# Characterizations of Tight Frame Wavelets with Special Dilation Matrices

Authors: Huang Yongdong; Zhu Fengjuan
Journal: Mathematical Problems in Engineering
 (2010)

---

## Abstract

We study all generalized low-pass filters and tight frame wavelets with special dilation
matrixM (M-TFW), where M satisfies Md=2Id and generates the checkerboard lattice.
Firstly, we study the pseudoscaling function, generalized low-pass filters and multiresolution analysis
tight frame wavelets with dilation matrix M (MRA M-TFW), and also give some important
characterizations about them. Then, we characterize all M-TFW by showing precisely their corresponding
dimension functions which are nonnegative integer valued. Finally, we also show that an
M-TFW arises from our MRA construction if and only if the dimension of a particular linear space
is either zero or one.


---

## Body

## 1. Introduction

Wavelet analysis with its fast algorithms is used in many fields of applied mathematics, such as image or signal analysis and numerical treatment of operator equations (see [1–4]). Moreover, wavelet bases, recently also wavelet frames, are applied to the characterization of the function space [5]. The classical MRA wavelets are probably the most important class of orthonormal wavelets. Many of the better known examples as well as those often used in applications belong to this class. However, Journe's wavelet is not an MRA wavelet. Thus, it was a natural question to find necessary and sufficient conditions for an orthonormal wavelet to be an MRA wavelet. An interesting approach to this involves the dimension function. On the other hand, there are useful filters, such as m(ξ)=(1/2)(1+e3iξ), that do not produce orthonormal basis; nevertheless, they produce systems that have the reconstruction property, as well as many other useful features.It is natural, therefore, to develop a theory involving more general filters that do, indeed, produce systems having these properties. A natural setting for such a theory is provided by frames [1]. Several authors have considered this problem and have shown how more general filters produce such frames. A successful development of these ideas is provided by the papers [6, 7]. These results, however, do involve certain restrictions and technical assumptions such as semiorthogonality. In particular, they exclude the use of the filter we described above. A related approach can be found in [8]. Note that the design of tight wavelet frames is still a challenging problem and a number of references have dealt with this subject (see [3–25]).In [20, 21], authors successfully developed a theory of univariate tight frame wavelets and MRA tight frame wavelets. In [24], authors revealed the deep and rich structure of the set of dyadic tight frame wavelets. In [9, 10, 25], authors extended the one-dimensional case to the multidimensional case with the expending dilation matrix of determinant 2, but it is a pity that authors only studied the pseudoscaling function, generalized low-pass filters, and the multiplier classes associated with M-TFWs, but they did neither study the relationship between MRA M-TFW and M-TFW, nor study the relationship between MRA M-TFW and dimension function. In [18], authors constructed multivariate compactly supported tight wavelet frames with dilation matrix 2I, unfortunately, they obtained that at least 2d-1 tight frame generators can generate a tight wavelet frame. Therefore, with the dimension of space increasing, the computational complexity increases. In [8], the author gave the characterizations of the abstract tight frames with arbitrary dilation matrix for L2(ℝd); due to arbitrariness of dilation matrix, the author did not give the expression of tight frame generators (in the time domain or the frequency domain). In fact, even in one-dimensional case, it is very difficult to give the explicit expression of tight frame generators with dilation factor a(a>2) by means of scaling function (see [16]). With the dimension d increasing, the computational complexity increases, with the absolute value of determinant of the dilation matrix M increasing, the computational complexity increases, that is, at least |detM|-1 generators can generate tight wavelet frames for L2(ℝd). Therefore, we have to recur some special dilation matrices to solve the problem in L2(ℝd). In the paper, we study all the generalized low-pass filters and tight frame wavelets with special dilation matrix M (M-TFW), where M satisfies Md=2Id and generates the checkerboard lattice [14], that is,
(1.1)MZd={k∈Zd:∑i=1dki∈2Z}.
In this case, we only need one function ψ such that the system {ψj,k,j∈ℤ,k∈ℤd} is a tight frame (with A=B=1) for L2(ℝd) (in other word, ψ is a M-TFW), then we can give an explicit expression of ψ by means of pseudoscaling function in the frequency domain; this is due to |detM|=2. Firstly, we study the pseudoscaling function, generalized low-pass filters, MRA M-TFW, and also give some important characterizations about them. Secondly, we characterize all M-TFW by showing precisely their corresponding dimension functions which are nonnegative integer valued. Finally, we also show that a M-TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one. Our result is a generalization of the construction of TFW from generalized low-pass filters that is introduced in [20, 21]. But, it is well known that the situation in higher dimension is more complicated than the situation in one dimension.Let us now describe the organization of the material as follows. Section2 presents preliminaries and basic definitions. In Section 3, we study the pseudoscaling function, generalized low-pass filters, and MRA M-TFW, and give some important characterizations about them. In Section 4, we characterize all M-TFW in L2(ℝd) by dimension function.


