-periodic Wiener functional. Substituting (25, 27) into (21) gives

$$
\hat{\rho}(\xi) = \exp\left(\mathrm{i}\langle h, e\_0 \rangle\_{\mathrm{H}}/2\right) \overline{\left(m\_0(\xi/2 + \pi e\_0)\right)} \nu(\xi) \hat{\Phi}(\xi/2). \tag{28}
$$

3.The general form (28) for the Fourier–Wiener transform of *φ*∈**W**<sup>0</sup> suggests that we take

$$\hat{\Psi}(\xi) = \exp\left(\mathrm{i}\langle h, e\_0 \rangle\_{\mathrm{H}}/2\right) \overline{\left(m\_0(\xi/2 + \pi e\_0)\right)} \,\hat{\Phi}(\xi/2) \tag{29}$$

as a candidate for our wavelet. Forgetting convergence questions, (28) can indeed be written as

$$\hat{\rho}(\xi) = \left(\sum\_{k} \nu\_k \exp\left(-\mathrm{ik}\langle \xi, e\_0 \rangle\_{\mathbf{H}}\right)\right) \hat{\Psi}(\xi)$$

or

$$
\varphi = \sum\_{k} \nu\_k \Psi(.-ke\_0),
$$

so that the Ψð Þ *:*–*ne*<sup>0</sup> are a good candidate for a basis of **W**0. We need to verify that the Ψ0,*<sup>k</sup>* are indeed an orthonormal basis for **W**0. First, the properties of m0 and Φ^ ensure that (29) defines an *L*<sup>2</sup> ð Þ *μ* -Wiener functional belonging to **V**�<sup>1</sup> and orthogonal to **V**<sup>0</sup> (by the analysis above), so that Ψ ∈**W**0. Orthonormality of the Ψ0,*<sup>k</sup>* is easy to check:

$$\begin{split} \int\_{\mathbf{H}} \Psi(h) \left( \overline{\Psi(h - m\varepsilon\_0)} \right) \mu(dh) &= \int\_{\mathbf{H}} \left| \dot{\Psi}(\xi) \right|^2 \exp \left( \text{im} \langle \xi, \varepsilon\_0 \rangle\_{\mathbf{H}} \right) \mu(d\xi) \\ &= \int\_{\{\langle \xi, \varepsilon\_0 \rangle \in [\mathbf{0}, 2\pi] \}} \exp \left( \text{im} \langle \xi, \varepsilon\_0 \rangle\_{\mathbf{H}} \right) \sum\_{l} \left| \dot{\Psi}(\xi + 2\pi l \varepsilon\_0) \right|^2 \mu(d\xi) . \end{split}$$

Now

$$\begin{split} \sum\_{l} \left| \hat{\Psi}(\xi + 2\pi l e\_{0}) \right|^{2} &= \sum\_{l} |m\_{0}(\xi/2 + \pi l e\_{0} + \pi e\_{0})|^{2} \left| \hat{\Phi}(\xi/2 + \pi l e\_{0}) \right|^{2} \\ &= \left| m\_{0}(\xi/2 + \pi e\_{0}) \right|^{2} \sum\_{\mathfrak{n}} \left| \hat{\Phi}(\xi/2 + 2\pi m e\_{0}) \right|^{2} \\ &\quad + \left| m\_{0}(\xi/2) \right|^{2} \sum\_{\mathfrak{n}} \left| \hat{\Phi}(\xi/2 + \pi e\_{0} + 2\pi m e\_{0}) \right|^{2} \\ &= (2\pi)^{-1} \left( |m\_{0}(\xi/2)|^{2} + |m\_{0}(\xi/2 + \pi e\_{0})|^{2} \right) \mu - \text{a.e.} \left( \text{by (1.19)} \right) \\ &= (2\pi)^{-1} \mu - \text{a.e.} \left( \text{by (1.20)} \right). \end{split}$$

Hence Ð **H** Ψð Þ *h* Ψð Þ *h*–*me*<sup>0</sup> � � *<sup>μ</sup>*ð Þ¼ *dh <sup>δ</sup>m*,0. In order to check that the <sup>Ψ</sup>0,*<sup>m</sup>* are indeed a basis for all **<sup>W</sup>**0, it then suffices to check that any *<sup>φ</sup>*∈**W**<sup>0</sup> can be written as *<sup>φ</sup>* <sup>¼</sup> <sup>P</sup> *<sup>n</sup>γn*Ψ0,*n*, with P *<sup>n</sup> <sup>γ</sup><sup>n</sup>* j j<sup>2</sup> <sup>&</sup>lt; <sup>∞</sup>, or

$$
\hat{\rho}(\xi) = \chi(\xi)\hat{\Psi}(\xi),
\tag{30}
$$

with *γ* a square integrable 2*πe*0-periodic Wiener functional. Let us return to (28). We have *φ ξ* ^ð Þ¼ *ν ξ*ð ÞΨ^ð Þ*<sup>ξ</sup>* , with *<sup>ν</sup>*<sup>∈</sup> *<sup>L</sup>*<sup>2</sup> ð Þ *μ* . By (22),

$$\int\_{\mathbf{H}} \left| m\_{\varrho}(\xi) \right|^2 \mu(d\xi) = \pi \sum\_{n} |\lambda\_n|^2 = \pi \left( ||\varrho||\_{\mu} \right)^2 < \infty.$$

On the other hand, by (25),

$$\int\_{\mathcal{H}} \left| m\_{\rho}(\xi) \right|^{2} \mu(d\xi) = \int\_{\mathcal{H}} \left| \Theta(\xi) \right|^{2} \left| m\_{0}(\xi + \pi e\_{0}) \right|^{2} \mu(d\xi)$$

$$= \int\_{\mathcal{H}} \left| \Theta(\xi + \pi e\_{0}) \right|^{2} \left| m\_{0}(\xi + \pi e\_{0}) \right|^{2} \mu(d\xi), \text{using (1.26)}\tag{31}$$

Now, with the change of variable *h* ¼ *ξ* þ *πe*<sup>0</sup> and with the help of stochastic calculus we find for this integral

$$\begin{split} &\mathbb{E}\exp\left[\left(\mathbf{1}-\boldsymbol{\pi}^{2}\right)/2\right] \int\_{\mathbf{H}} |\Theta(h)|^{2} |m\_{0}(h)|^{2} \mu(dh) \\ &= \mathbb{E}\exp\left[\left(\mathbf{1}-\boldsymbol{\pi}^{2}\right)/2\right] \int\_{\mathbf{H}} |\Theta(h)|^{2} \left(\mathbf{1}-|m\_{0}(h+\boldsymbol{\pi}e\_{0})|^{2}\right) \mu(dh), \text{using (1.20),} \\ &= \exp\left[\left(\mathbf{1}-\boldsymbol{\pi}^{2}\right)/2\right] \int\_{\mathbf{H}} |\Theta(h)|^{2} \mu(dh) \\ &= \exp\left[\left(\mathbf{1}-\boldsymbol{\pi}^{2}\right)/2\right] \int\_{\mathbf{H}} |\Theta(h+\boldsymbol{\pi}e\_{0})|^{2} |m\_{0}(h+\boldsymbol{\pi}e\_{0})|^{2} \mu(dh). \end{split}$$

Put: *I* ≔ Ð **H** j j Θð Þ *ξ* þ *πe*<sup>0</sup> 2 j j *m*0ð Þ *ξ* þ *πe*<sup>0</sup> 2 *μ*ð Þ *dξ* . We hence have, combining (31) and this last equality:

$$I = \exp\left[\left(\mathbf{1} - \boldsymbol{\pi}^2\right)/2\right] \int\_{\mathbf{H}} |\Theta(h)|^2 \mu(dh) - \exp\left[\left(\mathbf{1} - \boldsymbol{\pi}^2\right)/2\right] I,\text{ which implies}$$

$$I = \exp\left[\left(\mathbf{1} - \boldsymbol{\pi}^2\right)/2\right]. (\mathbf{1} + \exp\left[\left(\mathbf{1} - \boldsymbol{\pi}^2\right)/2\right]^{-1} \int\_{\mathbf{H}} |\Theta(h)|^2 \mu(dh).$$

Hence, k k*ν <sup>μ</sup>* � �<sup>2</sup> ¼ 2*π φ*k k*<sup>μ</sup>* � �<sup>2</sup> < ∞, and *φ* is of the form (30) with *γ* ∈*L*<sup>2</sup> ð Þ *μ* and 2*πe*0-periodic. We have thus prove the following theorem.

