Bases of Wavelets and Multiresolution in Analysis on Wiener Space

Abstract

The multiresolution analysis is applied into the space of square integrable Wiener functionals for extending well-known constructions of orthonormal wavelets in L2(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 L1-spaces do not have unconditional bases, wavelets still outperform Fourier analysis in some sense. We illustrate this by introducing periodized Wiener wavelets.

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 $L^P\left(\mu \right)$, $1<p<\infty ,\mu$ being the Wiener measure. We start it by first proving an extension of the well-known Calderon–Zygmund decomposition theorem. As L¹-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 $L^2\left(\mu \right)$, the space of μ-square integrable Wiener functionals, μ denoting the Wiener measure, consists of a sequence approximation spaces ${\left({\mathbf{V}}_j\right)}_{j\in \mathrm{\mathbb{Z}}}$, ${\mathbf{V}}_j\subset L^2\left(\mu \right)$; these subspaces are assumed to be closed and satisfy the following:

$${\mathbf{V}}_j\subset {\mathbf{V}}_{j-1},\text{for}\mathrm{all}j\in \mathrm{\mathbb{Z}},$$E1

$$\text{the closed space generated}\mathrm{by}\overline{{\bigcup }_{j\in \mathrm{\mathbb{Z}}}^{}{\mathbf{V}}_j}\text{is}{L}^2\left(\mu \right),$$E2

$${\bigcap }_{j\in \mathrm{\mathbb{Z}}}^{}{\mathbf{V}}_j=\left\{0\right\}.$$E3

If we denote by ${\prod }_j$ the orthogonal projection operator onto V_j, then (2) ensures that ${lim}_{j\to \infty }{\prod }_j\phi =\phi$ for all $\phi \in L^2\left(\mu \right)$. 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

$$\phi \in {\mathbf{V}}_j\iff \phi \left(2^j.\right)\in {\mathbf{V}}_0.$$E4

That is, all the spaces are scaled versions of the central space V₀. An example corresponding to the Haar multiresolution analysis in real analysis (see [8, 9]) is the following:

$${\mathbf{V}}_j≔\left\{\phi \in L^2\left(\mu \right);\forall k\in \mathrm{\mathbb{Z}}:{\phi }_{\left[2^j{e}_{\left(k\right)},2^{j+1}{e}_{\left(k+1\right)}\right[}=\mathit{\text{constant}}\right\},$$

where $e\left(k\right),k\in \mathrm{\mathbb{Z}}$ (or $e_k,k\in \mathrm{\mathbb{Z}}$, another notation) is a fixed orthonormal basis in the Cameron – Martin space H defined by

$$\mathbf{H}≔\left\{h:\left[0,1\right]\to {\mathrm{ℝ}}^d/h\left(t\right)={\int }_{\left[0,\mathrm{t}\right]}{h}^{\prime }\left(s\right)\mathit{ds},{\left({‖h‖}_{\mathbf{H}}\right)}^2≔{\int }_{\left[0,1\right]}{\left|h^{\prime }\left(s\right)\right|}^2\mathit{ds}<\infty \right\},$$

and the above interval in definition of V_j is in the sense of usual order relation of functions. This example exhibits another feature that we require from a multiresolution analysis: invariance of V₀ under “ integer ” translations,

$$\phi \in {\mathbf{V}}_0\Rightarrow \phi \left(.-{\mathit{ne}}_0\right)\in {\mathbf{V}}_0,\forall n\in \mathrm{\mathbb{Z}}.$$E5

Because of (4) this implies that if $\phi \in {\mathbf{V}}_j$, then $\phi \left(.–2^j{\mathit{ne}}_0\right)$ belongs to V_j for all $n\in \mathrm{\mathbb{Z}}$. Finally, we require also that there exists $\mathrm{\varphi }\in {\mathbf{V}}_0$ so that

$$\left\{{\mathrm{\varphi }}_{0,n};n\in \mathrm{\mathbb{Z}}\right\}\text{is an orthonormal basis in}{\mathbf{V}}_0,$$E6

where, for all $j,n\in \mathrm{\mathbb{Z}}$, ${\mathrm{\varphi }}_{j,n}\left(\omega \right)≔2^{-j/2}\mathrm{\varphi }\left(2^{-j}\omega –{\mathit{ne}}_0\right)$. Together, (4) and (6) imply that $\left\{{\mathrm{\varphi }}_{j,n};n\in \mathrm{\mathbb{Z}}\right\}$ is an orthonormal basis for V_j for all $j\in \mathrm{\mathbb{Z}}$. In the example given above, the possible choice for $\mathrm{\varphi }$ is the indicator Wiener functional for $\left[0,e_0\right]≔\left\{\omega \in \mathbf{W};0\le \omega \left(s\right)\le e_0\left(s\right),\forall s\in \left[0,1\right]\right\}$, that is, $\mathrm{\varphi }\left(\omega \right)=1$ if $\omega \in \left[0,e_0\right]$, interval in the lattice space (W,≤), and $\mathrm{\varphi }\left(\omega \right)=0$ otherwise. We call $\mathrm{\varphi }$ the scaling Wiener functional of the multiresolution analysis. Note that $\mathrm{\varphi }$ depends on the choice of e₀. Hence, we will say that $\mathrm{\varphi }$ 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 $\left\{{\psi }_{j,k};j,k\in \mathrm{\mathbb{Z}}\right\}$ of $L^2\left(\mu \right)$,

${\psi }_{j,k}\left(\omega \right)≔2^{-j/2}\psi \left(2^{-j}\omega –{\mathit{ke}}_0\right)$, such that, for all $\phi \in L^2\left(\mu \right)$,

$$\prod _{j–1}\phi ={\prod }_j\phi +\sum _{k\in \mathrm{\mathbb{Z}}}{⟨\phi ,{\psi }_{j,k}⟩}_{\mu }{\psi }_{j,k},$$E7

where the bracket ${⟨.,.⟩}_{\mu }$ denotes the scalar product in $L^2\left(\mu \right)$. For every $j\in \mathrm{\mathbb{Z}}$, define W_j to be the complement orthogonal of V_j in V_{j – 1}. We have

$${\mathbf{V}}_{j–1}={\mathbf{V}}_j\mathbf{\oplus }{\mathbf{W}}_j$$E8

and

$${\mathbf{W}}_j\perp {\mathbf{W}}_{j^{\prime }}\text{if}j\ne j^{\prime }.$$E9

It follows that, for j < j₀,

$${\mathbf{V}}_j={\mathbf{V}}_{j0}\mathbf{\oplus }\left(\underset{k=0}{\overset{j\left(0\right)-1}{\mathbf{\oplus }}}{\mathbf{W}}_{j\left(0\right)–k}\right),$$E10

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

$$L^2\left(\mu \right)=\underset{k\in \mathrm{\mathbb{Z}}}{\mathbf{\oplus }}{\mathbf{W}}_j,$$E11

A decomposition of $L^2\left(\mu \right)$ into mutually orthogonal subspaces. Furthermore, the W_j spaces inherit the scaling property (4) from the V_j:

$$\phi \in {\mathbf{W}}_j\iff \phi \left(2^j.\right)\in {\mathbf{W}}_0.$$E12

Formula (7) is equivalent to saying that, for fixed $j,\left\{{\psi }_{j,k};k\in \mathrm{\mathbb{Z}}\right\}$ constitutes an orthonormal basis for W_j. Because of (11), (2) and (3), this then automatically implies that the whole collection $\left\{{\psi }_{j,k};j,k\in \mathrm{\mathbb{Z}}\right\}$ is an orthonormal basis for $L^2\left(\mu \right)$. On the other hand, (12) ensures that if $\left\{{\psi }_{0,k};k\in \mathrm{\mathbb{Z}}\right\}$ is an orthonormal basis for W₀, then $\left\{{\psi }_{j,k};k\in \mathrm{\mathbb{Z}}\right\}$ will likewise be an orthonormal basis for W_j, for any $j\in \mathrm{\mathbb{Z}}$. Construction of $\psi$ 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 $\psi \in {\mathbf{W}}_0$ such that the $\psi \left(.–{\mathit{ke}}_0\right)$ constitute an orthonormal basis for W₀. Let us first write out some interesting properties of $\mathrm{\varphi }$ and W₀.