## 2. Preliminaries and Basic Definitions

In this paper, we denote byD the transpose of M:D=MT, and 𝕊=ℤd∖Dℤd, ε0 is a constant vector and ε0∈𝕊. For ψ∈L2(ℝd), and k∈ℤd, we denote
(2.1)ψj,k:=2j/2ψ(Mjx-k).Let us recall the concept of frame.Given a countable index setI, a collection {fi:i∈I} in a Hilbert space ℍ is called a frame in ℍ if there exist two constants A,B>0 such that
(2.2)A‖f‖H2≤∑i∈I|〈f,fi〉|2≤B‖f‖H2,∀f∈H.
If we can choose A=B in (2.2), then {fi:i∈I} is called a tight frame.Now, we give some basic definitions which will be used in this paper. In fact, they are some generalizations of the notations in [20, 21].Definition 2.1.
A functionψ∈L2(ℝd) is a tight frame wavelet with dilation matrixM(briefly: M-TFW) if and only if the system {ψj,k,j∈ℤ,k∈ℤd} is a tight frame (with A=B=1) for L2(ℝd).Definition2.1 implies that
(2.3)‖f‖22=∑j∈Z∑k∈Zd|〈f,ψj,k〉|2,∀f∈L2(Rd).
It is clearly that (2.3) is equivalent to
(2.4)f=∑j∈Z∑k∈Zd〈f,ψj,k〉ψj,k
for all f∈L2(ℝd), where the sum converges unconditionally in L2(ℝd).Bownik gave a deeper result as follows.Proposition 2.2 (see [8, Theorem  4.2]).
ψ∈L2(ℝd) is a M-TFW if and only if
(2.5)∑j∈Z|ψ̂(Djξ)|2=1,a.e.ξ∈Rd,(2.6)ϱk(ξ)=∑j=0+∞ψ̂(Djξ)ψ̂(Dj(ξ+2πk))¯=0,a.e.ξ∈Rd,k∈S.A M-TFW is said to besemi-orthogonal if and only if ψj,k is orthogonal to ψℓ,m whenever j≠ℓ (for all k,m∈ℤd). This is equivalent to the orthogonality of the subspaces Wj and Wℓ if j≠ℓ, where
(2.7)Wj=span⁡{ψj,k:k∈Zd}¯,
as j ranges throughout ℤ.Proposition 2.3 (see [3, Chapter  7]).
Supposeψ∈L2(ℝd) is a semi-orthogonal M-TFW, then, for each j⩾1,
(2.8)∑k∈Zdψ̂(Dj(ξ+2πk))ψ̂(ξ+2πk)¯=0,a.e.ξ∈Rd.Definition 2.4.
A measurable2π-ℤd periodic function H on ℝd is a generalized filter with dilation matrixM (briefly: M-GF) if it satisfies
(2.9)|H(ξ)|2+|H(ξ+2πD-1ε0)|2=1,a.e.ξ∈Rd.
As what was done in [20, 21], we will denote by F̃ the set of generalized filters with dilation matrix M and let F̃+={H∈F̃:H⩾0}. Observe that H∈F̃⇒|H|∈F̃+.Definition 2.5.
A functionφ∈L2(ℝd) is called a pseudoscaling function with dilation matrixM (briefly: M-PSF) if there exists a generalized filter H∈F̃ such that
(2.10)φ̂(Dξ)=H(ξ)φ̂(ξ),a.e.ξ∈Rd.Remark 2.6.
Notice thatH is not uniquely determined by the M-PSF φ. Therefore, we will denote by F̃φ the set of all H∈F̃ such that H satisfies (2.10) for φ. For example, if φ=0, then F̃φ=F̃. If φ is a scaling function of MRA wavelet, then F̃φ is a singleton. On the other hand, if φ is an M-PSF, then |φ̂|∨ is also an M-PSF, and if H∈F̃, then |H|∈F̃|φ̂|∨.Definition 2.7.
ForH∈F̃, let
(2.11)N0(|H|)={ξ∈Rd:lim⁡j→+∞⁡φ̂|H|(D-jξ)=0},φ̂|H|(ξ)=∏j=1∞|H(D-jξ)|.
We say that H∈F̃ is a generalized low-pass filter with dilation matrixM(briefly: M-GLPF) if |N0(|H|)|=0.Now, we give the definition of MRA M-TFW.Definition 2.8.
A M-TFWψ is an MRA M-TFW if there exists a M-PSF φ and H∈F̃φ such that
(2.12)ψ̂(Dξ)=eiε0⋅ξρ(Dξ)H(ξ+2πD-1ε0)¯φ̂(ξ),a.e.ξ∈Rd,
where ρ(ξ) is a measurable 2π-ℤd periodic function and |ρ(ξ)|=1.In Section3, we will prove that if ψ is an MRA M-TFW, then H has to be more than just a generalized filter; H has to be a M-GLPF.


## 3. The Characterizations of M-PSF, M-GF, and MRA M-TFW

The main purpose of this section is to study the M-PSF, the M-GF, and the MRA M-TFW inL2(ℝd). We will give some important characterizations about them.Lemma 3.1.