**Theorem 1.1.** If a ladder of closed subspaces **V***<sup>j</sup>* � � *<sup>j</sup>*<sup>∈</sup> <sup>ℤ</sup> in *<sup>L</sup>*<sup>2</sup> ð Þ *μ* satisfies the conditions (1)–(6), then there exists an orthonormal Wiener wavelet basis Ψ*j*,*k*; *j*, *k*∈ ℤ � � for *L*<sup>2</sup> ð Þ *μ* such that

$$
\Pi\_{\dot{\jmath}-1} = \Pi\_{\dot{\jmath}} + \sum\_{k} \langle \,, \Psi\_{\dot{\jmath},k} \rangle\_{\mu} \Psi\_{\dot{\jmath},k} . \tag{32}
$$

One possibility for the construction of the Wiener wavelet Ψ is,

$$
\hat{\Psi}(\xi) = \exp\left(\mathbf{i}\langle \xi, e\_0 \rangle\_{\mathbf{H}}/2\right) \overline{\left(m\_0(\xi/2 + \pi e\_0)\right)} \hat{\Phi}(\xi/2),
$$

(with m0 as defined by (14) and (18)), or equivalently

$$
\Psi = \sum\_{n} (-1)^{n} c\_{-n-1} \Phi\_{-1,n},
\tag{33}
$$

<sup>Ψ</sup>ð Þ¼ *<sup>ω</sup>* <sup>2</sup>�1*=*<sup>2</sup> <sup>P</sup> *n* ð Þ �<sup>1</sup> *<sup>n</sup> <sup>c</sup>*�*n*�<sup>1</sup> <sup>Φ</sup>ð Þ <sup>2</sup>*ω*–*ne*<sup>0</sup> , with convergence in this series in *<sup>L</sup>*<sup>2</sup> ð Þ *μ* –

sense.

All the argument we hold for the proof of the above theorem is exactly the same which can be found in any book on this subject (in the finite dimension case). We can hence follow the Mallat's construction [5], *via* a multiresolution analysis, of orthonormal wavelets for *μ*-square integrable Wiener functionals. The reader can also take a look on Meyer's books [12, 13]. An extension to wavelets generated by a finite set of Wiener functionals can easily be done following arguments of Goodman–Lee and Tang paper [6]. We give now in next section an example of multiresolution approximation generated by cardinal Hermite *B*-splines in *L*<sup>2</sup> ð Þ *μ* , as we can find it in [6] for the one-dimensional case.

## **1.2 Multiresolution approximation generated by cardinal B-splines in L<sup>2</sup>**ð Þ *<sup>μ</sup>*

Let us first beginning with some recalls. For n, r positive integers, n even, such that *n*≥ 2*r*, put

$$S\_n \coloneqq \left\{ f \in \mathcal{C}^{n-r-1}(\mathbb{R}) : f\_{|[p,p+1]} \in \mathbf{P}\_{n-1}, v \in \mathbb{Z} \right\},\tag{34}$$

where **P**n-1 is the class of polynomials of degree ≤*n*–1. Functions in S*<sup>r</sup> <sup>n</sup>* are called cardinal Hermite splines of degree≤ than n–1.

For *j* ¼ 0, … ,*r*–1, let

$$S\_{n,j}^r := \left\{ f \in S\_n^r : f^{(k)}(v) = 0, v \in \mathbb{Z}, k = 0, \dots, r - 1, k \neq j \right\}.\tag{35}$$

The space *S<sup>r</sup> <sup>n</sup>*,*<sup>j</sup>* has a basis consisting of integer translates of a function *Nn*,*<sup>j</sup>* ¼ *Nr <sup>n</sup>*,*<sup>j</sup>* ∈ *Sr n*,*j* , *j* ¼ 0, … ,*r*–1, with minimal support ½ � �*n=*2–1 þ *r*, *n=*2 þ 1 � *r* , in the sense that every *f* ∈ *Sr <sup>n</sup>*,*<sup>j</sup>* has a unique representation of the form

$$f(\mathbf{x}) = \sum\_{\nu \in \mathbb{Z}} c\_{\nu} N\_{n,j}(\mathbf{x} - \nu), \mathbf{x} \in \mathbb{R}, \tag{36}$$

(see [14]). The functions *N*n,j are called cardinal Hermite *B*-splines and their Fourier transforms are given by (see [14])

$$\hat{N}\_{n,j}(u) = \left[2\sin\left(u/2\right)\right]^n \left|H\_{rj}(a\_n(u))\right| \tag{37}$$

where *Hr*,*j*ð Þ *αn*ð Þ *u* denotes the matrix obtained from the Hankel matrix *Hr*ð Þ *αn*ð Þ *u* of order r by replacing the ð Þ *<sup>j</sup>* <sup>þ</sup> <sup>1</sup> th column by *un*, *un*�1, … , *un*�*r*þ<sup>1</sup> ð Þ*<sup>T</sup>* ,

*j* ¼ 0, … ,*r*–1, denoting by j j *Hr*ð Þ *αn*ð Þ *u* its determinant.

Consider the map *K* : ℓ<sup>2</sup> ð Þ <sup>ℤ</sup> *<sup>r</sup>* ! *<sup>L</sup>*<sup>2</sup> ð Þ *μ* defined by the following:

$$K(s)(\boldsymbol{\omega}(.)) \coloneqq \sum\_{\boldsymbol{j}:\boldsymbol{0}\to\boldsymbol{r}-\boldsymbol{1}} \sum\_{\boldsymbol{v}\in\mathbb{Z}} s\_{\boldsymbol{j}}(\boldsymbol{v}) \mathbf{T}^{\boldsymbol{v}} N\_{n\boldsymbol{j}}(\boldsymbol{\omega}(.)), \boldsymbol{\omega} \in \mathbf{C}([\boldsymbol{0},\boldsymbol{1}]),\tag{38}$$

**T** being a unitary operator on *L*<sup>2</sup> ð Þ *μ* .

**Theorem 2.1.** The map K defined by (38) is an isomorphism of ℓ<sup>2</sup> ð Þ <sup>ℤ</sup> *<sup>r</sup>* onto a subspace of *L*<sup>2</sup> ð Þ *μ* .

This above theorem is an easy consequence of Theorem 4.1 in [6]. Let us denote by ~ S*r <sup>n</sup>*ð Þ *<sup>μ</sup>* the range of K. This is a closed subspace of *<sup>L</sup>*<sup>2</sup> ð Þ *<sup>μ</sup>* . Furthermore, <sup>~</sup> S*r <sup>n</sup>*ð Þ¼ *μ* S*r <sup>n</sup>*ð Þ *<sup>μ</sup>* <sup>∩</sup> *<sup>L</sup>*<sup>2</sup> ð Þ *<sup>μ</sup>* where S*<sup>r</sup> <sup>n</sup>*ð Þ *<sup>μ</sup>* is the space deduced from S*<sup>r</sup> <sup>n</sup>* by the following: *φ*∈ *Sr <sup>n</sup>*ð Þ *<sup>μ</sup>* <sup>⇔</sup>*φ ω*ð Þ ð Þ*<sup>t</sup>* <sup>∈</sup>S*<sup>r</sup> <sup>n</sup>*, *ω*∈*C*ð Þ ½ � 0, 1 , *t*∈½ � 0, 1 . Therefore, if *Dφ ω*ð Þ ≔ *φ*ð Þ 2*ω* , we have as in real analysis:

The closed space generated by ∪ *m* ∈ ℤ *D<sup>m</sup>* ~ S*r <sup>n</sup>*ð Þ *<sup>μ</sup>* in *<sup>L</sup>*<sup>2</sup> ð Þ *μ* contains the one generated

by ∪ *m* ∈ ℤ *<sup>D</sup><sup>m</sup>* <sup>~</sup> S1 *<sup>n</sup>*ð Þ *<sup>μ</sup>* which is *<sup>L</sup>*<sup>2</sup> ð Þ *μ* . We also have:

$$\bigcap\_{m \in \mathbb{Z}} D^m \bar{\mathbf{S}}\_n^{\bar{r}}(\mu) = \{\mathbf{O}\}.$$

Let **V***<sup>m</sup>* ≔ *D<sup>m</sup>* ~ S*r <sup>n</sup>*ð Þ *μ* , *m* ∈ ℤ. Then, ð Þ **V***<sup>m</sup> <sup>m</sup>* <sup>∈</sup> <sup>ℤ</sup> is a multiresolution approximation of *L*2 ð Þ *μ* . We shall call ð Þ **V***<sup>m</sup> <sup>m</sup>* <sup>∈</sup> <sup>ℤ</sup> a Wiener Hermite spline multiresolution approximation of *L*<sup>2</sup> ð Þ *μ* . We could go on and hence extend all results of T.N.T. Goodman, S.L. Lee and W.S. Tang paper [6]. We now prefer in next section deal with unconditional bases of *<sup>L</sup><sup>p</sup>*ð Þ *<sup>μ</sup>* .