1. Since $\mathrm{\varphi }\in {\mathbf{V}}_0\subset {\mathbf{V}}_{-1}$, and the ${\mathrm{\varphi }}_{-1,n}$ are an orthonormal basis in V₋₁, we have

   $$\mathrm{\varphi }=\sum _n{c}_n{\mathrm{\varphi }}_{-1,n},$$E13

   with

   $$c_n={⟨\mathrm{\varphi },{\mathrm{\varphi }}_{-1,n}⟩}_{\mu }\text{and}\sum _n{\left|c_n\right|}^2=1.$$E14

   we can rewrite (13) as either

   $$\mathrm{\varphi }\left(\omega \right)=2^{1/2}\sum _n{c}_n\mathrm{\varphi }\left(2\omega –{\mathit{ne}}_0\right)$$E15

   or

   $$\widehat{\mathrm{\varphi }}\left(\xi \right)=2^{-1/2}\sum _{n}exp\left[-\text{in}{⟨\xi ,e_0⟩}_{\mathbf{H}}/2\right]\widehat{\mathrm{\varphi }}\left(\xi /2\right),$$E16

   where the convergence in either sum holds in $L^2\left(\mu \right)$—sense, $\widehat{\mathrm{\varphi }}$ denoting the Fourier–Wiener transform. Formula (16) can be rewritten as

   $$\widehat{\mathrm{\varphi }}\left(\xi \right)=m_0\left(\xi /2\right)\widehat{\mathrm{\varphi }}\left(\xi /2\right),$$E17

   where

   $$m_0\left(\xi \right)=2^{-1/2}\sum _n{c}_{n}exp\left[-\text{in}{⟨\xi ,e_0⟩}_{\mathbf{H}}\right].$$E18

   Equality in (17) holds pointwise $\mu$—almost everywhere.

2. The orthonormality of the $\mathrm{\varphi }\left(.–{\mathit{ke}}_0\right)$ leads to special properties for m₀. We have

$$\begin{array}{c}{\delta }_{k,0}=\underset{\mathbf{H}}{\int }{\left|\widehat{\mathrm{\varphi }}\left(\xi \right)\right|}^{2}exp\left[\mathrm{ik}{⟨\xi ,e_0⟩}_{\mathbf{H}}\right]\mu \left(\mathit{d\xi }\right)\\ =\sum _l\underset{\left\{⟨\xi ,e\left(0\right)⟩\in \left[2\mathit{\pi l},2\pi \left(l+1\right)\right[\right\}}{\int }{\left|\widehat{\mathrm{\varphi }}\left(\xi \right)\right|}^{2}exp\left[\mathrm{ik}{⟨\xi ,e_0⟩}_{\mathbf{H}}\right]\mu \left(\mathit{d\xi }\right)\\ =\sum _{l}exp\left[-2\pi l^2\right]\underset{\left\{⟨h,e\left(0\right)⟩\in \left[0,2\pi \right[\right\}}{\int }{\left|\widehat{\mathrm{\varphi }}\left(h+2\pi {\mathit{le}}_0\right)\right|}^{2}exp\left[\mathrm{ik}{⟨h,e_0⟩}_{\mathbf{H}}\right].\\ exp\left[-2\mathit{\pi l}\underset{\left[0,1\right[}{\int }\underset{0}{e}\left(s\right)\mathit{dh}\left(s\right)\right]\mu \left(\mathit{dh}\right)\\ =\underset{\left\{⟨h,e\left(0\right)⟩\in \left[0,2\pi \right[\right\}}{\int }\sum _l{\left|\widehat{\mathrm{\varphi }}\left(h+2\pi {\mathit{le}}_0\right)\right|}^{2}exp\left[\mathrm{ik}{⟨h,e_0⟩}_{\mathbf{H}}\right]\mu \left(\mathit{dh}\right),\end{array}$$

Implying

$$\sum _l{\left|\widehat{\mathrm{\varphi }}\left(h+2\pi {\mathit{le}}_0\right)\right|}^2={\left(2\pi \right)}^{-1}\mu -\mathrm{a}.\mathrm{e}..$$E19

Substituting (17) leads to

$$\sum _l{\left|m_0\left(\xi +\pi {\mathit{le}}_0\right)\right|}^2{\left|\widehat{\mathrm{\varphi }}\left(\xi +\pi {\mathit{le}}_0\right)\right|}^2={\left(2\pi \right)}^{-1};$$

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

$${\left|m_0\left(\xi \right)\right|}^2+{\left|m_0\left(\xi +\pi e_0\right)\right|}^2=1\mu -\mathrm{a}.\mathrm{e}..$$E20

1. Let us now characterize W₀. $\phi \in {\mathbf{W}}_0$ is equivalent to $\phi \in {\mathbf{V}}_{-1}$ and $\phi$ is orthogonal to V₀. Since $\phi \in {\mathbf{V}}_{-1}$, we have

$\phi =\sum _n{\lambda }_n{\mathrm{\Phi }}_{-1,n}$, with ${\lambda }_n={⟨\phi ,{\mathrm{\varphi }}_{-1,n}⟩}_{\mu }$. This implies

$$\widehat{\phi }\left(\xi \right)=2^{-1/2}\sum _n{\lambda }_{n}exp\left(-\text{in}{⟨\xi ,e_0⟩}_{\mathbf{H}}/2\right)\widehat{\mathrm{\Phi }}\left(2^{-1}\xi \right)=m_{\phi }\left(2^{-1}\xi \right)\widehat{\mathrm{\Phi }}\left(2^{-1}\xi \right),$$E21

where

$$m_{\phi }\left(\xi \right)=2^{-1}\sum _n{\lambda }_{n}exp\left(-\text{in}{⟨\xi ,e_0⟩}_{\mathbf{H}}\right);$$E22

$m_{\phi }$ is a $2\pi e_0$-periodic Wiener functional; convergence in (22) holds pointwise $\mu$—a. e.. The constraint “ $\phi$ orthogonal to V₀ ” implies that $\phi$ is orthogonal to ${\mathrm{\Phi }}_{0,k}$ for all k, that is,

$$\underset{\mathbf{H}}{\int }\left(\widehat{\phi }\left(h\right)\right)\left(\overline{\widehat{\mathrm{\Phi }}\left(h\right)}\right)exp\left(\mathit{ik}{⟨h,e_0⟩}_{\mathbf{H}}\right)\mu \left(\mathit{dh}\right)=0$$

where z^c denotes the conjugate complex of complex z, or

$$\underset{\left\{⟨h,e_0⟩\in \left[0,2\pi \right[\right\}}{\int }exp\left(\mathrm{ik}{⟨h,e_0⟩}_{\mathbf{H}}\right)\sum _l\left(\widehat{\phi }\left(h+2\pi {\mathit{le}}_0\right)\overline{\widehat{\mathrm{\Phi }}\left(h+2\pi {\mathit{le}}_0\right)}\right)\mu \left(\mathit{dh}\right)=0;$$

note that, out of the ordinary, we denoted by $h\in \mathbf{H}$ the integration variable.

Hence

$$\sum _l\left(\widehat{\phi }\left(h+2\pi {\mathit{le}}_0\right)\overline{\widehat{\mathrm{\varphi }}\left(h+2\pi {\mathit{le}}_0\right)}\right)=0,$$E23

where the series in (23) converges absolutely in $L^1\left(\mu \right)$. 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

$$m_{\phi }\left(h\right)\left(\overline{m_0\left(h\right)}\right)+m_{\phi }\left(h+\pi e_0\right)\left(\overline{m_0\left(h+\pi e_0\right)}\right)=0\mu -\mathrm{a}.\mathrm{e}..$$E24

Since $\left(\overline{m_0\left(h\right)}\right)$ and $\left(\overline{m_0\left(h+\pi e_0\right)}\right)$ cannot vanish together on a set of nonzero Wiener measure (because of (20)), this implies the existence of a $2\pi e_0$-periodic Wiener functional ϴ so that

$$m_{\phi }\left(h\right)=\mathrm{\Theta }\left(h\right)\left(\overline{m_0\left(h+\pi e_0\right)}\right)\mu -\mathrm{a}.\mathrm{e}.$$E25

and

$$\mathrm{\Theta }\left(h\right)+\mathrm{\Theta }\left(h+\pi e_0\right)=0\mu -\mathrm{a}.\mathrm{e}..$$E26

This last equation can be rewrite as

$$\mathrm{\Theta }\left(h\right)=e\left(i{⟨h,e_0⟩}_{\mathbf{H}}\right)\nu \left(2h\right),$$E27

where $\nu$ is a $2\pi e_0$-periodic Wiener functional.

Substituting (25, 27) into (21) gives

$$\widehat{\phi }\left(\xi \right)=exp\left(\mathrm{i}{⟨h,e_0⟩}_{\mathbf{H}}/2\right)\left(\overline{m_0\left(\xi /2+\pi e_0\right)}\right)\nu \left(\xi \right)\widehat{\mathrm{\Phi }}\left(\xi /2\right).$$E28

1. The general form (28) for the Fourier–Wiener transform of $\phi \in {\mathbf{W}}_0$ suggests that we take

$$\widehat{\mathrm{\Psi }}\left(\xi \right)=exp\left(\mathrm{i}{⟨h,e_0⟩}_{\mathbf{H}}/2\right)\left(\overline{m_0\left(\xi /2+\pi e_0\right)}\right)\widehat{\mathrm{\Phi }}\left(\xi /2\right)$$E29