Supposeφ is a M-PSF and H∈F̃φ. If
(3.1)lim⁡j→+∞⁡|φ̂(D-jξ)|=1,a.e.ξ,
then,
(3.2)|φ̂(ξ)|=|∏j=1∞H(D-jξ)|,a.e.ξ,
and |N0(|H|)|=0.Proof.
By (2.10), we have
(3.3)|φ̂(ξ)|=|∏j=1nH(D-jξ)||φ̂(D-nξ)|,a.e.ξ∈Rd.
Using (3.1), we obtain that |φ̂(ξ)|=φ̂|H| and, therefore (3.2) and |N0(|H|)|=0 are clearly satisfied.Lemma 3.2 (see [25, Lemma  3.2]).
Iff∈L1(ℝd), then, for a.e. ω∈ℝd, limj→+∞|f(Djω)|=0.Proof.
As the proof is simple, we omit it.Theorem 3.3.
Suppose thatψ is an MRA M-TFW and φ is a M-PSF satisfying (2.12). Then H defined by (2.12) is a generalized low-pass filter.Proof.
Suppose thatψ is an MRA M-TFW, by (2.5), (2.9) and (2.12), we can obtain
(3.4)1=∑j∈Z|ψ̂(Djξ)|2=∑j∈Z|H(Dj-1ξ+2πD-1ε0)|2|φ̂(Dj-1ξ)|2=lim⁡n→+∞⁡∑j=-nn|H(Dj-1ξ+2πD-1ε0)|2|φ̂(Dj-1ξ)|2=lim⁡n→+∞⁡∑j=-nn{1-|H(Dj-1ξ)|2}|φ̂(Dj-1ξ)|2=lim⁡n→+∞⁡{|φ̂(D-n-1ξ)|2-|φ̂(Dn-1ξ)|2}.
Sinceφ∈L2(ℝd), Lemma 3.2 implies limn→+∞|φ̂(Dnξ)|2=0 for a.e. ξ∈L2(ℝd). This shows that for a.e. ξ∈L2(ℝd), limn→+∞|φ̂(D-nξ)|2=1, thus, by Lemma 3.1, H is a generalized low-pass filter.Suppose thatψ is a M-TFW, let
(3.5)σψ(ξ)=∑k∈Zd|ψ̂(ξ+2πk)|2,Dimψ(ξ)=∑j=1∞∑k∈Zd|ψ̂(Dj(ξ+2πk))|2.
We say that Dimψ(ξ) is the dimension function of M-TFW.Lemma 3.4.
Suppose thatψ is an MRA M-TFW, the corresponding M-PSF of the ψ is φ, then
(3.6)|φ̂(ξ)|2=∑j=1∞|ψ̂(Djξ)|2,a.e.ξ∈Rd.Proof.
By Definition2.8, there exists a M-GF H∈F̃ such that
(3.7)φ̂(Dξ)=H(ξ)φ̂(ξ),ψ̂(Dξ)=eiε0⋅ξρ(Dξ)H(ξ+2πD-1ε0)¯φ̂(ξ)=H1(ξ)φ̂(ξ)
we obtain that
(3.8)|φ̂(ξ)|2=|H(ξ)|2|φ̂(ξ)|2+|H1(ξ)|2|φ̂(ξ)|2=|φ̂(Dξ)|2+|ψ̂(Dξ)|2.
Thus
(3.9)|φ̂(ξ)|2=|φ̂(Dξ)|2+|ψ̂(Dξ)|2=|φ̂(D2ξ)|2+|ψ̂(D2ξ)|2+|ψ̂(Dξ)|2=⋯=|φ̂(DNξ)|2+|ψ̂(DNξ)|2+⋯+|ψ̂(Dξ)|2=|φ̂(DNξ)|2+∑j=1N|ψ̂(Djξ)|2.
Notice that |φ̂(ξ)|⩽1, hence {∑j=1N|ψ̂(Djξ)|2}N is an increased sequence and bounded with 1. Moreover,
(3.10)∫Rd|φ̂(DNξ)|2dξ=12N∫Rd|φ̂(ξ)|2dξ.
Therefore, by Fatou lemma,
(3.11)∫Rdlim⁡N→∞⁡|φ̂(DNξ)|2dξ⩽lim⁡N→∞⁡∫Rd|φ̂(DNξ)|2dξ=lim⁡N→∞⁡12N∫Rd|φ̂(ξ)|2dξ=0.
So limN→∞|φ̂(DNξ)|2=0, consequently, |φ̂(ξ)|2=∑j=1∞|ψ̂(Djξ)|2,a.e.ξ∈ℝd.Lemma 3.5.
Suppose thatψ is an MRA M-TFW, then
(3.12)Dimψ(Dξ)+σψ(Dξ)=Dimψ(ξ)+Dimψ(ξ+2πD-1ε0).
Furthermore, if ψ is a M-TFW, then (3.12) is also valid.Proof.
We only prove the case ofψ is an MRA M-TFW. By Definition 2.8, there exists an M-GF H∈F̃ such that
(3.13)φ̂(Dξ)=H(ξ)φ̂(ξ),ψ̂(Dξ)=eiε0⋅ξρ(Dξ)H(ξ+2πD-1ε0)¯φ̂(ξ)=H1(ξ)φ̂(ξ).