## **1.3 Unconditional bases for L<sup>p</sup>**ð Þ **<sup>μ</sup>**

Orthonormal bases of wavelets in *L*<sup>2</sup> ð Þ *μ* give good (i.e., unconditional) bases for many other spaces than L<sup>2</sup> . We start by proving the following extension of Calderon– Zygmund decomposition theorem. We denote in this section by j j *ω* for *<sup>ω</sup>*∈W<sup>≔</sup> C0 ½ � 0, 1 ; <sup>ℝ</sup>*<sup>d</sup>* � � the sup norm [15, 16].

**Theorem 3.1.** Assume φ to be a positive Wiener functional in *L*<sup>1</sup> ð Þ *μ* . Fix *<sup>α</sup>*>0. Then the Wiener space **<sup>W</sup>**<sup>≔</sup> C0 ½ � 0, 1 ; <sup>ℝ</sup>*<sup>d</sup>* � � can be decomposed as follows:

1.**W** ¼ **G** ∪ **B** with **G** ∩ **B** ¼ ∅;

2.On the " good " set **G**, *φ ω*ð Þ≤*α μ* - a. e.;

3.The " bad " set **B** can be written as **B** ¼ ∪ *k* ∈ℕ **Q***k*, where the **Q**<sup>k</sup> are nonoverlapping intervals in the Banach lattice **W**, and

$$a \le \mu(\mathbf{Q}\_k)^{-1} \int\_{\mathbf{Q}\_k} \rho(a) \,\mu(dao) \le 2\alpha, \forall k \in \mathbb{N}.$$

The proof of this theorem is identical with the proof of Theorem 9.1.1, p.289 of chapter 9 in Daubechies book [8, 17]. Next, we define Calderon–Zygmund operators for square integrable Wiener functionals and extend a classical property [18].

**Definition3.2.** A Calderon–Zygmund operator T on the Wiener space **W** is an integral operator

$$(T\wp)(\omega) = \int\_{\mathbf{H}} K(\boldsymbol{\alpha}, \boldsymbol{\xi}) \mu(d\boldsymbol{\xi}) \tag{39}$$

for which the integral kernels satisfies

$$|K(\alpha, \xi)| \le \mathcal{C} / |\alpha - \xi|,\tag{40}$$

$$\left\|\left|\nabla\_{\alpha}K(\alpha,\xi)\right\|\right\|\_{\mathbf{H}} + \left\|\left|\nabla\_{\xi}K(\alpha,\xi)\right\|\right\|\_{\mathbf{H}} \leq C/|\alpha-\xi|,\tag{41}$$

where the derivation symbol ∇ is the Malliavin derivative, and which defines a bounded operator on *L*<sup>2</sup> ð Þ *μ* .

**Theorem 3.3.** A Calderon–Zygmund operator on **W** is also a bounded operator from *L*<sup>1</sup> ð Þ *<sup>μ</sup>* to *<sup>L</sup>*<sup>1</sup> *weak*ð Þ *μ* .

We recall below the definition of *L*<sup>1</sup> *weak*ð Þ *μ* . **Definition 3.4.** *φ* ∈*L*<sup>1</sup> *weak*ð Þ *μ* if there exists *C*>0 so that, for all *α*> 0,

$$\mu\{a \in \mathbf{W} / |\varrho(a)| \ge a\} \le \mathbf{C}/a. \tag{42}$$

Like the proof of Theorem 3.1, this theorem is the extension of Theorem 9.1.2, p.291, chapter 9 in Daubechies book [8, 17]. We do not reproduce it as it is identical to the proof of Theorem 9.1.2 in [8]. The infimum of all C for which (42) holds (for all *α* >0) will be called k k*φ <sup>L</sup>*<sup>1</sup> *weak* . Note that this notation is abusive as it is not a " true " norm.

Now, let T be a Calderon–Zygmund operator on **W**. As T maps *L*<sup>2</sup> ð Þ *<sup>μ</sup>* to *<sup>L</sup>*<sup>2</sup> ð Þ *μ* and *L*1 ð Þ *<sup>μ</sup>* to *<sup>L</sup>*<sup>1</sup> *weak*ð Þ *<sup>μ</sup>* , we can extend T to other *Lp*ð Þ *<sup>μ</sup>* -spaces by interpolation theorem of Marcinkiewicz.

**Theorem 3.5.** If an operator T satisfies

$$||T\rho||\_{Lq(1)(\mu)\\\\omega} \leq C\_1 ||\rho||\_{Lp(1)(\mu)},\tag{43}$$

$$||\varrho||\_{Lq(\mathfrak{L})(\mu)\\
weak \le C\_{\mathfrak{L}}||\varrho||\_{Lp(\mathfrak{L})(\mu)},\tag{44}$$

where *q*<sup>1</sup> ≤ *p*1, *q*<sup>2</sup> ≤*p*2, then for 1*=p* ¼ *t=p*<sup>1</sup> þ ð Þ 1–*t =p*2,

1*=q* ¼ *t=q*<sup>1</sup> þ ð Þ 1–*t =q*2, with 0<*t* <1, there exists a constant K, depending on p1,q1, p2,q2, and t, so that

$$\|\|Tq\|\|\_{Lq(\mu)} \le K \|q\rho\|\|\_{Lp(\mu)}.$$

Here *Lq*ð Þ *μ weak* stands for the space of all Wiener functionals *φ* for which k k*<sup>φ</sup> Lq*ð Þ *<sup>μ</sup> weak* <sup>≔</sup> inf *<sup>C</sup>=μ ω*f g ; j j *φ ω*ð Þ <sup>≥</sup>*<sup>α</sup>* <sup>≤</sup>*Cα*�*<sup>q</sup>* ð Þ f g for all *<sup>α</sup>* <sup>&</sup>gt;<sup>0</sup> <sup>1</sup>*=<sup>q</sup>* is finite.

The proof of this theorem can be found in E. Stein and G. Weiss [19] for the finite dimensional case. Extension to Wiener functionals can easily be done from this proof. Notice that with this theorem it only needs weaker bounds at the two extrema, and nevertheless derives bounds on *Lq* ð Þ *<sup>μ</sup>* -norms (not *<sup>L</sup><sup>q</sup> weak*ð Þ *μ* ) for intermediate values q. The Marcinkiewicz interpolation theorem implies that the *L*<sup>1</sup> ð Þ! *<sup>μ</sup> <sup>L</sup>*<sup>1</sup> *weak*ð Þ *μ* -boundedness proved in Theorem 3.3 is sufficient to derive *<sup>L</sup>p*ð Þ! *<sup>μ</sup> <sup>L</sup>p*ð Þ *<sup>μ</sup>* boundedness for 1<*p* < ∞, as follows.

**Theorem 3.6.** If T is an integral operator with integral kernel K satisfying (40, 41), and if T is bounded from *L*<sup>2</sup> ð Þ *<sup>μ</sup>* to *<sup>L</sup>*<sup>2</sup> ð Þ *μ* , then T extends to a bounded operator from *Lp*ð Þ *<sup>μ</sup>* to *Lp*ð Þ *<sup>μ</sup>* for all *<sup>p</sup>*∈� ½ 1, <sup>∞</sup> .

