Open access peer-reviewed chapter

Bases of Wavelets and Multiresolution in Analysis on Wiener Space

Written By

Claude Martias

Submitted: 23 December 2021 Reviewed: 28 March 2022 Published: 23 November 2022

DOI: 10.5772/intechopen.104713

From the Edited Volume

Recent Advances in Wavelet Transforms and Their Applications

Edited by Francisco Bulnes

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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.

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 LPμ, 1<p<,μ 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 L2μ, the space of μ-square integrable Wiener functionals, μ denoting the Wiener measure, consists of a sequence approximation spaces Vjj, VjL2μ; these subspaces are assumed to be closed and satisfy the following:

VjVj1,forallj,E1
the closed space generatedbyjVj¯isL2μ,E2
jVj=0.E3

If we denote by j the orthogonal projection operator onto Vj, then (2) ensures that limjjφ=φ for all φL2μ. 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

φVjφ2j.V0.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:

VjφL2μk:φ2jek2j+1ek+1=constant,

where ek,k (or ek,k, another notation) is a fixed orthonormal basis in the Cameron – Martin space H defined by

Hh:01d/ht=0thsdshH201hs2ds<,

and the above interval in definition of Vj 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,

φV0φ.ne0V0,n.E5

Because of (4) this implies that if φVj, then φ.2jne0 belongs to Vj for all n. Finally, we require also that there exists ϕV0 so that

ϕ0,nnis an orthonormal basis inV0,E6

where, for all j,n, ϕj,nω2j/2ϕ2jωne0. Together, (4) and (6) imply that ϕj,nn is an orthonormal basis for Vj for all j. In the example given above, the possible choice for ϕ is the indicator Wiener functional for 0e0ωW0ωse0ss01, that is, ϕω=1 if ω0e0, interval in the lattice space (W,≤), and ϕω=0 otherwise. We call ϕ the scaling Wiener 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,kjk of L2μ,

ψj,kω2j/2ψ2jωke0, such that, for all φL2μ,

j1φ=jφ+kφψj,kμψj,k,E7

where the bracket ..μ denotes the scalar product in L2μ. For every j, define Wj to be the complement orthogonal of Vj in Vj – 1. We have

Vj1=VjWjE8

and

WjWjifjj.E9

It follows that, for j < j0,

Vj=Vj0k=0j01Wj0k,E10

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

L2μ=kWj,E11

A decomposition of L2μ into mutually orthogonal subspaces. Furthermore, the Wj spaces inherit the scaling property (4) from the Vj:

φWjφ2j.W0.E12

Formula (7) is equivalent to saying that, for fixed j,ψj,kk constitutes an orthonormal basis for Wj. Because of (11), (2) and (3), this then automatically implies that the whole collection ψj,kjk is an orthonormal basis for L2μ. On the other hand, (12) ensures that if ψ0,kk is an orthonormal basis for W0, then ψj,kk will likewise be an orthonormal basis for Wj, for any j. 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 ψW0 such that the ψ.ke0 constitute an orthonormal basis for W0. Let us first write out some interesting properties of ϕ and W0.

  1. Since ϕV0V1, and the ϕ1,n are an orthonormal basis in V−1, we have

    ϕ=ncnϕ1,n,E13

    with

    cn=ϕϕ1,nμandncn2=1.E14

    we can rewrite (13) as either

    ϕω=21/2ncnϕ2ωne0E15

    or

    ϕ̂ξ=21/2nexpinξe0H/2ϕ̂ξ/2,E16

    where the convergence in either sum holds in L2μ—sense, ϕ̂ denoting the Fourier–Wiener transform. Formula (16) can be rewritten as

    ϕ̂ξ=m0ξ/2ϕ̂ξ/2,E17

    where

    m0ξ=21/2ncnexpinξe0H.E18

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

  2. The orthonormality of the ϕ.ke0 leads to special properties for m0. We have

δk,0=Hϕ̂ξ2expikξe0Hμ=lξe02πl2πl+1ϕ̂ξ2expikξe0Hμ=lexp2πl2he002πϕ̂h+2πle02expikhe0H.exp2πl01e0sdhsμdh=he002πlϕ̂h+2πle02expikhe0Hμdh,