By Lemma 3.4, we can obtain
(3.14)Dimψ(ξ)=∑j=1∞∑k∈Zd|ψ̂(Dj(ξ+2πk))|2=∑k∈Zd∑j=1∞|ψ̂(Dj(ξ+2πk))|2=∑k∈Zd|φ̂(ξ+2πk)|2.
Thus,
(3.15)Dimψ(Dξ)=∑k∈Zd|φ̂(Dξ+2πk)|2=∑k∈DZd|φ̂(Dξ+2πk)|2+∑k∈S|φ̂(Dξ+2πk)|2=∑k∈Zd|φ̂(D(ξ+2πk))|2+∑k∈Zd|φ̂(D(ξ+2πk+2πD-1ε0))|2=∑k∈Zd|H(ξ+2πk)|2|φ̂(ξ+2πk)|2+∑k∈Zd|H(ξ+2πk+2πD-1ε0)|2|φ̂(ξ+2πk+2πD-1ε0)|2=|H(ξ)|2∑k∈Zd|φ̂(ξ+2πk)|2+|H(ξ+2πD-1ε0)|2|∑k∈Zdφ̂(ξ+2πk+2πD-1ε0)|2=|H(ξ)|2Dimψ(ξ)+|H(ξ+2πD-1ε0)|2Dimψ(ξ+2πD-1ε0).
Analogously, we have
(3.16)σψ(Dξ)=|H1(ξ)|2Dimψ(ξ)+|H1(ξ+2πD-1ε0)|2Dimψ(ξ+2πD-1ε0).
Thus,
(3.17)Dimψ(Dξ)+σψ(Dξ)=Dimψ(ξ)+Dimψ(ξ+2πD-1ε0).Theorem 3.6.
Suppose thatψ is an MRA M-TFW, the corresponding M-PSF of the ψ is φ, then
(3.18)0⩽Dimψ(ξ)⩽1,a.e.ξ∈Rd.Proof.
LetH∈F̃φ, and it is the corresponding M-GLPF of the ψ. Let
(3.19)φ̂n(ξ)={∏j=1nH(D-jξ)χ2n/dTd(ξ),n=qd,q∈Z,∏j=1n+1H(D-jξ)χ2(n+1)/dTd(ξ),n=qd-1,q∈Z,⋮∏j=1n+(d-1)H(D-jξ)χ2(n+d-1)/dTd(ξ),n=qd-(d-1),q∈Z,
where Γ denotes all the vertices of cube [0,1]d, 𝕋=[0,2π]. It is clearly that limn→∞|φ̂n(ξ)|=|φ̂(ξ)|. Now, we need to prove that {φn(·-k)}k∈ℤd is an orthonormal system for L2(ℝd). Denote
(3.20)In=∫Rdφn(x)φn(x-k)¯dx.
So, if n=qd,q∈ℤ, then
(3.21)In=∫Rdφn(x)φn(x-k)¯dx=1(2π)d∫[0,2q2π]d∏j=1n|H(D-jξ)|2eikξdξ=1(2π)d∫[0,2q2π]d∏j=1(q-1)d|H(D-jξ)|2eikξ{∏l=0d-1|H(D-qd+lξ)|2}dξ=1(2π)d∫[0,2q-12π]d∏j=1(q-1)d|H(D-jξ)|2eikξ×{∑ω∈Γ∏l=0d-1|H(D-qd+l(ξ+2qπω))|2}dξ=1(2π)d∫[0,2q-12π]d∏j=1(q-1)d|H(D-jξ)|eikξ=I(q-1)d=⋯=Id.
However,
(3.22)Id=1(2π)d∫[0,4π]d∏j=1d|H(D-jξ)|2eikξdξ=1(2π)d∫[0,2π]deikξ×{∑ω∈Γ∏l=1d|H(D-l(ξ+2πω))|2}dξ=1(2π)d∫[0,2π]deikξ=δk,0.
Thus we deduce from (3.19) and the above equation that if n=qd,q∈ℤ, then
(3.23)〈φn(⋅),φn(⋅-k)〉=δk,0.
Similarly, by (3.19), (3.23) also holds if n=qd-1, or n=qd-2,…, or n=qd-(d-1). Therefore, for a.e. ξ∈ℝd(3.24)∑k∈Zd|φ̂n(ξ+2πk)|2=1,∀n∈N.