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

$$\widehat{\phi }\left(\xi \right)=\left(\sum _k{\nu }_{k}exp\left(-\mathrm{ik}{⟨\xi ,e_0⟩}_{\mathbf{H}}\right)\right)\widehat{\mathrm{\Psi }}\left(\xi \right)$$

or

$$\phi =\sum _k{\nu }_k\mathrm{\Psi }\left(.–{\mathit{ke}}_0\right),$$

so that the $\mathrm{\Psi }\left(.–{\mathit{ne}}_0\right)$ are a good candidate for a basis of W₀. We need to verify that the ${\mathrm{\Psi }}_{0,k}$ are indeed an orthonormal basis for W₀. First, the properties of m₀ and $\widehat{\mathrm{\Phi }}$ ensure that (29) defines an $L^2\left(\mu \right)$-Wiener functional belonging to V₋₁ and orthogonal to V₀ (by the analysis above), so that $\mathrm{\Psi }\in {\mathbf{W}}_0$. Orthonormality of the ${\mathrm{\Psi }}_{0,k}$ is easy to check:

$$\begin{array}{c}\underset{\mathbf{H}}{\int }\mathrm{\Psi }\left(h\right)\left(\overline{\mathrm{\Psi }\left(h–{\mathit{me}}_0\right)}\right)\mu \left(\mathit{dh}\right)=\underset{\mathbf{H}}{\int }{\left|\widehat{\mathrm{\Psi }}\left(\xi \right)\right|}^{2}exp\left(\mathrm{im}{⟨\xi ,e_0⟩}_{\mathbf{H}}\right)\mu \left(\mathit{d\xi }\right)\\ =\underset{\left\{⟨\xi ,e_0⟩\in \left[\mathbf{0},\mathbf{2}\mathbf{\pi }\right[\right\}}{\int }exp\left(\mathrm{im}{⟨\xi ,e_0⟩}_{\mathbf{H}}\right)\sum _l{\left|\widehat{\mathrm{\Psi }}\left(\xi +2\pi {\mathit{le}}_0\right)\right|}^2\mu \left(\mathit{d\xi }\right).\end{array}$$

Now

$$\begin{array}{c}\sum _l{\left|\widehat{\mathrm{\Psi }}\left(\xi +2\pi {\mathit{le}}_0\right)\right|}^2=\sum _l{\left|m_0\left(\xi /2+\pi {\mathit{le}}_0+\pi e_0\right)\right|}^2{\left|\widehat{\mathrm{\Phi }}\left(\xi /2+\pi {\mathit{le}}_0\right)\right|}^2\\ ={\left|m_0\left(\xi /2+\pi e_0\right)\right|}^2\sum _n{\left|\widehat{\mathrm{\Phi }}\left(\xi /2+2\pi {\mathit{ne}}_0\right)\right|}^2\\ +{\left|m_0\left(\xi /2\right)\right|}^2\sum _n{\left|\widehat{\mathrm{\Phi }}\left(\xi /2+\pi e_0+2\pi {\mathit{ne}}_0\right)\right|}^2\\ ={\left(2\pi \right)}^{-1}\left({\left|m_0\left(\xi /2\right)\right|}^2+{\left|m_0\left(\xi /2+\pi e_0\right)\right|}^2\right)\mu -\mathrm{a}.\mathrm{e}.\left(\mathrm{by}\left(1.19\right)\right)\\ ={\left(2\pi \right)}^{-1}\mu -\mathrm{a}.\mathrm{e}.\left(\mathrm{by}\left(1.20\right)\right).\end{array}$$

Hence $\underset{\mathbf{H}}{\int }\mathrm{\Psi }\left(h\right)\left(\overline{\mathrm{\Psi }\left(h–{\mathit{me}}_0\right)}\right)\mu \left(\mathit{dh}\right)={\delta }_{m,0}$. In order to check that the ${\mathrm{\Psi }}_{0,m}$ are indeed a basis for all W₀, it then suffices to check that any $\phi \in {\mathbf{W}}_0$ can be written as $\phi ={\sum }_n^{}{\gamma }_n{\mathrm{\Psi }}_{0,n}$, with ${\sum }_n^{}{\left|{\gamma }_n\right|}^2<\infty$, or

$$\widehat{\phi }\left(\xi \right)=\gamma \left(\xi \right)\widehat{\mathrm{\Psi }}\left(\xi \right),$$E30

with $\gamma$ a square integrable $2\pi e_0$-periodic Wiener functional. Let us return to (28).

We have $\widehat{\phi }\left(\xi \right)=\nu \left(\xi \right)\widehat{\mathrm{\Psi }}\left(\xi \right)$, with $\nu \in L^2\left(\mu \right)$. By (22),

$$\underset{\mathbf{H}}{\int }{\left|m_{\phi }\left(\xi \right)\right|}^2\mu \left(\mathit{d\xi }\right)=\pi \sum _n{\left|{\lambda }_n\right|}^2=\pi {\left({‖\phi ‖}_{\mu }\right)}^2<\infty .$$

On the other hand, by (25),

$$\underset{\mathbf{H}}{\int }{\left|m_{\phi }\left(\xi \right)\right|}^2\mu \left(\mathit{d\xi }\right)=\underset{\mathbf{H}}{\int }{\left|\mathrm{\Theta }\left(\xi \right)\right|}^2{\left|m_0\left(\xi +\pi e_0\right)\right|}^2\mu \left(\mathit{d\xi }\right)$$

$$=\underset{\mathbf{H}}{\int }{\left|\mathrm{\Theta }\left(\xi +\pi e_0\right)\right|}^2{\left|m_0\left(\xi +\pi e_0\right)\right|}^2\mu \left(\mathit{d\xi }\right),\text{using}\left(1.26\right)$$E31

Now, with the change of variable $h=\xi +\pi e_0$ and with the help of stochastic calculus we find for this integral

$$\begin{array}{l}exp\left[\left(1–{\pi }^2\right)/2\right]\underset{\mathbf{H}}{\int }{\left|\mathrm{\Theta }\left(h\right)\right|}^2{\left|m_0\left(h\right)\right|}^2\mu \left(\mathit{dh}\right)\\ =exp\left[\left(1–{\pi }^2\right)/2\right]\underset{\mathbf{H}}{\int }{\left|\mathrm{\Theta }\left(h\right)\right|}^2\left(1-{\left|m_0\left(h+\pi e_0\right)\right|}^2\right)\mu \left(\mathit{dh}\right),\text{using}\left(1.20\right),\\ =exp\left[\left(1–{\pi }^2\right)/2\right]\underset{\mathbf{H}}{\int }{\left|\mathrm{\Theta }\left(h\right)\right|}^2\mu \left(\mathit{dh}\right)\\ –exp\left[\left(1–{\pi }^2\right)/2\right]\underset{\mathbf{H}}{\int }{\left|\mathrm{\Theta }\left(h+\pi e_0\right)\right|}^2{\left|m_0\left(h+\pi e_0\right)\right|}^2\mu \left(\mathit{dh}\right).\end{array}$$

Put: $I≔\underset{\mathbf{H}}{\int }{\left|\mathrm{\Theta }\left(\xi +\pi e_0\right)\right|}^2{\left|m_0\left(\xi +\pi e_0\right)\right|}^2\mu \left(\mathit{d\xi }\right)$. We hence have, combining (31) and this last equality:

$I=exp\left[\left(1–{\pi }^2\right)/2\right]\underset{\mathbf{H}}{\int }{\left|\mathrm{\Theta }\left(h\right)\right|}^2\mu \left(\mathit{dh}\right)–exp\left[\left(1–{\pi }^2\right)/2\right]I$, which implies

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

Hence, ${\left({‖\nu ‖}_{\mu }\right)}^2=2\pi {\left({‖\phi ‖}_{\mu }\right)}^2<\infty$, and $\phi$ is of the form (30) with $\gamma \in L^2\left(\mu \right)$ and $2\pi e_0$-periodic. We have thus prove the following theorem.

Theorem 1.1. If a ladder of closed subspaces ${\left({\mathbf{V}}_j\right)}_{j\in \mathrm{\mathbb{Z}}}$ in $L^2\left(\mu \right)$ satisfies the conditions (1)–(6), then there exists an orthonormal Wiener wavelet basis $\left\{{\mathrm{\Psi }}_{j,k};j,k\in \mathrm{\mathbb{Z}}\right\}$ for $L^2\left(\mu \right)$ such that

$${\mathrm{\Pi }}_{j-1}={\mathrm{\Pi }}_j+\sum _k{⟨.,{\mathrm{\Psi }}_{j,k}⟩}_{\mu }{\mathrm{\Psi }}_{j,k}.$$E32

One possibility for the construction of the Wiener wavelet $\mathrm{\Psi }$ is,

$$\widehat{\mathrm{\Psi }}\left(\xi \right)=exp\left(\mathrm{i}{⟨\xi ,e_0⟩}_{\mathbf{H}}/2\right)\left(\overline{m_0\left(\xi /2+\pi e_0\right)}\right)\widehat{\mathrm{\Phi }}\left(\xi /2\right),$$

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

$$\mathrm{\Psi }=\sum _n{\left(-1\right)}^n{c}_{-n-1}{\mathrm{\Phi }}_{-1,n},$$E33

$\mathrm{\Psi }\left(\omega \right)=2^{-1/2}\sum _n{\left(-1\right)}^n{c}_{-n-1}\mathrm{\Phi }\left(2\omega –{\mathit{ne}}_0\right)$, with convergence in this series in $L^2\left(\mu \right)$ – 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 $\mu$-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 $L^2\left(\mu \right)$, as we can find it in [6] for the one-dimensional case.