Implying

lϕ̂h+2πle02=2π1μa.e..E19

Substituting (17) leads to

lm0ξ+πle02ϕ̂ξ+πle02=2π1;

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

m0ξ2+m0ξ+πe02=1μa.e..E20

  1. Let us now characterize W0. φW0 is equivalent to φV1 and φ is orthogonal to V0. Since φV1, we have

φ=nλnΦ1,n, with λn=φϕ1,nμ. This implies

φ̂ξ=21/2nλnexpinξe0H/2Φ̂21ξ=mφ21ξΦ̂21ξ,E21

where

mφξ=21nλnexpinξe0H;E22

mφ is a 2πe0-periodic Wiener functional; convergence in (22) holds pointwise μ—a. e.. The constraint “ φ orthogonal to V0 ” implies that φ is orthogonal to Φ0,k for all k, that is,

Hφ̂hΦ̂h¯expikhe0Hμdh=0

where zc denotes the conjugate complex of complex z, or

he002πexpikhe0Hlφ̂h+2πle0Φ̂h+2πle0¯μdh=0;

note that, out of the ordinary, we denoted by hH the integration variable.

Hence

lφ̂h+2πle0ϕ̂h+2πle0¯=0,E23

where the series in (23) converges absolutely in L1μ. 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

mφhm0h¯+mφh+πe0m0h+πe0¯=0μa.e..E24

Since m0h¯ and m0h+πe0¯ cannot vanish together on a set of nonzero Wiener measure (because of (20)), this implies the existence of a 2πe0-periodic Wiener functional ϴ so that

mφh=Θhm0h+πe0¯μa.e.E25

and

Θh+Θh+πe0=0μa.e..E26

This last equation can be rewrite as

Θh=eihe0Hν2h,E27

where ν is a 2πe0-periodic Wiener functional.

Substituting (25, 27) into (21) gives

φ̂ξ=expihe0H/2m0ξ/2+πe0¯νξΦ̂ξ/2.E28

  1. The general form (28) for the Fourier–Wiener transform of φW0 suggests that we take

Ψ̂ξ=expihe0H/2m0ξ/2+πe0¯Φ̂ξ/2E29

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

φ̂ξ=kνkexpikξe0HΨ̂ξ

or

φ=kνkΨ.ke0,

so that the Ψ.ne0 are a good candidate for a basis of W0. We need to verify that the Ψ0,k are indeed an orthonormal basis for W0. First, the properties of m0 and Φ̂ ensure that (29) defines an L2μ-Wiener functional belonging to V−1 and orthogonal to V0 (by the analysis above), so that ΨW0. Orthonormality of the Ψ0,k is easy to check:

HΨhΨhme0¯μdh=HΨ̂ξ2expimξe0Hμ=ξe002πexpimξe0HlΨ̂ξ+2πle02μ.

Now

lΨ̂ξ+2πle02=lm0ξ/2+πle0+πe02Φ̂ξ/2+πle02=m0ξ/2+πe02nΦ̂ξ/2+2πne02+m0ξ/22nΦ̂ξ/2+πe0+2πne02=2π1m0ξ/22+m0ξ/2+πe02μa.e.by1.19=2π1μa.e.by1.20.

Hence HΨhΨhme0¯μdh=δm,0. In order to check that the Ψ0,m are indeed a basis for all W0, it then suffices to check that any φW0 can be written as φ=nγnΨ0,n, with nγn2<, or

φ̂ξ=γξΨ̂ξ,E30

with γ a square integrable 2πe0-periodic Wiener functional. Let us return to (28).

We have φ̂ξ=νξΨ̂ξ, with νL2μ. By (22),

Hmφξ2μ=πnλn2=πφμ2<.