By Fatou lemma,
(3.25)Dimψ(ξ)=∑k∈Zd|φ̂(ξ+2πk)|2=∑k∈Zdlim⁡n→∞⁡|φ̂n(ξ+2πk)|2⩽lim⁡n→∞⁡∑k∈Zd|φ̂n(ξ+2πk)|2=1.Recall that the MRA TFW are precisely thoseψ∈L2(ℝ) that can be constructed from a generalized low-pass filter as described in [20, 21]. What are the properties of σψ when ψ∈L2(ℝd) is a M-TFW? We will see that the answers to this question are important for determining the properties of Dimψ.Forψ∈L2(ℝd) (not necessarily a M-TFW), let us consider the principal shift-invariant spaceW(ψ)≡span{ψ(·-k):k∈ℤd}¯. When ψ is an M-TFW, then the space W0 we defined in (2.7) is the space W(ψ). We will be interested in examining what type of spanning set for W0 is {ψ(·-k)}={ψ0k},k∈ℤd. If ψ is an orthonormal wavelet, then {ψ(·-k)} is an orthonormal basis for W0. What will happen if ψ∈L2(ℝd) is a M-TFW?It is not hard to see that we can findφ∈W(ψ) such that {φ(·-k),k∈ℤd} is a tight frame (with constant 1) for W(ψ). Indeed, let s(ξ)=1/σψ(ξ) if ξ∈Uψ and s(ξ)=0 if ξ∉Uψ, where Uψ=U={ξ∈𝕋,σψ>0}. Straight forward calculations show that φ defined by φ̂(ξ)=s(ξ)ψ̂(ξ) provides us with the desired function. Moreover,
(3.26)σφ=χUψ(ξ)
and Uφ=Uψ. This result can be extended to the following known result.Lemma 3.7 (see [5,Theorem  7.2.3]).
{ψ(·-k)},k∈ℤd, is a tight frame with constant A,B for W(ψ) if and only if
(3.27)AχU⩽σψ⩽BχU.Whenψ is a M-TFW, for the space W0, one has the following results.Lemma 3.8 (see [21,Theorem  2.7]).
Suppose thatψ is a M-TFW. The followings are equivalent:(a)
{ψ(·-k)}k∈ℤd is a tight frame (of constant 1) for W0 (i.e., ψ is a W0-TF),(b)
∥ψ∥22=∑k∈ℤd|〈ψ,ψ0k〉|2,(c)
ψ is semi-orthogonal,(d)
σψ=χU a.e. on ℝd.Proof.
It is clear that (a) implies (b), this is (2.3) with f=ψ,H=W0. Since
(3.28)‖ψ‖22=∑k∈Zd|〈ψ,ψ0k〉|2
and {ψjk} is a tight frame, (b) implies 〈ψ,ψjk〉=0 whenever j≠0. This clearly implies the semiorthogonality. Thus, (b) implies (c). If we assume (c), so that 〈f,ψjk〉=0 whenever f∈W0 and j≠0, an application of (2.3) gives us
(3.29)‖f‖22=∑j∈Z∑k∈Zd|〈f,ψjk〉|2=∑k∈Zd|〈f,ψ0k〉|2
and we see that (c) implies (a).
Lemma3.7 with A=B=1 tells us that (a) and (d) are equivalent.Although the proof of Lemma3.8 is rather simple, the result is not obvious. Some aspects of this lemma are counter intuitive: the properties (a), (b), and (d) are tied to the inner structure of W(ψ)=W0. On the other hand, (c) provides information about the relationship between W0 and the other space Wj,j≠0. The assumption that ψ is an M-TFW is very important. If we do not assume this to be the case, (a) and (d) are still equivalent. However, (b) is not equivalent to (d) (see [21]).Now, we need to make some facts clear. For Theorem3.3, Lemma 3.4, and Lemma 3.8, the assumption of Md=2Id can be weakened into |detM|=2, and the assumption of integer-valued dilation matrix can be weakened into noninteger-valued dilation matrix, Theorem 3.3, Lemma 3.4, and Lemma 3.8 still hold; their proofs work more or less in an unchanged form from the original ones. On the other hand, for Lemma 3.5 and Theorem 3.6, the assumption of integer-valued dilation matrix cannot be weakened into noninteger-valued dilation matrix, because we can obtain Dℤd⊂ℤd and ℤd=Dℤd+ℤd∖Dℤd if and only if dilation matrix is integer valued, in fact, during the course of these proofs we always use ∑k∈ℤd=∑k∈Dℤd+∑ℤd∖Dℤd.


## 4. Dimension Function of M-TFW

The dimension function of a multivariate orthonormal waveletψ is integer value; moreover, unless ψ is an MRA wavelet, it attains each of the integer values in the interval [0,M], where M is the supremum of Dimψ, on sets of positive measure (see [8, 21]). We will investigate the properties of the dimension function for M-TFW.Theorem 4.1.
Suppose thatψ is a M-TFW. Then ψ is semi-orthogonal if and only if Dimψ is integer valued a.e.Proof.
First, let us prove the sufficiency. Suppose thatDimψ is integer valued a.e. From Lemma 3.5, we see that σψ must also be integer valued a.e. By (3.18) and σψ(ξ)⩽1 a.e., we have σψ(ξ)=χUψ(ξ) a.e. By Lemma 3.8, part (c) and (d), we conclude that ψ is semi-orthogonal.
Now, we prove the necessity. Let us assume thatψ is semi-orthogonal, denote
(4.1)Gn(ξ)=∑j=1∞ψ̂(Djξ)∑k∈Zdψ̂(Dn(ξ+2πk))ψ̂(Dj(ξ+2πk))¯,n⩾1.