1.2 Multiresolution approximation generated by cardinal B-splines in ${\mathbf{L}}^{\mathbf{2}}\left(\mu \right)$

Let us first beginning with some recalls. For n, r positive integers, n even, such that $n\ge 2r$, put

$$\underset{n}{S}≔\left\{f\in C^{n-r-1}\left(\mathrm{ℝ}\right):f_{\left|\left[v,v+1\right[\right.}\in {\mathbf{P}}_{n-1},v\in \mathrm{\mathbb{Z}}\right\},$$E34

where P_n-1 is the class of polynomials of degree $\le n–1$. Functions in ${\mathrm{S}}_n^r$ are called cardinal Hermite splines of degree≤ than n–1.

For $j=0,\dots ,r–1$, let

$$S_{n,j}^r≔\left\{f\in S_n^r:f^{\left(k\right)}\left(v\right)=0,v\in \mathrm{\mathbb{Z}},k=0,\dots ,r–1,k\ne j\right\}.$$E35

The space $S_{n,j}^r$ has a basis consisting of integer translates of a function $N_{n,j}=N_{n,j}^r\in S_{n,j}^r,j=0,\dots ,r–1$, with minimal support $\left[-n/2–1+r,n/2+1-r\right]$, in the sense that every $f\in S_{n,j}^r$ has a unique representation of the form

$$f\left(x\right)=\sum _{v\in \mathrm{\mathbb{Z}}}{c}_v{N}_{n,j}\left(x–v\right),x\in \mathrm{ℝ},$$E36

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

$${\widehat{N}}_{n,j}\left(u\right)={\left[2sin\left(u/2\right)\right]}^n\left|H_{r,j}\left({\alpha }_n\left(u\right)\right)\right|$$E37

where $H_{r,j}\left({\alpha }_n\left(u\right)\right)$ denotes the matrix obtained from the Hankel matrix $H_r\left({\alpha }_n\left(u\right)\right)$ of order r by replacing the $\left(j+1\right)$ th column by ${\left(u^n,u^{n-1},\dots ,u^{n-r+1}\right)}^T$,

$j=0,\dots ,r–1$, denoting by $\left|H_r\left({\alpha }_n\left(u\right)\right)\right|$ its determinant.

Consider the map $K:{\mathrm{\ell }}^2{\left(\mathrm{\mathbb{Z}}\right)}^r\to L^2\left(\mu \right)$ defined by the following:

$$K\left(s\right)\left(\omega \left(.\right)\right)≔\sum _{j:0\to r-1}\sum _{v\in \mathrm{\mathbb{Z}}}{s}_j\left(v\right){\mathbf{T}}^v{N}_{n,j}\left(\omega \left(.\right)\right),\omega \in C\left(\left[0,1\right]\right),$$E38

T being a unitary operator on $L^2\left(\mu \right)$.

Theorem 2.1. The map K defined by (38) is an isomorphism of ${\mathrm{\ell }}^2{\left(\mathrm{\mathbb{Z}}\right)}^r$ onto a subspace of $L^2\left(\mu \right)$.

This above theorem is an easy consequence of Theorem 4.1 in [6]. Let us denote by $\stackrel{\sim }{{\mathrm{S}}_n^r}\left(\mu \right)$ the range of K. This is a closed subspace of $L^2\left(\mu \right)$. Furthermore, $\stackrel{\sim }{{\mathrm{S}}_n^r}\left(\mu \right)={\mathrm{S}}_n^r\left(\mu \right)\cap L^2\left(\mu \right)$ where ${\mathrm{S}}_n^r\left(\mu \right)$ is the space deduced from ${\mathrm{S}}_n^r$ by the following: $\phi \in S_n^r\left(\mu \right)\iff \phi \left(\omega \left(t\right)\right)\in {\mathrm{S}}_n^r$, $\omega \in C\left(\left[0,1\right]\right)$, $t\in \left[0,1\right]$. Therefore, if $\mathit{D\phi }\left(\omega \right)≔\phi \left(2\omega \right)$, we have as in real analysis:

The closed space generated by $\overline{\underset{m\in \mathrm{\mathbb{Z}}}{\cup }{D}^m\stackrel{\sim }{{\mathrm{S}}_n^r}\left(\mu \right)}$ in $L^2\left(\mu \right)$ contains the one generated by $\overline{\underset{m\in \mathrm{\mathbb{Z}}}{\cup }{D}^m\stackrel{\sim }{{\mathrm{S}}_n^1}\left(\mu \right)}$ which is $L^2\left(\mu \right)$.

We also have:

$$\underset{m\in \mathrm{\mathbb{Z}}}{\cap }{D}^m\stackrel{\sim }{{\mathrm{S}}_n^r}\left(\mu \right)=\left\{0\right\}.$$

Let ${\mathbf{V}}_m≔D^m\stackrel{\sim }{{\mathrm{S}}_n^r}\left(\mu \right)$, $m\in \mathrm{\mathbb{Z}}$. Then, ${\left({\mathbf{V}}_m\right)}_{m\in \mathrm{\mathbb{Z}}}$ is a multiresolution approximation of $L^2\left(\mu \right)$. We shall call ${\left({\mathbf{V}}_m\right)}_{m\in \mathrm{\mathbb{Z}}}$ a Wiener Hermite spline multiresolution approximation of $L^2\left(\mu \right)$. 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 $L^p\left(\mu \right)$.

1.3 Unconditional bases for ${\mathbf{L}}^{\mathbf{p}}\left(\mathbf{\mu }\right)$

Orthonormal bases of wavelets in $L^2\left(\mu \right)$ give good (i.e., unconditional) bases for many other spaces than L². We start by proving the following extension of Calderon–Zygmund decomposition theorem. We denote in this section by $\left|\omega \right|$ for $\omega \in \mathrm{W}≔{\mathrm{C}}_0\left(\left[0,1\right];{\mathrm{ℝ}}^d\right)$ the sup norm [15, 16].

Theorem 3.1. Assume φ to be a positive Wiener functional in $L^1\left(\mu \right)$.

Fix $\alpha >0$. Then the Wiener space $\mathbf{W}≔{\mathrm{C}}_0\left(\left[0,1\right];{\mathrm{ℝ}}^d\right)$ can be decomposed as follows:

1. $\mathbf{W}=\mathbf{G}\cup \mathbf{B}$ with $\mathbf{G}\cap \mathbf{B}=\varnothing$;

2. On the “ good ” set G, $\phi \left(\omega \right)\le \alpha \mu$ - a. e.;

3. The “ bad ” set B can be written as $\mathbf{B}=\underset{k\in \mathrm{\mathbb{N}}}{\cup }{\mathbf{Q}}_k$, where the Q_k are non-overlapping intervals in the Banach lattice W, and

$$\alpha \le \mu {\left({\mathbf{Q}}_k\right)}^{-1}\underset{{\mathbf{Q}}_k}{\int }\phi \left(\omega \right)\mu \left(\mathit{d\omega }\right)\le 2\alpha ,\forall k\in \mathrm{\mathbb{N}}.$$

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

$$\left(\mathit{T\phi }\right)\left(\omega \right)=\underset{\mathbf{H}}{\int }K\left(\omega ,\xi \right)\mu \left(\mathit{d\xi }\right)$$E39

for which the integral kernels satisfies

$$\left|K\left(\omega ,\xi \right)\right|\le C/\left|\omega -\xi \right|,$$E40

$${‖{\nabla }_{\omega }K\left(\omega ,\xi \right)‖}_{\mathbf{H}}+{‖{\nabla }_{\xi }K\left(\omega ,\xi \right)‖}_{\mathbf{H}}\le C/\left|\omega -\xi \right|,$$E41

where the derivation symbol $\nabla$ is the Malliavin derivative, and which defines a bounded operator on $L^2\left(\mu \right)$.

Theorem 3.3. A Calderon–Zygmund operator on W is also a bounded operator from $L^1\left(\mu \right)$ to $L_{\mathit{\text{weak}}}^1\left(\mu \right)$.

We recall below the definition of $L_{\mathit{\text{weak}}}^1\left(\mu \right)$.

Definition 3.4.$\phi \in L_{\mathit{\text{weak}}}^1\left(\mu \right)$ if there exists $C>0$ so that, for all $\alpha >0$,

$$\mu \left\{\omega \in \mathbf{W}/\left|\phi \left(\omega \right)\right|\ge \alpha \right\}\le C/\alpha .$$E42

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 $\alpha >0$) will be called ${‖\phi ‖}_{L_{\mathit{\text{weak}}}^1}$. 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 $L^2\left(\mu \right)$ to $L^2\left(\mu \right)$ and $L^1\left(\mu \right)$ to $L_{\mathit{\text{weak}}}^1\left(\mu \right)$, we can extend T to other $L^p\left(\mu \right)$-spaces by interpolation theorem of Marcinkiewicz.