On the other hand, by (25),

Hmφξ2μ=HΘξ2m0ξ+πe02μ
=HΘξ+πe02m0ξ+πe02μ,using1.26E31

Now, with the change of variable h=ξ+πe0 and with the help of stochastic calculus we find for this integral

exp1π2/2HΘh2m0h2μdh=exp1π2/2HΘh21m0h+πe02μdh,using1.20,=exp1π2/2HΘh2μdhexp1π2/2HΘh+πe02m0h+πe02μdh.

Put: IHΘξ+πe02m0ξ+πe02μ. We hence have, combining (31) and this last equality:

I=exp1π2/2HΘh2μdhexp1π2/2I, which implies

I=exp1π2/2.1+exp1π2/21HΘh2μdh.

Hence, νμ2=2πφμ2<, and φ is of the form (30) with γL2μ and 2πe0-periodic. We have thus prove the following theorem.

Theorem 1.1. If a ladder of closed subspaces Vjj in L2μ satisfies the conditions (1)(6), then there exists an orthonormal Wiener wavelet basis Ψj,kjk for L2μ such that

Πj1=Πj+k.Ψj,kμΨj,k.E32

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

Ψ̂ξ=expiξe0H/2m0ξ/2+πe0¯Φ̂ξ/2,

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

Ψ=n1ncn1Φ1,n,E33

Ψω=21/2n1ncn1Φ2ωne0, with convergence in this series in L2μ – 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 L2μ, as we can find it in [6] for the one-dimensional case.

1.2 Multiresolution approximation generated by cardinal B-splines in L2μ

Let us first beginning with some recalls. For n, r positive integers, n even, such that n2r, put

SnrfCnr1:fvv+1Pn1v,E34

where Pn-1 is the class of polynomials of degree n1. Functions in Snr are called cardinal Hermite splines of degree≤ than n–1.

For j=0,,r1, let

Sn,jrfSnr:fkv=0vk=0r1kj.E35

The space Sn,jr has a basis consisting of integer translates of a function Nn,j=Nn,jrSn,jr,j=0,,r1, with minimal support n/21+rn/2+1r, in the sense that every fSn,jr has a unique representation of the form

fx=vcvNn,jxv,x,E36

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

N̂n,ju=2sinu/2nHr,jαnuE37

where Hr,jαnu denotes the matrix obtained from the Hankel matrix Hrαnu of order r by replacing the j+1 th column by unun1unr+1T,

j=0,,r1, denoting by Hrαnu its determinant.

Consider the map K:2rL2μ defined by the following:

Ksω.j:0r1vsjvTvNn,jω.,ωC01,E38

T being a unitary operator on L2μ.

Theorem 2.1. The map K defined by (38) is an isomorphism of 2r onto a subspace of L2μ.

This above theorem is an easy consequence of Theorem 4.1 in [6]. Let us denote by Snrμ the range of K. This is a closed subspace of L2μ. Furthermore, Snrμ=SnrμL2μ where Snrμ is the space deduced from Snr by the following: φSnrμφωtSnr, ωC01, t01. Therefore, if ωφ2ω, we have as in real analysis:

The closed space generated by mDmSnrμ¯ in L2μ contains the one generated by mDmSn1μ¯ which is L2μ.

We also have:

mDmSnrμ=0.

Let VmDmSnrμ, m. Then, Vmm is a multiresolution approximation of L2μ. We shall call Vmm a Wiener Hermite spline multiresolution approximation of L2μ. 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 Lpμ.

1.3 Unconditional bases for Lpμ

Orthonormal bases of wavelets in L2μ 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 ω for ωWC001d the sup norm [15, 16].

Theorem 3.1. Assume φ to be a positive Wiener functional in L1μ.

Fix α>0. Then the Wiener space WC001d can be decomposed as follows:

  1. W=GB with GB=;

  2. On the “ good ” set G, φωαμ - a. e.;

  3. The “ bad ” set B can be written as B=kQk, where the Qk are non-overlapping intervals in the Banach lattice W, and

αμQk1Qkφωμ2α,k.