**Proof.**


$$\int (T^\*\,\rho)(\alpha) \left(\overline{\psi(\alpha)}\right) \,\mu(d\alpha) = \int \rho(\alpha) \left(\overline{(T\psi)(\alpha)}\right) \,\mu(d\alpha).$$

It is associated with the integral kernel *K*0ð Þ¼ *ω*, *ξ K*ð Þ *ξ*,*ω* � �, which also satisfies the conditions (40, 41). This adjoint operator T\* is exactly the adjoint operator used in operator theory on Hilbert spaces. It follows from Theorem 3.3 that T\* is bounded from *L*<sup>1</sup> ð Þ *<sup>μ</sup>* to *<sup>L</sup>*<sup>1</sup> *weak*ð Þ *<sup>μ</sup>* , and hence by Theorem 3.5, that it is bounded from *Lp*ð Þ *<sup>μ</sup>* to *Lp*ð Þ *<sup>μ</sup>* for 1<*p*<sup>≤</sup> 2. Since for 1*=<sup>p</sup>* <sup>þ</sup> <sup>1</sup>*=<sup>q</sup>* <sup>¼</sup> 1, T\* : *<sup>L</sup><sup>p</sup>*ð Þ! *<sup>μ</sup> <sup>L</sup><sup>p</sup>*ð Þ *<sup>μ</sup>* is the adjoint operator of T: *Lq* ð Þ! *<sup>μ</sup> Lq* ð Þ *μ* , it follows that T is bounded for 2 ≤*q*< ∞. More explicitly, for readers unfamiliar with adjoints on Banach spaces,

$$\|T\varphi\|\_{q} = \sup\_{\Psi \in L^{r}, \|\varphi\|=1} \left| \left( T\varphi \right)(a) \left( \overline{\varphi(a)} \right) \,\mu(da) \right| \left( \text{if } 1/p + 1/q = 1 \right).$$

$$= \sup\_{\Psi \in L^{r}, \|\varphi\|=1} \left| \left( \int \left( \rho(\xi) K(a), \xi \right) \overline{\varphi(a\cdot)} \right) \,\mu(d\xi) \right| \,\mu(da) \right| $$

$$= \sup\_{\Psi \in L^{r}, \|\varphi\|=1} \left| \int \rho(\xi) (T^{\*}\,\varphi)(\xi) \,\mu(d\varphi) \right| $$

$$\leq \sup\_{\Psi \in L^{r}, \|\varphi\|=1} ||\rho||\_{L^{q}(\mu)} \quad \|T^{\*}\,\varphi\|\_{L^{r}(\mu)} \leq C ||\varphi||\_{L^{q}(\mu)}.$$

We now apply this to prove that a Wiener functional has some decay and some regularity and if the <sup>Ψ</sup>*<sup>j</sup>*,*<sup>k</sup>*ð Þ *<sup>ω</sup>* <sup>≔</sup> <sup>2</sup>�*j=*<sup>2</sup> <sup>2</sup>�*<sup>j</sup> ω*–*ke*<sup>0</sup> � �, *j*, *k*∈ ℤ, *ω* ∈**W**, and where e0 belongs to the Cameron–Martin space **H**, constitute an orthonormal basis for *L*<sup>2</sup> ð Þ *μ* , then the <sup>Ψ</sup>*<sup>j</sup>*,*<sup>k</sup>* also provide unconditional bases for *<sup>L</sup><sup>p</sup>*ð Þ *<sup>μ</sup>* , 1<*<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>.

What we need to prove is that if *<sup>φ</sup>* <sup>¼</sup> <sup>P</sup> *<sup>j</sup>*,*k*<sup>∈</sup> <sup>ℤ</sup>*cj*,*<sup>k</sup>*Ψ*<sup>j</sup>*,*<sup>k</sup>* <sup>∈</sup>*Lp*ð Þ *<sup>μ</sup>* , then P *<sup>j</sup>*,*k*<sup>∈</sup> <sup>ℤ</sup>*ε<sup>j</sup>*,*kcj*,*<sup>k</sup>*Ψ*<sup>j</sup>*,*<sup>k</sup>* <sup>∈</sup> *Lp*ð Þ *<sup>μ</sup>* for any choice of the *<sup>ε</sup><sup>j</sup>*,*<sup>k</sup>* ¼ �1 (see Preliminaries in [8]).

We will assume that *ψ* is continuously Malliavin differentiable on the lattice space (**W**, |.|, ≤) where |.| denotes the sup norm, and that both *ψ*, ∇*ψ* decay faster than ð Þ <sup>1</sup> <sup>þ</sup> j j *<sup>ω</sup>* �1:

$$|\psi(\alpha)|, \|\nabla\psi(\alpha)\|\_{\mathbf{H}} \le \mathbf{C} (\mathbf{1} + |\alpha|)^{-1-r}.\tag{45}$$

Then <sup>Ψ</sup> <sup>∈</sup>*Lp*ð Þ *<sup>μ</sup>* for 1<*<sup>p</sup>* <sup>&</sup>lt; <sup>∞</sup>, and *<sup>φ</sup>* <sup>¼</sup> <sup>P</sup> *j*, *k cj*,*kψj*,*<sup>k</sup>* implies that *cj*,*<sup>k</sup>* ¼

Ð **H** *φ ω*ð Þ*ψj*,*k*ð Þ *ω μ*ð Þ *dω* because of the orthonormality of the *ψj*,*k*. We therefore want to show that, for aby choice of the *εj*,*<sup>k</sup>* ¼ �1, *T<sup>ε</sup>* defined by

$$T\_{\mathfrak{e}}\mathfrak{q} = \sum\_{j,k} \mathfrak{e}\_{j,k} \left\langle \mathfrak{q}, \left. \Psi\_{j,k} \right\rangle\_{L^2(\mu)} \mathfrak{w}\_{j,k} \right.$$

Is a bounded operator from *Lp*ð Þ *<sup>μ</sup>* to *<sup>L</sup><sup>p</sup>*ð Þ *<sup>μ</sup>* . First, we know that *<sup>T</sup><sup>ε</sup>* is bounded from *L*2 ð Þ *<sup>μ</sup>* to *<sup>L</sup>*<sup>2</sup> ð Þ *μ* , since, denoting by k k *: <sup>μ</sup>* (resp. <.,. > <sup>μ</sup>) the norm (resp. the scalar product) in *L*<sup>2</sup> ð Þ *μ* ,

$$\left(\|\|T\_{\epsilon}\rho\|\|\_{\mu}\right)^{2} = \sum\_{j,k} \left|e\_{j,k}\left\langle\rho,\psi\_{j,k}\right\rangle\_{\mu}\right|^{2} = \sum\_{j,k} \left|\left\langle\rho,\psi\_{j,k}\right\rangle\_{\mu}\right|^{2} = \left(\left\|\|\rho\|\right\|\_{\mu}\right)^{2}\lambda$$

so the *Lp*ð Þ *<sup>μ</sup>* -boundedness will follow by Theorem 3.5 if we can prove that *<sup>T</sup><sup>ε</sup>* is an integral operator with kernel satisfying (40, 41). This is the content of the following lemma.

**Lemma 3.7.** Choose *<sup>ε</sup><sup>j</sup>*,*<sup>k</sup>* ¼ �1, and define *<sup>K</sup>*ð Þ *<sup>ω</sup>*, *<sup>ξ</sup>* <sup>≔</sup><sup>P</sup> *j*, *k ε<sup>j</sup>*,*<sup>k</sup>ψ<sup>j</sup>*,*<sup>k</sup>*ð Þ *ω ψ<sup>j</sup>*,*<sup>k</sup>*ð Þ*ξ* � �. Then there exists C > 0 so that

$$|K(\boldsymbol{\alpha}, \boldsymbol{\xi})| \le \mathsf{C} |\boldsymbol{\alpha} - \boldsymbol{\xi}|^{-1}$$

and

$$\left\| \left| \nabla\_{\boldsymbol{\alpha}} \boldsymbol{K}(\boldsymbol{\alpha}, \boldsymbol{\xi}) \right| \right\|\_{\mathbf{H}} + \left\| \left| \nabla\_{\boldsymbol{\xi}} \boldsymbol{K}(\boldsymbol{\alpha}, \boldsymbol{\xi}) \right| \right\|\_{\mathbf{H}} \leq \mathsf{C} |\boldsymbol{\alpha} - \boldsymbol{\xi}|^{-2}.$$