Theorem 3.5. If an operator T satisfies

$${‖\mathit{T\phi }‖}_{\mathit{Lq}\left(1\right)\left(\mu \right)\mathit{\text{weak}}}\le C_1{‖\phi ‖}_{\mathit{Lp}\left(1\right)\left(\mu \right)},$$E43

$${‖\phi ‖}_{\mathit{Lq}\left(2\right)\left(\mu \right)\mathit{\text{weak}}}\le C_2{‖\phi ‖}_{\mathit{Lp}\left(2\right)\left(\mu \right)},$$E44

where $q_1\le p_1$, $q_2\le p_2$, then for $1/p=t/p_1+\left(1–t\right)/p_2$,

$1/q=t/q_1+\left(1–t\right)/q_2$, with $0<t<1$, there exists a constant K, depending on p₁,q₁,p₂,q₂, and t, so that

$${‖\mathit{T\phi }‖}_{\mathit{Lq}\left(\mu \right)}\le K{‖\phi ‖}_{\mathit{Lp}\left(\mu \right)}.$$

Here $L_q{\left(\mu \right)}_{\mathit{\text{weak}}}$ stands for the space of all Wiener functionals $\phi$ for which ${‖\phi ‖}_{\mathit{Lq}\left(\mu \right)\mathit{\text{weak}}}≔{\left(inf\left\{C/\mu \left\{\omega ;\left|\phi \left(\omega \right)\right|\ge \alpha \right\}\le C{\alpha }^{-q}\text{for}\mathrm{all}\alpha >0\right\}\right)}^{1/q}$ 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 $L^q\left(\mu \right)$-norms (not $L_{\mathit{\text{weak}}}^q\left(\mu \right)$) for intermediate values q. The Marcinkiewicz interpolation theorem implies that the $L^1\left(\mu \right)\to L_{\mathit{\text{weak}}}^1\left(\mu \right)$-boundedness proved in Theorem 3.3 is sufficient to derive $L^p\left(\mu \right)\to L^p\left(\mu \right)$ boundedness for $1<p<\infty$, as follows.

Theorem 3.6. If T is an integral operator with integral kernel K satisfying (40, 41), and if T is bounded from $L^2\left(\mu \right)$ to $L^2\left(\mu \right)$, then T extends to a bounded operator from $L^p\left(\mu \right)$ to $L^p\left(\mu \right)$ for all $p\in \left]1,\infty \right[$.

Proof.

1. Theorem 3.3 proves that T is bounded from $L^1\left(\mu \right)$ to $L_{\mathit{\text{weak}}}^1\left(\mu \right)$; by Marcinkiewicz’s theorem, T extends to a bounded operator from $L^p\left(\mu \right)$ to $L^p\left(\mu \right)$ for $1<p\le 2$.

2. For the range $2\le p<\infty$, we use the adjoint T^* of T, defined by

$$\int \left(T^{\ast }\phi \right)\left(\omega \right)\left(\overline{\psi \left(\omega \right)}\right)\mu \left(\mathit{d\omega }\right)=\int \phi \left(\omega \right)\left(\overline{\left(\mathit{T\psi }\right)\left(\omega \right)}\right)\mu \left(\mathit{d\omega }\right).$$

It is associated with the integral kernel $K_0\left(\omega ,\xi \right)=\left(\overline{K\left(\xi ,\omega \right)}\right)$, 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 $L^1\left(\mu \right)$ to $L_{\mathit{\text{weak}}}^1\left(\mu \right)$, and hence by Theorem 3.5, that it is bounded from $L^p\left(\mu \right)$ to $L^p\left(\mu \right)$ for $1<p\le 2$. Since for $1/p+1/q=1$, T^*: $L^p\left(\mu \right)\to L^p\left(\mu \right)$ is the adjoint operator of T: $L^q\left(\mu \right)\to L^q\left(\mu \right)$, it follows that T is bounded for $2\le q<\infty$. More explicitly, for readers unfamiliar with adjoints on Banach spaces,

$$\begin{array}{c}{‖\mathit{T\phi }‖}_q={sup}_{\psi \in L^p,,‖\psi ‖=1}\left|\int \left(\mathit{T\phi }\right)\left(\omega \right)\left(\overline{\psi \left(\omega \right)}\right)\mu \left(\mathit{d\omega }\right)\right|\left(\text{if}1/p+1/q=1\right)\\ ={sup}_{\psi \in L^p,‖\psi ‖=1}\left|\int \left(\int \phi \left(\xi \right)K\left(\omega ,\xi \right)\left(\overline{\psi \left(\omega \right)}\right)\mu \left(\mathit{d\xi }\right)\right)\mu \left(\mathit{d\omega }\right)\right|\\ ={sup}_{\psi \in L^p,‖\psi ‖=1}\left|\int \phi \left(\xi \right)\left(T^{\ast }\psi \right)\left(\xi \right)\mu \left(\mathit{d\psi }\right)\right|\\ \le {sup}_{\psi \in L^p,‖\psi ‖=1}{‖\phi ‖}_{L^q\left(\mu \right)}{‖T^{\ast }\psi ‖}_{L^p\left(\mu \right)}\le C{‖\phi ‖}_{L^q\left(\mu \right)}.\end{array}$$

We now apply this to prove that a Wiener functional has some decay and some regularity and if the ${\mathrm{\Psi }}_{j,k}\left(\omega \right)≔2^{-j/2}\left(2^{-j}\omega –{\mathit{ke}}_0\right)$, $j,k\in \mathrm{\mathbb{Z}}$, $\omega \in \mathbf{W}$, and where e₀ belongs to the Cameron–Martin space H, constitute an orthonormal basis for $L^2\left(\mu \right)$, then the ${\mathrm{\Psi }}_{j,k}$ also provide unconditional bases for $L^p\left(\mu \right)$, $1<p<\infty$.

What we need to prove is that if $\phi ={\sum }_{j,k\in \mathrm{\mathbb{Z}}}^{}{c}_{j,k}{\mathrm{\Psi }}_{j,k}\in L^p\left(\mu \right)$, then ${\sum }_{j,k\in \mathrm{\mathbb{Z}}}^{}{\epsilon }_{j,k}{c}_{j,k}{\mathrm{\Psi }}_{j,k}\in L^p\left(\mu \right)$ for any choice of the ${\epsilon }_{j,k}=\pm 1$ (see Preliminaries in [8]).

We will assume that $\psi$ is continuously Malliavin differentiable on the lattice space (W, |.|, ≤) where |.| denotes the sup norm, and that both $\psi$, $\nabla \psi$ decay faster than ${\left(1+\left|\omega \right|\right)}^{-1}$^:

$$\left|\psi \left(\omega \right)\right|,{‖\nabla \psi \left(\omega \right)‖}_{\mathbf{H}}\le C{\left(1+\left|\omega \right|\right)}^{-1-\epsilon }.$$E45

Then $\mathrm{\Psi }\in L^p\left(\mu \right)$ for $1<p<\infty$, and $\phi =\sum _{j,k}{c}_{j,k}{\psi }_{j,k}$ implies that $c_{j,k}=\underset{\mathbf{H}}{\int }\phi \left(\omega \right){\psi }_{j,k}\left(\omega \right)\mu \left(\mathit{d\omega }\right)$ because of the orthonormality of the ${\psi }_{j,k}$. We therefore want to show that, for aby choice of the ${\epsilon }_{j,k}=\pm 1$, $T_{\epsilon }$ defined by

$$T_{\epsilon }\phi =\sum _{j,k}{\epsilon }_{j,k}{⟨\phi ,{\psi }_{j,k}⟩}_{L^2\left(\mu \right)}{\psi }_{j,k}$$

Is a bounded operator from $L^p\left(\mu \right)$ to $L^p\left(\mu \right)$. First, we know that $T_{\epsilon }$ is bounded from $L^2\left(\mu \right)$ to $L^2\left(\mu \right)$, since, denoting by ${‖.‖}_{\mu }$ (resp. <.,. > _μ) the norm (resp. the scalar product) in $L^2\left(\mu \right)$,

$${\left({‖T_{\epsilon }\phi ‖}_{\mu }\right)}^2=\sum _{j,k}{\left|{\epsilon }_{j,k}{⟨\phi ,{\psi }_{j,k}⟩}_{\mu }\right|}^2=\sum _{j,k}{\left|{⟨\phi ,{\psi }_{j,k}⟩}_{\mu }\right|}^2={\left({‖\phi ‖}_{\mu }\right)}^2,$$

so the $L^p\left(\mu \right)$-boundedness will follow by Theorem 3.5 if we can prove that $T_{\epsilon }$ is an integral operator with kernel satisfying (40, 41). This is the content of the following lemma.

Lemma 3.7. Choose ${\epsilon }_{j,k}=\pm 1$, and define $K\left(\omega ,\xi \right)≔\sum _{j,k}{\epsilon }_{j,k}{\psi }_{j,k}\left(\omega \right)\left(\overline{{\psi }_{j,k}\left(\xi \right)}\right)$. Then there exists C > 0 so that

$$\left|K\left(\omega ,\xi \right)\right|\le C{\left|\omega -\xi \right|}^{-1}$$

and

$${‖{\nabla }_{\omega }K\left(\omega ,\xi \right)‖}_{\mathbf{H}}+{‖{\nabla }_{\xi }K\left(\omega ,\xi \right)‖}_{\mathbf{H}}\le C{\left|\omega -\xi \right|}^{-2}.$$

Proof.