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

ω=HKωξμE39

for which the integral kernels satisfies

KωξC/ωξ,E40
ωKωξH+ξKωξHC/ωξ,E41

where the derivation symbol is the Malliavin derivative, and which defines a bounded operator on L2μ.

Theorem 3.3. A Calderon–Zygmund operator on W is also a bounded operator from L1μ to Lweak1μ.

We recall below the definition of Lweak1μ.

Definition 3.4.φLweak1μ if there exists C>0 so that, for all α>0,

μωW/φωαC/α.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 α>0) will be called φLweak1. 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 L2μ to L2μ and L1μ to Lweak1μ, we can extend T to other Lpμ-spaces by interpolation theorem of Marcinkiewicz.

Theorem 3.5. If an operator T satisfies

Lq1μweakC1φLp1μ,E43
φLq2μweakC2φLp2μ,E44

where q1p1, q2p2, then for 1/p=t/p1+1t/p2,

1/q=t/q1+1t/q2, with 0<t<1, there exists a constant K, depending on p1,q1,p2,q2, and t, so that

LqμKφLpμ.

Here Lqμweak stands for the space of all Wiener functionals φ for which φLqμweakinfC/μωφωαCαqforallα>01/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 Lqμ-norms (not Lweakqμ) for intermediate values q. The Marcinkiewicz interpolation theorem implies that the L1μLweak1μ-boundedness proved in Theorem 3.3 is sufficient to derive LpμLpμ 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 L2μ to L2μ, then T extends to a bounded operator from Lpμ to Lpμ for all p1.

Proof.

  1. Theorem 3.3 proves that T is bounded from L1μ to Lweak1μ; by Marcinkiewicz’s theorem, T extends to a bounded operator from Lpμ to Lpμ for 1<p2.

  2. For the range 2p<, we use the adjoint T* of T, defined by

Tφωψω¯μ=φωω¯μ.

It is associated with the integral kernel K0ωξ=Kξω¯, 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 L1μ to Lweak1μ, and hence by Theorem 3.5, that it is bounded from Lpμ to Lpμ for 1<p2. Since for 1/p+1/q=1, T*: LpμLpμ is the adjoint operator of T: LqμLqμ, it follows that T is bounded for 2q<. More explicitly, for readers unfamiliar with adjoints on Banach spaces,

q=supψLp,,ψ=1ωψω¯μif1/p+1/q=1=supψLp,ψ=1φξKωξψω¯μμ=supψLp,ψ=1φξTψξμsupψLp,ψ=1φLqμTψLpμCφLqμ.

We now apply this to prove that a Wiener functional has some decay and some regularity and if the Ψj,kω2j/22jωke0, j,k, ωW, and where e0 belongs to the Cameron–Martin space H, constitute an orthonormal basis for L2μ, then the Ψj,k also provide unconditional bases for Lpμ, 1<p<.

What we need to prove is that if φ=j,kcj,kΨj,kLpμ, then j,kεj,kcj,kΨj,kLpμ for any choice of the εj,k=±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 1+ω1:

ψω,ψωHC1+ω1ε.E45

Then ΨLpμ for 1<p<, and φ=j,kcj,kψj,k implies that cj,k=Hφωψj,kωμ because of the orthonormality of the ψj,k. We therefore want to show that, for aby choice of the εj,k=±1, Tε defined by

Tεφ=j,kεj,kφψj,kL2μψj,k

Is a bounded operator from Lpμ to Lpμ. First, we know that Tε is bounded from L2μ to L2μ, since, denoting by .μ (resp. <.,. > μ) the norm (resp. the scalar product) in L2μ,

Tεφμ2=j,kεj,kφψj,kμ2=j,kφψj,kμ2=φμ2,

so the Lpμ-boundedness will follow by Theorem 3.5 if we can prove that Tε is an integral operator with kernel satisfying (40, 41). This is the content of the following lemma.

Lemma 3.7. Choose εj,k=±1, and define Kωξj,kεj,kψj,kωψj,kξ¯. Then there exists C > 0 so that

KωξCωξ1

and

ωKωξH+ξKωξHCωξ2.