**Proof.**

$$\begin{split} |K(\boldsymbol{\alpha},\boldsymbol{\xi})| &\leq \sum\_{j,k} \left| \boldsymbol{\nu}\_{j,k}(\boldsymbol{\alpha}) \right| \cdot \left| \boldsymbol{\nu}\_{j,k}(\boldsymbol{\xi}) \right| \\ \text{1.} &\quad \leq C \sum\_{j,k} 2^{-j} \left( 1 + \left| 2^{-j} \boldsymbol{\alpha} \cdot \boldsymbol{k} \boldsymbol{\epsilon}\_{0} \right| \right)^{-1-\varepsilon} \left( 1 + \left| 2^{-j} \boldsymbol{\xi} - \boldsymbol{k} \boldsymbol{\epsilon}\_{0} \right| \right)^{-1-\varepsilon} \text{by (3.13)}. \end{split}$$

Find *<sup>j</sup>*ð Þ <sup>0</sup> <sup>∈</sup> <sup>ℤ</sup> so that 2*<sup>j</sup>*ð Þ <sup>0</sup> <sup>≤</sup>j j *<sup>ω</sup>* � *<sup>ξ</sup>* <sup>≤</sup>2*<sup>j</sup>*ð Þþ <sup>0</sup> <sup>1</sup> . We split the sum over j into two parts: *j*<*j* <sup>0</sup> and *j*≥ *j*ð Þ 0 .

$$\begin{aligned} \text{2.} &\sum\_{k} (\mathbf{1} + |a - k|)^{-1-\varepsilon} (\mathbf{1} + |b - k|)^{-1-\varepsilon} \text{ is uniformly bounded for all values of } \\ &a, b \in \mathbb{R}: \text{in fact,} \end{aligned}$$

$$\sum\_{k} (\mathbf{1} + |a - k|)^{-1 - \varepsilon} (\mathbf{1} + |b - k|)^{-1 - \varepsilon} \le \sum\_{k} (\mathbf{1} + |a - k|)^{-1 - \varepsilon}$$

$$\le \sup\_{0 \le a' \le 1} \sum\_{k} (\mathbf{1} + |a' - k|)^{-1 - \varepsilon} \le 2 \sum\_{l \in \mathbb{N}} (\mathbf{1} + l)^{-1 - \varepsilon} < \infty.$$

$$\begin{aligned} &\sum\_{j\geq j(0)}\sum\_{k} 2^{-j} \left(\mathbf{1} + \left|2^{-j}a\nuke(\mathbf{0})\right|\right)^{-1-\varepsilon} \left(\mathbf{1} + \left|2^{-j}\xi - ke(\mathbf{0})\right|\right)^{-1-\varepsilon} \\ &\leq \mathbf{C} \sum\_{j\geq j(0)} 2^{-j} \leq \mathbf{C} \cdot 2^{-j(0)+1} \leq 4\mathbf{C}|a-\xi|^{-1}. \end{aligned}$$

$$\sum\_{-\boldsymbol{\alpha}\prec\boldsymbol{j}\leq j(0)-1}2^{-j}\sum\_{k}\Big[\big(\mathbbm{1}+\big|2^{-j}\boldsymbol{\alpha}\cdot\ker(\mathbf{0})\big)\big).\big(\mathbbm{1}+\big|2^{-j}\boldsymbol{\xi}-\ker(\mathbf{0})\big)\Big]^{-1-\varepsilon}$$

$$=\sum\_{j\geq -j(0)+1}2^{j}\sum\_{k}\Big[\big(\mathbbm{1}+\big|2^{j}\boldsymbol{\alpha}\cdot\ker(\mathbf{0})\big)\big).\big(\mathbbm{1}+\big|2^{j}\boldsymbol{\xi}-\ker(\mathbf{0})\big)\Big]^{-1-\varepsilon}$$

$$\leq 2^{1+\varepsilon}\sum\_{j\geq j(0)+1}2^{j}\sum\_{k}\Big[\big(\mathbbm{2}+\big|2^{j}\boldsymbol{\alpha}\cdot\ker(\mathbf{0})\big)\big).\big(\mathbbm{2}+\big|2^{j}\boldsymbol{\xi}-\ker(\mathbf{0})\big)\Big]^{-1-\varepsilon}.\tag{46}$$

$$\begin{aligned} \left| \mathbf{2} + \left| \mathbf{2}^{j} \boldsymbol{\alpha} \text{-} \boldsymbol{k} \mathbf{e}(\mathbf{0}) \right| &= \mathbf{2} + \left| \mathbf{2}^{j} (\boldsymbol{\alpha} - \boldsymbol{\xi})/2 - \mathrm{l} \mathbf{e}(\mathbf{0}) + \left( \mathbf{2}^{j} (\boldsymbol{\alpha} + \boldsymbol{\xi})/2 \mathbf{-} \mathbf{k}\_{0} \mathbf{e}(\mathbf{0}) \right) \right| \\ &\geq \mathbf{1} + \left| \mathbf{2}^{j} (\boldsymbol{\alpha} - \boldsymbol{\xi})/2 \mathbf{-} \mathrm{l} \mathbf{e}(\mathbf{0}) \right|; \end{aligned}$$

$$
\mathcal{Q} + \left| \mathcal{Q}^{\circ} \xi - ke(\mathbf{0}) \right| \ge \mathbf{1} + \left| \mathcal{Q}^{\circ} (\xi - a) / \mathfrak{L} \cdot le(\mathbf{0}) \right|.
$$

$$\begin{aligned} &\sum\_{k} \left[ \left( 2 + \left| 2^{j}o - ke(\mathbf{0}) \right| \right) . \left( 2 + \left| 2^{j}\xi - ke(\mathbf{0}) \right| \right) \right]^{-1-\varepsilon} \\ &\leq \sum\_{l} \left[ (\mathbf{1} + \left| o \mathbf{1} + le(\mathbf{0}) \right|). (\mathbf{1} + \left| o \mathbf{1} + le(\mathbf{0}) \right|) \right]^{-1-\varepsilon} \leq \mathbf{C} (\mathbf{1} + \left| o \mathbf{1} \right|)^{-1-\varepsilon}, \end{aligned}$$

$$\begin{split} & \leq \mathsf{C} \sum\_{j \geq -j(0)+1} \mathsf{2}^{j} \left( \mathsf{1} + \mathsf{2}^{j} |\boldsymbol{\omega} - \boldsymbol{\xi}| / 2 \right)^{-1-\varepsilon} \\ & \leq \mathsf{C} \sum\_{j \geq 1} \mathsf{2}^{j' - j(0)} \left( \mathsf{1} + \mathsf{2}^{j' + j(0)} (\mathsf{1}/2) \mathsf{2}^{j(0)+1} \right)^{-1-\varepsilon} \text{ as we have } |\boldsymbol{\omega} - \boldsymbol{\xi}| \leq 2^{j(0)+1}, \\ & \leq \mathsf{C} \mathsf{2}^{-j(0)} \sum\_{j' \geq 1} \mathsf{2}^{j'} \left( \mathsf{1} + \mathsf{2}^{j'} \right)^{-1-\varepsilon} \\ & \leq \mathsf{C} \mathsf{2}^{-j(0)} \leq 2 \mathsf{C}' |\boldsymbol{\omega} - \boldsymbol{\xi}|^{-1}. \end{split} \tag{47}$$

4.For the estimates on ∇*ωK* and ∇*ξK*, we write

$$\begin{split} \left||\nabla\_{\boldsymbol{w}}K(\boldsymbol{\alpha},\boldsymbol{\xi})\right||\_{\mathbf{H}} &\leq \sum\_{j,k} 2^{-j} \left||\nabla\boldsymbol{\nu}\left(2^{-j}\boldsymbol{\alpha}\cdot\boldsymbol{k}\boldsymbol{e}\_{0}\right)\right||\_{\mathbf{H}} \left|\boldsymbol{\nu}\left(2^{-j}\boldsymbol{\alpha}\cdot\boldsymbol{k}\boldsymbol{e}\_{0}\right)\right| \\ &\leq \mathbf{C} \sum\_{j,k} 2^{-2j} \left[\left(1+\left|2^{-j}\boldsymbol{\alpha}\cdot\boldsymbol{k}\boldsymbol{e}\_{0}\right|\right)\left(1+\left|2^{-j}\boldsymbol{\xi}-\boldsymbol{k}\boldsymbol{e}\_{0}\right|\right)\right]^{-1-\varepsilon} \end{split}$$

and we follow the same technique; we obtain.

k k <sup>∇</sup>*ωK*ð Þ *<sup>ω</sup>*, *<sup>ξ</sup>* **<sup>H</sup>**, <sup>∇</sup>*ξK*ð Þ *<sup>ω</sup>*, *<sup>ξ</sup>* � � � � **<sup>H</sup>** <sup>≤</sup>*C*j j *<sup>ω</sup>* � *<sup>ξ</sup>* �<sup>2</sup> *:*∎

It therefore follows from the lines we write before the lemma the following theorem [20, 21].