1. $\begin{array}{c}\left|K\left(\omega ,\xi \right)\right|\le \sum _{j,k}\left|{\psi }_{j,k}\left(\omega \right)\right|\cdot \left|{\psi }_{j,k}\left(\xi \right)\right|\\ \le C\sum _{j,k}{2}^{-j}{\left(1+\left|2^{-j}\omega –{\mathit{ke}}_0\right|\right)}^{-1-\epsilon }{\left(1+\left|2^{-j}\xi -{\mathit{ke}}_0\right|\right)}^{-1-\epsilon }\mathrm{by}\left(3.13\right).\end{array}$

Find $j\left(0\right)\in \mathrm{\mathbb{Z}}$ so that $2^{j\left(0\right)}\le \left|\omega -\xi \right|\le 2^{j\left(0\right)+1}$. We split the sum over j into two parts:

$j<j_0$ and $j\ge j\left(0\right)$.

1. $\sum _k{\left(1+\left|a-k\right|\right)}^{-1-\epsilon }{\left(1+\left|b-k\right|\right)}^{-1-\epsilon }$ is uniformly bounded for all values of $a,b\in \mathrm{ℝ}$: in fact,

$$\begin{array}{l}\sum _k{\left(1+\left|a-k\right|\right)}^{-1-\epsilon }{\left(1+\left|b-k\right|\right)}^{-1-\epsilon }\le \sum _k{\left(1+\left|a-k\right|\right)}^{-1-\epsilon }\\ \le {sup}_{0\le a^{\prime }\le 1}\sum _k{\left(1+\left|a^{\prime }-k\right|\right)}^{-1-\epsilon }\le 2\sum _{l\in \mathrm{\mathbb{N}}}{\left(1+l\right)}^{-1-\epsilon }<\infty .\end{array}$$

Hence,

$$\begin{array}{l}\sum _{j\ge j\left(0\right)}\sum _k{2}^{-j}{\left(1+\left|2^{-j}\omega –\mathit{ke}\left(0\right)\right|\right)}^{-1-\epsilon }{\left(1+\left|2^{-j}\xi -\mathit{ke}\left(0\right)\right|\right)}^{-1-\epsilon }\\ \le C\sum _{j\ge j\left(0\right)}{2}^{-j}\le C{.2}^{-j\left(0\right)+1}\le 4C{\left|\omega -\xi \right|}^{-1}.\end{array}$$

1. The part $j<j\left(0\right)$ is a little less easy.

$$\begin{array}{l}\sum _{-\infty <j\le j\left(0\right)-1}{2}^{-j}\sum _k{\left[\left(1+\left|2^{-j}\omega –\mathit{ke}\left(0\right)\right|\right).\left(1+\left|2^{-j}\xi -\mathit{ke}\left(0\right)\right|\right)\right]}^{-1-\epsilon }\\ =\sum _{j\ge -j\left(0\right)+1}{2}^j\sum _k{\left[\left(1+\left|2^j\omega –\mathit{ke}\left(0\right)\right|\right).\left(1+\left|2^j\xi -\mathit{ke}\left(0\right)\right|\right)\right]}^{-1-\epsilon }\end{array}$$

$$\le 2^{1+\epsilon }\sum _{j\ge j\left(0\right)+1}{2}^j\sum _k{\left[\left(2+\left|2^j\omega –\mathit{ke}\left(0\right)\right|\right).\left(2+\left|2^j\xi -\mathit{ke}\left(0\right)\right|\right)\right]}^{-1-\epsilon }.$$E46

Find $k_0\in \mathrm{\mathbb{Z}}$ so that $k_0\le 2^j\left(\left|\omega \right|+\left|\xi \right|\right)/2\le k_0+1$ and define $l=k–k_0$. Then

$$\begin{array}{c}2+\left|2^j\omega –\mathit{ke}\left(0\right)\right|=2+\left|2^j\left(\omega -\xi \right)/2-\mathit{le}\left(0\right)+\left(2^j\left(\omega +\xi \right)/2–k_{0}e\left(0\right)\right)\right|\\ \ge 1+\left|2^j\left(\omega -\xi \right)/2–\mathit{le}\left(0\right)\right|;\end{array}$$

similarly,

$$2+\left|2^j\xi -\mathit{ke}\left(0\right)\right|\ge 1+\left|2^j\left(\xi -\omega \right)/2–\mathit{le}\left(0\right)\right|.$$

Consequently, with ${\omega }_1≔2^j\left(\omega -\xi \right)/2$,

$$\begin{array}{c}\sum _k{\left[\left(2+\left|2^j\omega –\mathit{ke}\left(0\right)\right|\right).\left(2+\left|2^j\xi -\mathit{ke}\left(0\right)\right|\right)\right]}^{-1-\epsilon }\\ \le \sum _l{\left[\left(1+\left|{\omega }_1+\mathit{le}\left(0\right)\right|\right).\left(1+\left|{\omega }_1+\mathit{le}\left(0\right)\right|\right)\right]}^{-1-\epsilon }\le C{\left(1+\left|{\omega }_1\right|\right)}^{-1-\epsilon },\end{array}$$

so that

$$\begin{array}{l}\le C\sum _{j\ge -j\left(0\right)+1}{2}^j{\left(1+2^j\left|\omega -\xi \right|/2\right)}^{-1-\epsilon }\\ \le C\sum _{j^{\prime }\ge 1}{2}^{j^{\prime }-j\left(0\right)}{\left(1+2^{j^{\prime }–j\left(0\right)}\left(1/2\right)2^{j\left(0\right)+1}\right)}^{-1-\epsilon }\mathrm{as}\text{we have}\left|\omega -\xi \right|\le 2^{j\left(0\right)+1},\\ \le C{2}^{-j\left(0\right)}\sum _{j^{\prime }\ge 1}{2}^{j^{\prime }}{\left(1+2^{j^{\prime }}\right)}^{-1-\epsilon }\\ \le C^{\prime }{2}^{-j\left(0\right)}\le 2C^{\prime }{\left|\omega -\xi \right|}^{-1}.\end{array}$$E47

It therefore follows that $\left|K\left(\omega ,\xi \right)\right|\le C{\left|\omega -\xi \right|}^{-1}$

1. For the estimates on ${\nabla }_{\omega }K$ and ${\nabla }_{\xi }K$, we write

$$\begin{array}{c}{‖{\nabla }_{\omega }K\left(\omega ,\xi \right)‖}_{\mathbf{H}}\le \sum _{j,k}{2}^{-j}{‖\nabla \psi \left(2^{-j}\omega –{\mathit{ke}}_0\right)‖}_{\mathbf{H}}\left|\psi \left(2^{-j}\omega –{\mathit{ke}}_0\right)\right|\\ \le C\sum _{j,k}{2}^{-2j}{\left[\left(1+\left|2^{-j}\omega –{\mathit{ke}}_0\right|\right)\left(1+\left|2^{-j}\xi -{\mathit{ke}}_0\right|\right)\right]}^{-1-\epsilon }\end{array}$$

and we follow the same technique; we obtain.

${‖{\nabla }_{\omega }K\left(\omega ,\xi \right)‖}_{\mathbf{H}},{‖{\nabla }_{\xi }K\left(\omega ,\xi \right)‖}_{\mathbf{H}}\le C{\left|\omega -\xi \right|}^{-2}.$∎

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

Theorem3.8. If $\psi$ is a Wiener functional continuously Malliavin differentiable and $\left|\psi \left(\omega \right)\right|,{‖\nabla \psi \left(\omega \right)‖}_{\mathbf{H}}\le C{\left(1+\left|\omega \right|\right)}^{-1-\epsilon }$, and if the ${\psi }_{j,k}\left(\omega \right)≔2^{-j/2}\psi \left(2^{-j}\omega –{\mathit{ke}}_0\right)$ constitute an orthonormal basis for $L^2\left(\mu \right)$, $e_0\in \mathbf{H}$ being given, then the $\left\{{\psi }_{j,k};j,k\in \mathrm{\mathbb{Z}}\right\}$ also constitute an unconditional basis for all $L^p\left(\mu \right)$ – spaces, $1<p<\infty$.