Proof.

  1. Kωξj,kψj,kωψj,kξCj,k2j1+2jωke01ε1+2jξke01εby3.13.

Find j0 so that 2j0ωξ2j0+1. We split the sum over j into two parts:

j<j0 and jj0.

  1. k1+ak1ε1+bk1ε is uniformly bounded for all values of a,b: in fact,

k1+ak1ε1+bk1εk1+ak1εsup0a1k1+ak1ε2l1+l1ε<.

Hence,

jj0k2j1+2jωke01ε1+2jξke01εCjj02jC.2j0+14Cωξ1.

  1. The part j<j0 is a little less easy.

<jj012jk1+2jωke0.1+2jξke01ε=jj0+12jk1+2jωke0.1+2jξke01ε
21+εjj0+12jk2+2jωke0.2+2jξke01ε.E46

Find k0 so that k02jω+ξ/2k0+1 and define l=kk0. Then

2+2jωke0=2+2jωξ/2le0+2jω+ξ/2k0e01+2jωξ/2le0;

similarly,

2+2jξke01+2jξω/2le0.

Consequently, with ω12jωξ/2,

k2+2jωke0.2+2jξke01εl1+ω1+le0.1+ω1+le01εC1+ω11ε,

so that

Cjj0+12j1+2jωξ/21εCj12jj01+2jj01/22j0+11εaswe haveωξ2j0+1,C2j0j12j1+2j1εC2j02Cωξ1.E47

It therefore follows that KωξCωξ1

  1. For the estimates on ωK and ξK, we write

ωKωξHj,k2jψ2jωke0Hψ2jωke0Cj,k22j1+2jωke01+2jξke01ε

and we follow the same technique; we obtain.

ωKωξH,ξKωξHCωξ2.

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 ψω,ψωHC1+ω1ε, and if the ψj,kω2j/2ψ2jωke0 constitute an orthonormal basis for L2μ, e0H being given, then the ψj,kjk also constitute an unconditional basis for all Lpμ – spaces, 1<p<.

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 ϕ and Wiener wavelet ψ, both with reasonable decay (say, ϕω,ψωC1+ω1ε), we define

ϕj,kperωlϕj,kω+le0,ψj,kperωlψj,kω+le0;

and

Vjperclosure of spanϕj,kperk,Wjperclosure of spanψj,kperk

First, notice that we have: lϕω+le0=1. In fact, put φωlϕω+le0. The conditions on ϕ (continuity and its “ reasonable decay ”) ensure that φ is well defined and continuous. Moreover, we can write:

ϕω=ncnϕ2ωne0withcnn2.

Then, φω=lncnϕ2ω2le0ne0=lmcm2lϕ2ωme0=mjcm2jϕ2ωme0=mϕ2ωme0=φ2ω.

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

φω=φ2ω==φ2nω=

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

lϕωle0=c

Since

ϕ=1, this constant is necessarily equal to 1. We deduce, for j0, ϕj,kperω=2j/2lϕ2jωke0+2jle0=2j/2, so that the Vjper, for j0, are all identical one-dimensional spaces, containing the constant functionals.

Similarly, lψω+l/2e0=0. In fact, lψω+l/2e0=ln1ncn+1ϕ2ω+le0ne0 where the cn are the coefficients appearing in (13), that is,

cn=ϕϕ1,n=k,m1m+1cmϕ2ω+ke0k=lnm=n+1=0becausemc2m=mc2m+1.

Hence, for j1, Wjper=0. We therefore restrict our attention to the Vjper, Wjper with j0. Obviously Vjper, WjperVj1per, a property inherited from the non-periodized spaces. Moreover, Wjper is still orthogonal to Vjper, because

HΨj,kperωϕj,kperωμ=l,l2jHψ2jω+2jle0ke0ϕ2jω+2jle0ke0¯μ=l,l2jle0l+1e0Ψ2jω+2jlle0ke0ϕ2jξke0¯μ,

because j0,

=rψj,k+2jrϕj,kμ=0.