Apply Schwart's inequality to the sum in (4.1) with respect to k, we obtain
(4.2)∑j=1∞|ψ̂(Djξ)|∑k∈Zd|ψ̂(Dn(ξ+2πk))|⋅|ψ̂(Dj(ξ+2πk))|⩽∑j=1∞|ψ̂(Djξ)|(∑k∈Zd|ψ̂(Dn(ξ+2πk))|2)1/2(∑k∈Zd|ψ̂(Dj(ξ+2πk))|2)1/2⩽∑j=1∞|ψ̂(Djξ)|(∑k∈Zd|ψ̂(Dj(ξ+2πk))|2)1/2(σψ(Dnξ))1/2.
Using the fact that σψ(Dnξ)⩽1 a.e. and appling Schwart's inequality to the sum over j, we obtain that
(4.3)(∑j=1∞|ψ̂(Djξ)|2)1/2(∑j=1∞∑k∈Zd|ψ̂(Dj(ξ+2πk))|2)1/2⩽Dimψ(ξ).
Therefore, the sum in (4.1) is absolutely convergent. The fact allows us to interchange the sums in the expression Gn(ξ). By Proposition 2.3, we have
(4.4)Gn(ξ)=∑k∈Zdψ̂(Dn(ξ+2πk))∑j=1∞ψ̂(Djξ)ψ̂(Dj(ξ+2πk))¯+ψ̂(ξ)∑k∈Zdψ̂(Dn(ξ+2πk))ψ̂(ξ+2πk)¯=∑k∈Zdψ̂(Dn(ξ+2πk))∑j=0∞ψ̂(Djξ)ψ̂(Dj(ξ+2πk))¯=∑k∈Zdψ̂(Dn(ξ+2πk))ϱk(ξ),
where ϱk(ξ) is defined as in (2.6). Since ϱk(ξ)=0, a.e. ξ∈ℝd, k∈𝕊, we obtain
(4.5)Gn(ξ)=∑k∈Zdψ̂(Dn(ξ+2πDk))∑j=0∞ψ̂(Djξ)ψ̂(Dj(ξ+2πDk))¯.
On the other hand,
(4.6)Gn+1(D-1ξ)=∑k∈Zdψ̂(Dn+1(D-1ξ+2πk))∑j=1∞ψ̂(DjD-1ξ)ψ̂(Dj(D-1ξ+2πk))¯=∑k∈Zdψ̂(Dn(ξ+2πDk))∑j=0∞ψ̂(Djξ)ψ̂(Dj(ξ+2πDk))¯=Gn(ξ).
Therefore, we have shown that Gn(ξ)=Gn-1(Dξ), and consequently, Gn(ξ)=G1(Dn-1ξ) for n⩾1. By Propositions 2.2 and 2.3, we have
(4.7)G1(ξ)=∑k∈Zdψ̂(D(ξ+2πk))∑j=1∞ψ̂(Djξ)ψ̂(Dj(ξ+2πk))¯=∑k∈Zdψ̂(Dξ+2πDk)∑j=0∞ψ̂(Dj⋅Dξ)ψ̂(Dj(Dξ+2πDk))¯=∑k∈Zdψ̂(Dξ+2πDk)∑j=0∞ψ̂(Dj⋅Dξ)ψ̂(Dj(Dξ+2πDk))¯+∑k∈Sψ̂(Dξ+2πk)∑j=0∞ψ̂(Dj⋅Dξ)ψ̂(Dj(Dξ+2πk))¯=∑k∈Zdψ̂(Dξ+2πk)∑j=0∞ψ̂(Dj⋅Dξ)ψ̂(Dj(Dξ+2πk))¯=∑j=0∞ψ̂(Dj⋅Dξ)∑k∈Zdψ̂(Dξ+2πk)ψ̂(Dj(Dξ+2πk))¯=∑j=1∞ψ̂(Dj⋅Dξ)∑k∈Zdψ̂(Dξ+2πk)ψ̂(Dj(Dξ+2πk))¯+ψ̂(Dξ)σψ(Dξ)=ψ̂(Dξ)σψ(Dξ).
This shows that(4.8)Gn(ξ)=ψ̂(Dnξ)σψ(Dnξ).
Finally, sinceψ is a semi-orthogonal M-TFW, σψ(Dnξ) is either 0 or 1 a.e.; this and the last fact, which implies that ψ̂(Dnξ)=0 when σψ(Dnξ)=0, give us
(4.9)ψ̂(Dnξ)=Gn(ξ),a.e.,n⩾1.
Now, let(4.10)Ψj(ξ)={ψ̂(Dj(ξ+2kπ)):k∈Zd},j⩾1.
Using the fact that σψ(ξ)⩽1 once again, we see that Ψj∈ℓ2(ℤd) for a.e. ξ∈ℝd. But, from (2.6), (2.7), and (2.8), we see that
(4.11)Ψj(ξ)=∑n=1∞〈Ψn(ξ),Ψj(ξ)〉l2(Zd)Ψn(ξ)
for j⩾1.
By the definition ofDimψ, we have
(4.12)Dimψ(ξ)=∑j=1∞‖Ψj(ξ)‖l2(Zd)2.