1.4 Periodized Wiener wavelets

Even if L¹-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 $\mathrm{\varphi }$ and Wiener wavelet $\psi$, both with reasonable decay (say, $\left|\varphi \left(\omega \right)\right|,\left|\psi \left(\omega \right)\right|\le C{\left(1+\left|\omega \right|\right)}^{-1-\epsilon }$), we define

$${\mathrm{\varphi }}_{j,k}^{\mathit{per}}\left(\omega \right)≔\sum _{l\in \mathrm{\mathbb{Z}}}{\mathrm{\varphi }}_{j,k}\left(\omega +{\mathit{le}}_0\right),{\psi }_{j,k}^{\mathit{per}}\left(\omega \right)≔\sum _{l\in \mathrm{\mathbb{Z}}}{\psi }_{j,k}\left(\omega +{\mathit{le}}_0\right);$$

and

$$\begin{array}{l}{\mathbf{V}}_j^{\mathit{per}}≔\mathit{\text{closure of span}}\left\{{\mathrm{\varphi }}_{j,k}^{\mathit{per}};k\in \mathrm{\mathbb{Z}}\right\},\\ {\mathbf{W}}_j^{\mathit{per}}≔\mathit{\text{closure of span}}\left\{{\psi }_{j,k}^{\mathit{per}};k\in \mathrm{\mathbb{Z}}\right\}\end{array}$$

First, notice that we have: ${\sum }_{l\in \mathrm{\mathbb{Z}}}^{}\mathrm{\varphi }\left(\omega +{\mathit{le}}_0\right)=1$. In fact, put $\phi \left(\omega \right)≔{\sum }_{l\in \mathrm{\mathbb{Z}}}^{}\mathrm{\varphi }\left(\omega +{\mathit{le}}_0\right)$. The conditions on $\mathrm{\varphi }$ (continuity and its “ reasonable decay ”) ensure that $\phi$ is well defined and continuous. Moreover, we can write:

$$\mathrm{\varphi }\left(\omega \right)=\sum _{n\in \mathrm{\mathbb{Z}}}{c}_n\mathrm{\varphi }\left(2\omega –{\mathit{ne}}_0\right)\text{with}{\left(c_n\right)}_n\in {\mathrm{\ell }}^2\left(\mathrm{\mathbb{Z}}\right).$$

Then, $\begin{array}{c}\phi \left(\omega \right)=\sum _l\sum _n{c}_n\mathrm{\varphi }\left(2\omega –2{\mathit{le}}_0–{\mathit{ne}}_0\right)=\sum _l\sum _m{c}_{m-2l}\mathrm{\varphi }\left(2\omega –{\mathit{me}}_0\right)\\ =\sum _m\left(\sum _j{c}_{m-2j}\right)\mathrm{\varphi }\left(2\omega –{\mathit{me}}_0\right)=\sum _m\mathrm{\varphi }\left(2\omega –{\mathit{me}}_0\right)=\phi \left(2\omega \right).\end{array}$

Hence, $\phi$ is continuous, periodic with period e₀, and

$$\phi \left(\omega \right)=\phi \left(2\omega \right)=\dots =\phi \left(2^n\omega \right)=\dots$$

It follows that $\phi$ is a constant Wiener functional. We then put:

$$\sum _l\mathrm{\varphi }\left(\omega –{\mathit{le}}_0\right)=c$$

Since

$\int \mathrm{\varphi }\mathit{d\mu }=1$, this constant is necessarily equal to 1. We deduce, for $j\ge 0$, ${\mathrm{\varphi }}_{j,k}^{\mathit{per}}\left(\omega \right)=2^{-j/2}\sum _l\mathrm{\varphi }\left(2^{-j}\omega –{\mathit{ke}}_0+2^{-j}{\mathit{le}}_0\right)=2^{j/2}$, so that the ${\mathbf{V}}_j^{\mathit{per}}$, for $j\ge 0$, are all identical one-dimensional spaces, containing the constant functionals.

Similarly, $\sum _l\psi \left(\omega +\left(l/2\right)e_0\right)=0$. In fact, $\sum _l\psi \left(\omega +\left(l/2\right)e_0\right)=\sum _l\sum _n{\left(-1\right)}^n{c}_{-n+1}\mathrm{\varphi }\left(2\omega +{\mathit{le}}_0–{\mathit{ne}}_0\right)$ where the c_n are the coefficients appearing in (13), that is,

$$\begin{array}{c}{c}_n=⟨\mathrm{\varphi },{\mathrm{\varphi }}_{-1,n}⟩=\sum _{k,m}{\left(-1\right)}^{m+1}{c}_m\mathrm{\varphi }\left(2\omega +{\mathit{ke}}_0\right)\left(k=l–n,m=-n+1\right)\\ =0\left(\text{because}\sum _m{c}_{2m}=\sum _m{c}_{2m+1}\right).\end{array}$$

Hence, for $j\ge 1\ge$, ${\mathbf{W}}_j^{\mathit{per}}=\left\{0\right\}$. We therefore restrict our attention to the ${\mathbf{V}}_j^{\mathit{per}}$, ${\mathbf{W}}_j^{\mathit{per}}$ with $j\le 0$. Obviously ${\mathbf{V}}_j^{\mathit{per}}$, ${\mathbf{W}}_j^{\mathit{per}}\subset {\mathbf{V}}_{j-1}^{\mathit{per}}$, a property inherited from the non-periodized spaces. Moreover, ${\mathbf{W}}_j^{\mathit{per}}$ is still orthogonal to ${\mathbf{V}}_j^{\mathit{per}}$, because

$$\begin{array}{c}\underset{\mathbf{H}}{\int }{\mathrm{\Psi }}_{j,k}^{\mathit{per}}\left(\omega \right){\mathrm{\varphi }}_{j,k^{\prime }}^{\mathit{per}}\left(\omega \right)\mu \left(\mathit{d\omega }\right)=\sum _{l,l^{\prime }\in \mathrm{\mathbb{Z}}}{2}^{-j}\underset{\mathbf{H}}{\int }\psi \left(2^{-j}\omega +2^{-j}{\mathit{le}}_0–{\mathit{ke}}_0\right)\left(\overline{\mathrm{\varphi }\left(2^{-j}\omega +2^{-j}{l}^{\prime }{e}_0–k^{\prime }{e}_0\right)}\right)\mu \left(\mathit{d\omega }\right)\\ =\sum _{l,l^{\prime }\in \mathrm{\mathbb{Z}}}{2}^{\left|j\right|}\underset{\left]l^{\prime }{e}_0,\left(l^{\prime }+1\right)e\left(0\right)\right]}{\int }\mathrm{\Psi }\left(2^{\left|j\right|}\omega +2^{\left|j\right|}\left(l–l^{\prime }\right)e_0–{\mathit{ke}}_0\right)\overline{\mathrm{\varphi }\left(2^{\left|j\right|}\xi -k^{\prime }{e}_0\right)}\mu \left(\mathit{d\xi }\right),\end{array}$$

because $j\le 0$,

$$=\sum _{r\in \mathrm{\mathbb{Z}}}{⟨{\psi }_{j,k+2^{\left|j\right|}r},{\mathrm{\varphi }}_{j,k^{\prime }}⟩}_{\mu }=0.$$

It follows that, as in the non-periodized case, ${\mathbf{V}}_{j-1}^{\mathit{per}}={\mathbf{V}}_j^{\mathit{per}}\oplus {\mathbf{W}}_j^{\mathit{per}}$. The spaces ${\mathbf{V}}_j^{\mathit{per}}$, ${\mathbf{W}}_j^{\mathit{per}}$ are all finite-dimensional: since ${\mathrm{\Phi }}_{j,k+m{2}^{\left|j\right|}}={\mathrm{\Phi }}_{j,k}$ for $m\in \mathrm{\mathbb{Z}}$, and the same is true for $\mathrm{\Psi }$, both ${\mathbf{V}}_j^{\mathit{per}}$ and ${\mathbf{W}}_j^{\mathit{per}}$ are spanned by the 2^|j| Wiener functionals obtained from $k=0,1,\dots ,2^{\left|j\right|}-1$. These 2^|j| Wiener functionals are moreover orthonormal; in, for example, ${\mathbf{W}}_j^{\mathit{per}}$ we have, for $0\le k,k^{\prime }\le 2^{\left|j\right|}-1$,

$${⟨{\mathrm{\Psi }}_{j,k}^{\mathit{per}},{\mathrm{\Psi }}_{j,k^{\prime }}^{\mathit{per}}⟩}_{\mu }=\sum _{r\in \mathrm{\mathbb{Z}}}{⟨{\psi }_{j,k+2^{\left|j\right|}r},{\psi }_{j,k^{\prime }}⟩}_{\mu }={\delta }_{k;k^{\prime }}.$$

We have therefore a ladder of multiresolution spaces, ${\mathbf{V}}_0^{\mathit{per}}\subset {\mathbf{V}}_{-1}^{\mathit{per}}\subset {\mathbf{V}}_{-2}^{\mathit{per}}\subset \dots$, with successive orthogonal complements ${\mathbf{W}}_0^{\mathit{per}}\left(\text{of}{\mathbf{V}}_0^{\mathit{per}}\text{in}{\mathbf{V}}_{-1}^{\mathit{per}}\right)$, ${\mathbf{W}}_1^{\mathit{per}},\dots ,$ and orthonormal bases $\left\{{\mathrm{\varphi }}_{j,k};k=0,\dots ,2^{\left|j\right|}-1\right\}$ in ${\mathbf{V}}_j^{\mathit{per}},\left\{{\psi }_{j,k};k=0,\dots ,2^{\left|j\right|}-1\right\}$ in ${\mathbf{W}}_j^{\mathit{per}}$. Since the closed space spanned by $\overline{\underset{j\in -\mathrm{\mathbb{N}}}{\cup }{\mathbf{V}}_j^{\mathit{per}}}$ is $L^2\left(\mu \right)$ (this follows from the corresponding non-periodized version), the Wiener functionals in $\left\{{\mathrm{\varphi }}_{0,0}^{\mathit{per}}\right\}\cup \left\{{\psi }_{j,k}^{\mathit{per}};-j\in \mathrm{\mathbb{N}},k=0,\dots ,2^{\left|j\right|}-1\right\}$ constitute an orthonormal basis in $L^2\left(\mu \right)$. We will relabel this basis as follows:

$$\begin{array}{l}{\psi }_0\left(\omega \right)=1={\mathrm{\varphi }}_{0,0}^{\mathit{per}}\left(\omega \right),{\psi }_1\left(\omega \right)={\psi }_{0,0}^{\mathit{per}}\left(\omega \right),{\psi }_2\left(\omega \right)={\psi }_{-1,0}^{\mathit{per}}\left(\omega \right),\\ {\psi }_3\left(\omega \right)={\psi }_{-1,1}^{\mathit{per}}\left(\omega \right),\dots ,\\ {\psi }_{2^j}\left(\omega \right)={\psi }_{-j,0}^{\mathit{per}}\left(\omega \right),\dots ,{\psi }_{2^j+k}\left(\omega \right)={\psi }_{-j,k}^{\mathit{per}}\left(\omega \right)={\psi }_{2^j}\left(\omega –k{2}^{-j}{e}_0\right)\\ \text{for}0\le k\le 2^j–1,\dots .\end{array}$$

Then this basis has the following property.