**Theorem3.8.** If *ψ* is a Wiener functional continuously Malliavin differentiable and j j *ψ ω*ð Þ , k k <sup>∇</sup>*ψ ω*ð Þ **<sup>H</sup>** <sup>≤</sup>*C*ð Þ <sup>1</sup> <sup>þ</sup> j j *<sup>ω</sup>* �1�*<sup>ε</sup>* , and if the *<sup>ψ</sup><sup>j</sup>*,*<sup>k</sup>*ð Þ *<sup>ω</sup>* <sup>≔</sup> <sup>2</sup>�*j=*<sup>2</sup> *ψ* 2�*<sup>j</sup> ω*–*ke*<sup>0</sup> � � constitute an orthonormal basis for *L*<sup>2</sup> ð Þ *μ* , *e*<sup>0</sup> ∈ **H** being given, then the *ψ<sup>j</sup>*,*<sup>k</sup>*; *j*, *k*∈ ℤ n o also constitute an unconditional basis for all *<sup>L</sup><sup>p</sup>*ð Þ *<sup>μ</sup>* – spaces, 1<*p*<sup>&</sup>lt; <sup>∞</sup>.

#### **1.4 Periodized Wiener wavelets**

Even if L<sup>1</sup> -spaces do not have unconditional bases, Wiener wavelets still outperform Fourier Wiener analysis in some sense. To illustrate this, let us first introduce " periodized Wiener wavelets ". Given a multiresolution Wiener analysis with scaling Wiener functional ϕ and Wiener wavelet *ψ*, both with reasonable decay (say, j j *ϕ ω*ð Þ , j j *ψ ω*ð Þ <sup>≤</sup>*C*ð Þ <sup>1</sup> <sup>þ</sup> j j *<sup>ω</sup>* �1�*<sup>ε</sup>* ), we define

$$\phi\_{j,k}^{per}(\boldsymbol{\omega}) \coloneqq \sum\_{l \in \mathbb{Z}} \phi\_{j,k}(\boldsymbol{\omega} + l\boldsymbol{\varepsilon}\_0),\\\psi\_{j,k}^{per}(\boldsymbol{\omega}) \coloneqq \sum\_{l \in \mathbb{Z}} \psi\_{j,k}(\boldsymbol{\omega} + l\boldsymbol{\varepsilon}\_0);$$

and

$$\begin{aligned} \mathbf{V}\_j^{per} &:= closure \text{ of } \operatorname{span} \left\{ \boldsymbol{\phi}\_{j,k}^{per} ; k \in \mathbb{Z} \right\}, \\\\ \mathbf{W}\_j^{per} &:= closure \text{ of } \operatorname{span} \left\{ \boldsymbol{\nu}\_{j,k}^{per} ; k \in \mathbb{Z} \right\} \end{aligned}$$

First, notice that we have: P *<sup>l</sup>*<sup>∈</sup> <sup>ℤ</sup>ϕð Þ¼ *ω* þ *le*<sup>0</sup> 1. In fact, put *φ ω*ð Þ <sup>≔</sup><sup>P</sup> *<sup>l</sup>*<sup>∈</sup> <sup>ℤ</sup>ϕð Þ *ω* þ *le*<sup>0</sup> . The conditions on ϕ (continuity and its " reasonable decay ") ensure that *φ* is well defined and continuous. Moreover, we can write:

$$\Phi(\boldsymbol{\alpha}) = \sum\_{n \in \mathbb{Z}} c\_n \phi(2\boldsymbol{\alpha} \cdot \boldsymbol{n} e\_0) \text{ with } (c\_n)\_n \in \ell^2(\mathbb{Z}).$$

$$\rho(\rho) = \sum\_{l} \sum\_{n} c\_n \Phi(2\rho - 2le\_0 - n\varepsilon\_0) = \sum\_{l} \sum\_{m} c\_{m-2l} \Phi(2\rho - m\varepsilon\_0)$$

$$\sum \left(\sum\_{n} c\_{n-1}\right)\_{d(\Im\_{2n-1}\varpi\_{2n})} = \sum \rho(\Im\_{2n}\varpi\_{2n}) = \rho(\Im\_{2n}\varpi\_{2n})$$

Then,

$$\delta = \sum\_{m} \left( \sum\_{j} c\_{m-2j} \right) \phi(2\alpha - m\varepsilon\_0) = \sum\_{m} \phi(2\alpha - m\varepsilon\_0) = \rho(2\alpha).$$

Hence, *φ* is continuous, periodic with period e0, and

*Bases of Wavelets and Multiresolution in Analysis on Wiener Space DOI: http://dx.doi.org/10.5772/intechopen.104713*

$$
\rho(a) = \rho(2a) = \dots = \rho(2^n a) = \dots
$$

It follows that *φ* is a constant Wiener functional. We then put:

$$\sum\_{l} \Phi(o \cdot le\_{0}) = c$$

Since

Ð <sup>ϕ</sup> *<sup>d</sup><sup>μ</sup>* <sup>¼</sup> 1, this constant is necessarily equal to 1. We deduce, for *<sup>j</sup>*≥0, <sup>ϕ</sup>*per <sup>j</sup>*,*<sup>k</sup>* ð Þ¼ *ω*

$$\mathfrak{I}^{-j/2} \sum\_{l} \Phi(\mathfrak{I}^{-j} o \mathfrak{e} \mathfrak{A} e\_0 + \mathfrak{I}^{-j} l e\_0) = \mathfrak{I}^{j/2}, \text{ so that the } \mathbf{V}\_j^{per} \text{, for } j \ge 0 \text{, are all identical one-to-one.}$$

dimensional spaces, containing the constant functionals.

Similarly, P *l ψ ω*ð Þ¼ <sup>þ</sup> ð Þ *<sup>l</sup>=*<sup>2</sup> *<sup>e</sup>*<sup>0</sup> 0. In fact, <sup>P</sup> *l ψ ω*ð Þ¼ þ ð Þ *l=*2 *e*<sup>0</sup>

P *l* P *n* ð Þ �<sup>1</sup> *<sup>n</sup> c*�*n*þ<sup>1</sup>ϕð Þ 2*ω* þ *le*0–*ne*<sup>0</sup> where the cn are the coefficients appearing in (13), that is,

$$c\_n = \langle \phi, \phi\_{-1,n} \rangle = \sum\_{k,m} (-1)^{m+1} c\_m \phi(2\alpha + k e\_0) (k = l - n, m = -n + 1)$$

$$= 0 \left( \text{because } \sum\_m c\_{2m} = \sum\_m c\_{2m+1} \right).$$

Hence, for *j*≥ 1≥ , **W***per <sup>j</sup>* <sup>¼</sup> f g<sup>0</sup> . We therefore restrict our attention to the **<sup>V</sup>***per <sup>j</sup>* , **W***per <sup>j</sup>* with *<sup>j</sup>*≤0. Obviously **<sup>V</sup>***per <sup>j</sup>* , **<sup>W</sup>***per <sup>j</sup>* <sup>⊂</sup> **<sup>V</sup>***per <sup>j</sup>*�<sup>1</sup>, a property inherited from the nonperiodized spaces. Moreover, **W***per <sup>j</sup>* is still orthogonal to **<sup>V</sup>***per <sup>j</sup>* , because

$$\begin{split} \int\_{\mathbf{H}} \Psi\_{j,k}^{per}(a) \Phi\_{j,k'}^{per}(a) \,\mu(dao) &= \sum\_{l,l' \in \mathbb{Z}} 2^{-j} \int\_{\mathbf{H}} \Psi\left(2^{-j}a + 2^{-j}le\_{0} - k\varepsilon\_{0}\right) \overline{\left(\Phi(2^{-j}a + 2^{-j}l'\varepsilon\_{0} - k'\varepsilon\_{0})\right)} \,\mu(dao) \\ &= \sum\_{l,l' \in \mathbb{Z}} 2^{jl} \int\_{\left[l'\varepsilon\_{0}, \left(l'+1\right)\varepsilon(0)\right]} \Psi\left(2^{jl}a + 2^{jl}\left(l - l'\right)\varepsilon\_{0} - k\varepsilon\_{0}\right) \overline{\Phi\left(2^{jl}\xi - k'\varepsilon\_{0}\right)} \,\mu(d\xi) ,\end{split}$$

because *j*≤0,

$$=\sum\_{r\in\mathbb{Z}} \left\langle \Psi\_{j,k+2^{|\boldsymbol{\beta}|}r}, \Phi\_{j,k'} \right\rangle\_{\mu} = \mathbf{0}.$$