Hence, we conclude that
(4.13)Dimψ(ξ)=dim⁡Fψ(ξ),a.e.ξ∈Rd,
where 𝔽ψ(ξ)=span{Ψj(ξ):j⩾1}¯; this is a well-defined subspace of ℓ2(ℤd). Obviously, (4.13) implies that Dimψ is integer valued a.e. The proof of Theorem 4.1 is completed.Theorem4.1 provides us with the following interesting statement.Corollary 4.2.
Suppose thatψ is a M-TFW. Then Dimψ(ξ)=dim𝔽ψ(ξ), for a.e. ξ∈ℝd, if and only if ψ is semi-orthogonal.The dimension function of an orthonormal wavelet which attains valueN>1 on a set of positive measure, must also attain value N-1 on a set of positive measure (for details, see [14]). Equation (3.12) is used there to prove that
(4.14)Dimψ(Dξ)⩾Dimψ(ξ)-1,a.e.ξ.
Fortunately, (4.14) is also valid for semi-orthogonal M-TFWs. More precisely, one has the following result.Proposition 4.3.
Suppose thatψ is a semi-orthogonal M-TFW. Let N>1 be an integer. If there exits a set A⊆ℝd of positive Lebesgue measure, such that Dimψ(ξ)⩾N for all ξ∈A, then there exits a set B⊆ℝd of positive Lebesgue measure such that Dimψ(ξ)=N-1 for all ξ∈B.There is another interesting consequence of Theorem4.1. Notice that a consequence of (2.5) is that for every M-TFW ψ, one has
(4.15)|ψ̂(ξ)|⩽1,a.e.
An interesting class consists of those M-TFW for which |ψ̂(ξ)| attains only values 0 and 1; in accordance with the orthonormal wavelet terminology one will call such M-TFW, MSF M-TFW. One may expect that MSF M-TFW may or may not be semi-orthogonal. Theorem 4.1, however, implies that they have to lie within the realm of semi-orthogonal M-TFW since the corresponding dimension function must be integer-valued. Let us state this in the form of a corollary (see [21]).Corollary 4.4.
Ifψ is an MSF M-TFW, then ψ is semi-orthogonal.Theorem 4.5.
Suppose thatψ is a M-TFW, then, ψ is an MRA M-TFW if and only if dim𝔽ψ(ξ)∈{0,1}, for a.e. ξ∈ℝd.Proof of Theorem4.5.
First, we prove the necessity. Suppose thatψ is an MRA M-TFW, thus, it satisfies (2.12) for an appropriate H,ρ and φ. If dim𝔽ψ(ξ)⩾2, for some ξ∈ℝd, then there exist k,ℓ such that Ψk(ξ) and Ψℓ(ξ) are linearly independent in ℓ2(ℤd). It is then impossible to have a vector ν∈ℓ2(ℤd) such that both Ψk(ξ) and Ψℓ(ξ) are in the one-dimensional subspace span{ν}. However, by (2.10) and (2.12), for every j⩾1, we have
(4.16)ψ̂(Dj(ξ+2πk))=eiε0⋅Dj-1(ξ+2πk)ρ(Dj(ξ+2πk))H(Dj-1(ξ+2πk)+2πD-1ε0)¯φ̂(Dj-1(ξ+2πk))=eiε0⋅Dj-1ξρ(Djξ)H(Dj-1ξ+2πD-1ε0)¯∏n=0j-2H(Dnξ)φ̂(ξ+2πk),k∈Zd.
Since {φ̂(ξ+2πk):k∈ℤd} is in ℓ2(ℤd), for a.e. ξ∈ℝd, by Lemma 3.5 and Theorem 3.6, we conclude that dim𝔽ψ(ξ) is at most 1, for a.e. ξ∈ℝd.
Now, we will prove the sufficiency. In other words, we have to prove thatψ is an MRA M-TFW, assuming that ψ is an M-TFW such that dim𝔽ψ(ξ) is either 0 or 1, for a.e. ξ∈ℝd. We noticed that 𝔽ψ(ξ) is always defined for a M-TFW ψ and is a closed subspace of ℓ2(ℤd). Theorem 4.1 implies that, for j⩾1,Ψj(ξ)∈ℓ2(ℤd), for a.e. ξ∈ℝd.
Therefore, we need to prove that there exist a generalized filterH and a corresponding pseudoscaling function φ such that (2.12) is satisfied, that is, a.e. ξ∈ℝd(4.17)ψ̂(Dξ)=eiε0⋅ξρ(Dξ)H(ξ+2πD-1ε0)¯φ̂(ξ).
Thus, we begin to prove that it is enough to show that there exist a generalized filterH0, a corresponding pseudoscaling function φ0, and a 2π-ℤd periodic, unimodular function ρ such that
(4.18)ψ̂(Dξ)=eiε0⋅ξρ(Dξ)H0(ξ+2πD-1ε0)¯φ0̂(ξ).
Let,(4.19)Z:={ξ∈Rd:Dimψ(ξ)=0}
and, for j∈ℕ, we define the set 𝒫j to be
(4.20)Pj:={ξ∈Rd:‖Ψj(ξ)‖l2(Zd)≠0,‖Ψl(ξ)‖2=0for1⩽l⩽j-1}.