Theorem 4.1. If $\phi$ is a continuous periodic Wiener functional with period e₀, then there exist ${\alpha }_n\in \mathrm{ℂ}$ so that

$${lim}_N{‖\phi -\sum _{n=0,\dots ,N}{\alpha }_n{\psi }_n‖}_{\infty }=0,$$E48

where ${‖.‖}_{\infty }$ denotes the norm of $L^{\infty }\left(\mu \right)$.

Proof.

1. Since the ${\psi }_n$ are orthonormal, we necessarily have ${\alpha }_n={⟨\phi ,{\psi }_n⟩}_{\mu }$. Define

$$S_N\phi =\sum _{n=0,\dots ,N}{⟨\phi ,{\psi }_n⟩}_{\mu }.$$

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

$${‖S_N\phi ‖}_{\infty }\le C{‖\phi ‖}_{\infty },$$E49

with C independent of $\phi$ or N.

1. If $N=2^j$, then ${S_2}^j$ is the orthogonal projection operator on ${\mathbf{V}}_{-j}^{\mathit{per}}$; hence

$$\left({S_2}^j\phi \right)\left(\omega \right)=\sum _{k=0,\dots ,2^{\left|j\right|}-1}{⟨\phi ,{\mathrm{\varphi }}_{-j,k}^{\mathit{per}}⟩}_{\mu }{\mathrm{\varphi }}_{-j,k}^{\mathit{per}}\left(\omega \right)=\underset{\mathbf{H}}{\int }{K}_j\left(\omega ,\xi \right)\phi \left(\xi \right)\mu \left(\mathit{d\xi }\right),$$

with $K_j\left(\omega ,\xi \right)=\sum _{k=0,\dots ,2^{\left|j\right|}-1}{\mathrm{\varphi }}_{-j,k}^{\mathit{per}}\left(\omega \right)\left(\overline{{\mathrm{\varphi }}_{-j,l}^{\mathit{per}}\left(\xi \right)}\right).$.

Consequently,

$${‖S_2\phi ‖}_{\infty }\le \left({sup}_{\omega \in \mathbf{H}}\underset{\mathbf{H}}{\int }\left|K_j\left(\omega ,\xi \right)\right|\mu \left(\mathit{d\xi }\right)\right){‖\phi ‖}_{\infty }.$$

Now,

$$\begin{array}{l}{sup}_{\omega \in \mathbf{H}}\underset{\mathbf{H}}{\int }\left|K_j\left(\omega ,\xi \right)\right|\mu \left(\mathit{d\xi }\right)\\ \le {sup}_{\omega \in \mathbf{H}}\underset{\mathbf{H}}{\int }\sum _{k=,,2^{\left|j\right|}-1}\sum _{l,l^{\prime }\in \mathrm{\mathbb{Z}}}\left|{\mathrm{\varphi }}_{-j,k}\left(\omega +{\mathit{le}}_0\right)\right|.\left|{\mathrm{\varphi }}_{-j,k}\left(\xi +l^{\prime }{e}_0\right)\right|\mu \left(\mathit{d\xi }\right)\\ \le {sup}_{\omega \in \mathbf{H}}\underset{\mathbf{H}}{\int }\sum _{k=0,\dots ,2^{\left|j\right|}-1}\sum _{l\in \mathrm{\mathbb{Z}}}{2}^j\left|\mathrm{\varphi }\left(2^j\left(\omega +{\mathit{le}}_0\right)–{\mathit{ke}}_0\right)\right|.\left|\mathrm{\varphi }\left(2^j\xi –{\mathit{ke}}_0\right)\right|\mu \left(\mathit{d\xi }\right)\\ \le C{sup}_{{\omega }^{\prime }\in \mathbf{H}}\sum _{k=0,\dots ,2^{\left|j\right|}-1}\sum _{l\in \mathrm{\mathbb{Z}}}\left|\mathrm{\varphi }\left({\omega }^{\prime }+2^j\mathit{le}\right.\right.\le C{sup}_{{\omega }^{\prime }\in \mathbf{H}}\sum _{m\in \mathrm{\mathbb{Z}}}\left|\mathrm{\varphi }\left({\omega }^{\prime }+{\mathit{me}}_0\right)\right|,\end{array}$$

and this is uniformly bounded if $\left|\mathrm{\varphi }\left(\omega \right)\right|\le C{\left(1+\left|\omega \right|\right)}^{-1-\epsilon }$. This establishes (49) for $N=2^j$.

1. If $N=2^j+m$, $0\le m\le 2^j–1$, then

$$\left(S_N\phi \right)\left(\omega \right)=\left({S_2}^j\phi \right)\left(\omega \right)+\sum _{k=0,\dots ,m}{⟨\phi ,{\psi }_{-j,k}^{\mathit{per}}⟩}_{\mu }{\psi }_{-j,k}^{\mathit{per}}\left(\omega \right).$$

Estimates exactly similar to those in point 2 show that the $L^{\infty }\left(\mu \right)$-norm of the second sum is also bounded by $C{‖\phi ‖}_{\infty }$, uniformly in j, which proves (49) for all N.

1. Take now $\phi \in \mathbf{E}=\underset{j\in -\mathrm{\mathbb{N}}}{\cup }{\mathbf{V}}_j^{\mathit{per}}$. Then $\phi \in {\mathbf{V}}_{-J}^{\mathit{per}}$ for some J > 0,

so that ${⟨\phi ,{\psi }_{-j,k}^{\mathit{per}}⟩}_{\mu }=0$ for $j^{\prime }\ge J$, i.e., ${⟨\phi ,{\psi }_l⟩}_{\mu }=0$ for $l\ge 2^J$. Consequently, $\phi =S_N\phi$ if $N\ge 2^J$, so that (48) clearly holds. Since E is dense in the space of continuous periodic Wiener functionals equipped with the ${‖.‖}_{\infty }$-norm, the theorem follows.∎.

We deduce a similar theorem for $L^1\left(\mu \right)$.

Theorem 4.2. If $\phi \in L^1\left(\mu \right)$, then ${lim}_N{‖\phi -\sum _{n=0,\dots ,N}{⟨\phi ,{\psi }_n⟩}_{\mu }{\psi }_n‖}_{\mu }=0$.

Proof.

As we have the following:

$${‖\phi ‖}_{L^1\left(\mu \right)}=sup\left\{\left|{⟨\phi ,\psi ⟩}_{\mu }\right|/\psi \text{continuous},\text{periodic with period}{e}_0,{‖\psi ‖}_{\infty }\le 1\right\},$$

this leads immediately to

$$\begin{array}{c}{‖S_N\phi ‖}_{L^1\left(\mu \right)}=sup\left\{\left|{⟨S_N\phi ,\psi ⟩}_{\mu }\right|\psi \text{continuous},e_0–\text{periodic},{‖\psi ‖}_{\infty }\le 1\right\}\\ =sup\left\{\left|{⟨\phi ,S_N\psi ⟩}_{\mu }\right|\psi \text{continuous},e_0–\text{periodic},{‖\psi ‖}_{\infty }\le 1\right\}\end{array}$$

$$\le C{‖\phi ‖}_{L^1\left(\mu \right)}$$E50

by the uniform bound (49) and because $\left|{⟨\phi ,\psi ⟩}_{\mu }\right|\le {‖\phi ‖}_{L^1\left(\mu \right)}{‖\psi ‖}_{\infty }$.

Since $\mathbf{E}={\cup }_{j\in -\mathrm{\mathbb{N}}}^{}{\mathbf{V}}_j^{\mathit{per}}$ is dense in $L^1\left(\mu \right)$, the uniform bound (50) is sufficient to prove the theorem.∎.

Remark. The ordering of the ${\psi }_n$ is important in Theorems 4.1 and 4.2: we have a Schauder basis, but not an unconditional basis.