It follows that, as in the non-periodized case, **V***per <sup>j</sup>*�<sup>1</sup> <sup>¼</sup> **<sup>V</sup>***per <sup>j</sup>* <sup>⊕</sup>**W***per <sup>j</sup>* . The spaces **<sup>V</sup>***per <sup>j</sup>* , **W***per <sup>j</sup>* are all finite-dimensional: since <sup>Φ</sup>*<sup>j</sup>*,*k*þ*m*2j j*<sup>j</sup>* <sup>¼</sup> <sup>Φ</sup>*<sup>j</sup>*,*<sup>k</sup>* for *<sup>m</sup>* <sup>∈</sup> <sup>ℤ</sup>, and the same is true for Ψ, both **V***per <sup>j</sup>* and **<sup>W</sup>***per <sup>j</sup>* are spanned by the 2|j| Wiener functionals obtained from *<sup>k</sup>* <sup>¼</sup> 0, 1, … , 2j j*<sup>j</sup>* � 1. These 2|j| Wiener functionals are moreover orthonormal; in, for example, **W***per <sup>j</sup>* we have, for 0≤ *k*, *k*<sup>0</sup> <sup>≤</sup>2j j*<sup>j</sup>* � 1,

$$\left\langle \Psi\_{j,k}^{per}, \Psi\_{j,k'}^{per} \right\rangle\_{\mu} = \sum\_{r \in \mathbb{Z}} \left\langle \Psi\_{j,k+2^{|\mathbb{J}|}r}, \left. \Psi\_{j,k'} \right\rangle\_{\mu} = \delta\_{k;k'} \right\rangle$$

We have therefore a ladder of multiresolution spaces, **V***per* <sup>0</sup> <sup>⊂</sup> **<sup>V</sup>***per* �<sup>1</sup> <sup>⊂</sup> **<sup>V</sup>***per* �<sup>2</sup> <sup>⊂</sup> … , with successive orthogonal complements **W***per* <sup>0</sup> of **<sup>V</sup>***per* <sup>0</sup> in **<sup>V</sup>***per* �1 � �, **W***per* <sup>1</sup> , … , and orthonormal bases <sup>ϕ</sup>*j*,*k*; *<sup>k</sup>* <sup>¼</sup> 0, … , 2j j*<sup>j</sup>* � <sup>1</sup> n o in **<sup>V</sup>***per <sup>j</sup>* , *<sup>ψ</sup>j*,*k*; *<sup>k</sup>* <sup>¼</sup> 0, … , 2j j*<sup>j</sup>* � <sup>1</sup> n o in **<sup>W</sup>***per <sup>j</sup>* . Since the closed space spanned by ∪ *j*∈ �ℕ **V***per <sup>j</sup>* is *L*<sup>2</sup> ð Þ *μ* (this follows from the corresponding nonperiodized version), the Wiener functionals in ϕ*per* 0,0 n o<sup>∪</sup> *<sup>ψ</sup>per <sup>j</sup>*,*<sup>k</sup>* ;�*j*∈ℕ,*<sup>k</sup>* <sup>¼</sup> 0, …,2j j*<sup>j</sup>* � <sup>1</sup> n o constitute an orthonormal basis in *L*<sup>2</sup> ð Þ *μ* . We will relabel this basis as follows:

$$\begin{aligned} \boldsymbol{\nu}\_{0}(\boldsymbol{\alpha}) &= \mathbf{1} = \boldsymbol{\Phi}^{\mathrm{per}}\_{0,0}(\boldsymbol{\alpha}), \boldsymbol{\nu}\_{1}(\boldsymbol{\alpha}) = \boldsymbol{\nu}^{\mathrm{per}}\_{0,0}(\boldsymbol{\alpha}), \boldsymbol{\nu}\_{2}(\boldsymbol{\alpha}) = \boldsymbol{\nu}^{\mathrm{per}}\_{-1,0}(\boldsymbol{\alpha}), \\ \boldsymbol{\nu}\_{3}(\boldsymbol{\alpha}) &= \boldsymbol{\nu}^{\mathrm{per}}\_{-1,1}(\boldsymbol{\alpha}), \dots, \\ \boldsymbol{\nu}\_{2^{j}}(\boldsymbol{\alpha}) &= \boldsymbol{\nu}^{\mathrm{per}}\_{-j,0}(\boldsymbol{\alpha}), \dots, \boldsymbol{\nu}\_{2^{j}+k}(\boldsymbol{\alpha}) = \boldsymbol{\nu}^{\mathrm{per}}\_{-j,k}(\boldsymbol{\alpha}) = \boldsymbol{\nu}\_{2}(\boldsymbol{\alpha} - k \mathbf{2}^{-j} \boldsymbol{\varepsilon} \boldsymbol{\alpha}) \\ \text{for } 0 \le k \le 2^{j} - 1, \dots \end{aligned}$$

Then this basis has the following property.

**Theorem 4.1.** If *φ* is a continuous periodic Wiener functional with period e0, then there exist *α<sup>n</sup>* ∈ℂ so that

$$\lim\_{N} \left\| \rho - \sum\_{n=0,\ldots,N} a\_n \varphi\_n \right\|\_{\infty} = 0,\tag{48}$$

where k k*:* <sup>∞</sup> denotes the norm of *<sup>L</sup>*<sup>∞</sup>ð Þ *<sup>μ</sup>* .

#### **Proof.**

1.Since the *ψ<sup>n</sup>* are orthonormal, we necessarily have *α<sup>n</sup>* ¼ *φ*, *ψ<sup>n</sup>* h i*μ*. Define

$$\mathcal{S}\_N \varphi = \sum\_{\mathfrak{n}=0,\ldots,N} \langle \varphi, \varphi\_\mathfrak{n} \rangle\_\mu.$$

In a first step we prove that the SN are uniformly bounded, that is,

$$\|\|\mathbf{S}\_N\boldsymbol{\rho}\|\|\_{\infty} \leq \mathbf{C} \|\|\boldsymbol{\rho}\|\|\_{\infty},\tag{49}$$

with C independent of *φ* or N.

2. If *<sup>N</sup>* <sup>¼</sup> <sup>2</sup>*<sup>j</sup>* , then *S*<sup>2</sup> *<sup>j</sup>* is the orthogonal projection operator on **V***per* �*<sup>j</sup>* ; hence

$$(\mathcal{S}\_{\mathbf{Z}}^{j}\boldsymbol{\varrho})(\boldsymbol{\alpha}) = \sum\_{k=0,\ldots,2^{|\boldsymbol{\beta}|}-1} \left\langle \boldsymbol{\varrho}, \boldsymbol{\Phi}\_{-j,k}^{\mathrm{per}} \right\rangle\_{\boldsymbol{\mu}} \boldsymbol{\Phi}\_{-j,k}^{\mathrm{per}}(\boldsymbol{\alpha}) = \int\_{\mathbf{H}} \mathcal{K}\_{j}(\boldsymbol{\alpha}, \boldsymbol{\xi}) \boldsymbol{\varrho}(\boldsymbol{\xi}) \,\boldsymbol{\mu}(d\boldsymbol{\xi}),$$

with *Kj*ð Þ¼ *<sup>ω</sup>*, *<sup>ξ</sup>* <sup>P</sup> *<sup>k</sup>*¼0, … , 2j j*<sup>j</sup>* �<sup>1</sup> ϕ*per* �*j*,*<sup>k</sup>*ð Þ *<sup>ω</sup>* <sup>ϕ</sup>*per* �*j*,*l* ð Þ*ξ* � �*:*.