It follows that, as in the non-periodized case, Vj1per=VjperWjper. The spaces Vjper, Wjper are all finite-dimensional: since Φj,k+m2j=Φj,k for m, and the same is true for Ψ, both Vjper and Wjper are spanned by the 2|j| Wiener functionals obtained from k=0,1,,2j1. These 2|j| Wiener functionals are moreover orthonormal; in, for example, Wjper we have, for 0k,k2j1,

Ψj,kperΨj,kperμ=rψj,k+2jrψj,kμ=δk;k.

We have therefore a ladder of multiresolution spaces, V0perV1perV2per, with successive orthogonal complements W0perofV0perinV1per, W1per,, and orthonormal bases ϕj,kk=02j1 in Vjper,ψj,kk=02j1 in Wjper. Since the closed space spanned by jVjper¯ is L2μ (this follows from the corresponding non-periodized version), the Wiener functionals in ϕ0,0perψj,kperjk=02j1 constitute an orthonormal basis in L2μ. We will relabel this basis as follows:

ψ0ω=1=ϕ0,0perω,ψ1ω=ψ0,0perω,ψ2ω=ψ1,0perω,ψ3ω=ψ1,1perω,,ψ2jω=ψj,0perω,,ψ2j+kω=ψj,kperω=ψ2jωk2je0for0k2j1,.

Then this basis has the following property.

Theorem 4.1. If φ is a continuous periodic Wiener functional with period e0, then there exist αn so that

limNφn=0,,Nαnψn=0,E48

where . denotes the norm of Lμ.

Proof.

  1. Since the ψn are orthonormal, we necessarily have αn=φψnμ. Define

SNφ=n=0,,Nφψnμ.

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

SNφCφ,E49

with C independent of φ or N.

  1. If N=2j, then S2j is the orthogonal projection operator on Vjper; hence

S2jφω=k=0,,2j1φϕj,kperμϕj,kperω=HKjωξφξμ,

with Kjωξ=k=0,,2j1ϕj,kperωϕj,lperξ¯..

Consequently,

S2φsupωHHKjωξμφ.

Now,

supωHHKjωξμsupωHHk=,,2j1l,lϕj,kω+le0.ϕj,kξ+le0μsupωHHk=0,,2j1l2jϕ2jω+le0ke0.ϕ2jξke0μCsupωHk=0,,2j1lϕω+2jleCsupωHmϕω+me0,

and this is uniformly bounded if ϕωC1+ω1ε. This establishes (49) for N=2j.

  1. If N=2j+m, 0m2j1, then

SNφω=S2jφω+k=0,,mφψj,kperμψj,kperω.

Estimates exactly similar to those in point 2 show that the Lμ-norm of the second sum is also bounded by Cφ, uniformly in j, which proves (49) for all N.

  1. Take now φE=jVjper. Then φVJper for some J > 0,

so that φψj,kperμ=0 for jJ, i.e., φψlμ=0 for l2J. Consequently, φ=SNφ if N2J, so that (48) clearly holds. Since E is dense in the space of continuous periodic Wiener functionals equipped with the .-norm, the theorem follows.∎.

We deduce a similar theorem for L1μ.

Theorem 4.2. If φL1μ, then limNφn=0,,Nφψnμψnμ=0.

Proof.

As we have the following:

φL1μ=supφψμ/ψcontinuousperiodic with periode0ψ1,

this leads immediately to

SNφL1μ=supSNφψμψcontinuouse0periodicψ1=supφSNψμψcontinuouse0periodicψ1
CφL1μE50

by the uniform bound (49) and because φψμφL1μψ.

Since E=jVjper is dense in L1μ, the uniform bound (50) is sufficient to prove the theorem.∎.

Remark. The ordering of the ψn is important in Theorems 4.1 and 4.2: we have a Schauder basis, but not an unconditional basis.

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Written By

Claude Martias

Submitted: 23 December 2021 Reviewed: 28 March 2022 Published: 23 November 2022