Consequently,

$$||\mathbb{S}\_2\rho||\_{\infty} \le \left(\sup\_{\theta \in \mathbb{H}} \int\_{\mathbb{H}} |K\_{\boldsymbol{\xi}}(\rho, \xi)| \, |\, \mu(d\xi) \right) ||\rho||\_{\infty}.$$

*Bases of Wavelets and Multiresolution in Analysis on Wiener Space DOI: http://dx.doi.org/10.5772/intechopen.104713*

Now,

$$\begin{split} &\sup \sup\_{\boldsymbol{\alpha}\in\mathbf{H}} \int\_{\mathbf{H}} \left| K\_{j}(\boldsymbol{\alpha},\boldsymbol{\xi}) \right| \, \mu(d\boldsymbol{\xi}) \\ &\leq \sup\_{\boldsymbol{\alpha}\in\mathbf{H}} \int\_{\mathbf{H}^{k}} \sum\_{\boldsymbol{\mathsf{x}}'=\boldsymbol{\mathsf{L}},\boldsymbol{\mathcal{Z}}'} \sum\_{l'\in\mathbb{Z}} \left| \phi\_{-j,k}(\boldsymbol{\alpha}+l\boldsymbol{e}\_{0}) \right| \cdot \left| \Phi\_{-j,k} \left( \boldsymbol{\xi}+l'\boldsymbol{e}\_{0} \right) \right| \, \mu(d\boldsymbol{\xi}) \\ &\leq \sup\_{\boldsymbol{\alpha}\in\mathbf{H}} \int\_{\mathbf{H}} \sum\_{k=0,\ldots,2^{|\boldsymbol{\mathsf{x}}|}} \sum\_{l'\in\mathbb{Z}} 2^{l} \left| \phi \big( 2^{j}(\boldsymbol{\alpha}+l\boldsymbol{e}\_{0}) \big) \right| \, \left| \Phi \big( 2^{j}\boldsymbol{\xi}-l\boldsymbol{e}\_{0} \big) \right| \, \mu(d\boldsymbol{\xi}) \\ &\leq C \sup\_{\boldsymbol{\alpha}'\in\mathbf{H}} \sum\_{k=0,\ldots,2^{|\boldsymbol{\mathsf{x}}|}} \sum\_{l'\in\mathbb{Z}} \left| \phi \big( \boldsymbol{\alpha}'+2^{j}l\boldsymbol{e} \leq C \sup\_{\boldsymbol{\alpha}'\in\mathbf{H}} \sum\_{m\in\mathbb{Z}} \left| \phi (\boldsymbol{\alpha}'+m\boldsymbol{e}\_{0}) \right| \,, \end{split}$$

and this is uniformly bounded if j j <sup>ϕ</sup>ð Þ *<sup>ω</sup>* <sup>≤</sup>*C*ð Þ <sup>1</sup> <sup>þ</sup> j j *<sup>ω</sup>* �1�*<sup>ε</sup>* . This establishes (49) for *<sup>N</sup>* <sup>¼</sup> <sup>2</sup>*<sup>j</sup>* .

3. If *<sup>N</sup>* <sup>¼</sup> <sup>2</sup>*<sup>j</sup>* <sup>þ</sup> *<sup>m</sup>*, 0 <sup>≤</sup> *<sup>m</sup>* <sup>≤</sup>2*<sup>j</sup>* –1, then

$$(\mathcal{S}\_N \rho)(\alpha) = \left(\mathcal{S}\_2^j \rho\right)(\alpha) + \sum\_{k=0,\ldots,m} \left\langle \rho, \psi\_{-j,k}^{per} \right\rangle\_{\mu} \psi\_{-j,k}^{per}(\alpha).$$

Estimates exactly similar to those in point 2 show that the *<sup>L</sup>*<sup>∞</sup>ð Þ *<sup>μ</sup>* -norm of the second sum is also bounded by *C*k k*φ* <sup>∞</sup>, uniformly in j, which proves (49) for all N.

$$\text{4. Take now } \boldsymbol{\varrho} \in \mathbf{E} = \bigcup\_{j \in -\mathbb{N}} \mathbf{V}\_j^{per}. \text{ Then } \boldsymbol{\varrho} \in \mathbf{V}\_{-f}^{per} \text{ for some } \mathbf{J} > \mathbf{0},$$

so that *φ*, *ψper* �*j*,*k* D E *<sup>μ</sup>* <sup>¼</sup> 0 for *<sup>j</sup>* <sup>0</sup> <sup>≥</sup>*J*, i.e., *<sup>φ</sup>*, *<sup>ψ</sup><sup>l</sup>* h i*<sup>μ</sup>* <sup>¼</sup> 0 for *<sup>l</sup>*≥2*<sup>J</sup>* . Consequently, *φ* ¼ *SNφ* if *N* ≥2*<sup>J</sup>* , so that (48) clearly holds. Since **E** is dense in the space of continuous periodic Wiener functionals equipped with the k k *:* <sup>∞</sup>-norm, the theorem follows.∎.

We deduce a similar theorem for *L*<sup>1</sup> ð Þ *μ* .

$$\begin{aligned} \text{Theorem 4.2. If } \boldsymbol{\rho} \in L^1(\boldsymbol{\mu}) \text{, then } \lim\_{N \to \infty} \left\| \boldsymbol{\rho} - \sum\_{n=0,\ldots,N} \langle \boldsymbol{\rho}, \boldsymbol{\nu}\_n \rangle\_{\boldsymbol{\mu}} \boldsymbol{\nu}\_n \right\|\_{\boldsymbol{\mu}} = \mathbf{0}. \end{aligned}$$

#### **Proof.**

As we have the following:

$$\|\|\boldsymbol{\varphi}\|\|\_{L^{1}(\mu)} = \sup \left\{ \left| \langle \boldsymbol{\varphi}, \boldsymbol{\psi} \rangle\_{\mu} \right| / \boldsymbol{\psi} \text{ continuous, periodic with period } \epsilon\_{0}, \|\|\boldsymbol{\psi}\|\|\_{\infty} \le 1 \right\},$$

this leads immediately to

$$\|\mathbb{S}\_N \wp\|\_{L^1(\mu)} = \sup \left\{ \left| \langle \mathbb{S}\_N \wp, \wp \rangle\_{\mu} \right| \nu \text{ continuous}, e\_0-\text{periodic}, \|\!|\wp\|\!|\_{\\*\*\infty} \le 1 \right\} $$

$$= \sup \left\{ \left| \langle \wp, \mathbb{S}\_N \wp \rangle\_{\mu} \right| \nu \text{ continuous}, e\_0-\text{periodic}, \|\!|\wp\|\!|\_{\\*\*\infty} \le 1 \right\} $$

$$\le C \|\!|\wp\|\!|\_{L^1(\mu)} \tag{50} $$

by the uniform bound (49) and because h i *φ*, *ψ <sup>μ</sup>* � � � � � �<sup>≤</sup> k k*<sup>φ</sup> <sup>L</sup>*1ð Þ *<sup>μ</sup>* k k *<sup>ψ</sup>* <sup>∞</sup>.

Since **<sup>E</sup>** <sup>¼</sup> <sup>∪</sup> *<sup>j</sup>*<sup>∈</sup> �<sup>ℕ</sup>**V***per <sup>j</sup>* is dense in *L*<sup>1</sup> ð Þ *μ* , the uniform bound (50) is sufficient to prove the theorem.∎.

**Remark.** The ordering of the *ψ<sup>n</sup>* is important in Theorems 4.1 and 4.2: we have a Schauder basis, but not an unconditional basis.

## **Author details**

Claude Martias French West Indies University, French West Indies

\*Address all correspondence to: claude.martias0157@orange.fr

© 2022 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.

*Bases of Wavelets and Multiresolution in Analysis on Wiener Space DOI: http://dx.doi.org/10.5772/intechopen.104713*

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## Section 3