It is clearly that𝒵 is the set where all vectors Ψj(ξ),j⩾1, are zero (or, equivalently, where dim𝔽ψ(ξ)=0). Moreover, the set 𝒵 and the 𝒫j,j⩾1, are 2π-ℤd periodic, measurable and they form a partition of ℝd.
We defineφ0̂ as follows:
(4.21)φ0̂(ξ)={0,ξ∈Z,Dimψ(ξ)∑k∈Zd|ψ̂(Dj(ξ+2πk))|2⋅ψ̂(Djξ),ξ∈Pj.
Observe that (4.21) makes sense and defines a measurable function φ0̂:ℝd→ℂ. Furthermore, the function clearly satisfies
(4.22)∑k∈Zd|φ0̂(ξ+2πk)|2=Dimψ(ξ),a.e.
Using (4.22), we obtain that ∥φ0̂∥22=∫𝕋dDimψ(ξ)dξ. It follows that φ0̂∈L2(ℝd). Thus, φ0∈L2(ℝd).
Next, we divide the argument into six steps (six lemmas).
Lemma 4.6.
For a.e.ξ∈ℝd, one has
(4.23)|φ0̂(ξ)|2=∑j=1∞|ψ̂(Djξ)|2.Proof.
Ifξ∈𝒵, then, (4.23) is trivially true. We only need to consider ξ∈𝒫ℓ,ℓ∈ℕ. Thus, Ψℓ(ξ)≠0 and, by assumption, dim𝔽ψ(ξ)=1. This implies that for every j⩾1 there exists a 2π-ℤd periodic, measurable function λjℓ:𝒫ℓ→ℂ, such that, for a.e. ξ∈𝒫ℓ,
(4.24)Ψj(ξ)=λjl(ξ)Ψl(ξ).
Coordinatewise, this means that for every k∈ℤd, a.e. ξ∈𝒫ℓ(4.25)ψ̂(Dj(ξ+2πk))=λjl(ξ)ψ̂(Dl(ξ+2πk)).
It follows that the right hand side of (4.23) satisfies
(4.26)∑j=1∞|ψ̂(Djξ)|2=[∑j=1∞|λjl(ξ)|2]⋅|ψ̂(Dlξ)|2
for a.e. ξ∈𝒫ℓ. Keeping in mind formula (4.21) for ξ∈𝒫ℓ, (4.26) implies that
(4.27)∑j=1∞|λjl(ξ)|2=∑j=1∞|ψ̂(Djξ)|2|ψ̂(Dlξ)|2,
for a.e. ξ∈𝒫ℓ. On the other hand, we obtain
(4.28)Dimψ(ξ)=∑j=1∞∑k∈Zd|ψ̂(Dj(ξ+2πk))|2=∑j=1∞∑k∈Zd|λjl(ξ)|2|ψ̂(Dl(ξ+2πk))|2=∑j=1∞|λjl(ξ)|2∑k∈Zd|ψ̂(Dl(ξ+2πk))|2.
Using (4.26), (4.27), and the above equation, we obtain
(4.29)∑j=1∞|ψ̂(Djξ)|2=Dimψ(ξ)∑k∈Zd|ψ̂(Dl(ξ+2πk))|2|ψ̂(Dlξ)|2.
Combine (4.29) with definition of φ0, for a.e. ξ∈𝒫ℓ, we obtain
(4.30)|φ0̂(ξ)|2=∑j=1∞|ψ̂(Djξ)|2.
This completes the proof of Lemma 4.6 since 𝒵 and 𝒫j,j⩾1 cover all of ℝd.Lemma 4.7.
There exists a2π-ℤd periodic, measurable function H1:𝒵𝒞→ℂ such that
(4.31)|H1(ξ)|⩽1,a.e.ξ∈ZC,(4.32)ψ̂(Dξ)=eiε0⋅ξH1(ξ)¯φ0̂(ξ),a.e.ξ∈ZC.Proof.
Notice that the2π-ℤd periodicity of H1 is consistent with the 2π-ℤd periodicity of the measurable set
(4.33)ZC=⋃l∈NPl.
DefineH1 as the following formula
(4.34)H1(ξ)={0,ξ∈Pl,l⩾2,eiε0⋅ξ∑k∈Zd|ψ̂(D(ξ+2πk))|2Dimψ(ξ),ξ∈P1.
Since all the functions and sets involved in (4.34) are measurable and 2π-ℤd periodic, it is obvious that (4.34) defines a 2π-ℤd periodic, measurable function H1:𝒵𝒞→ℂ. Furthermore,
(4.35)∑k∈Zd|ψ̂(D(ξ+2πk))|2⩽Dimψ(ξ)
implies that (4.31) is valid. For ξ∈𝒫1, (4.32) is the direct consequence of (4.21) and (4.34). For ξ∈𝒫ℓ,ℓ⩾2,Ψ1(ξ)=0. In particular, ψ̂(Dξ)=0; thus (4.32) is trivially satisfied. The proof of Lemma 4.7 is completed.Lemma 4.8.
There exists a2π-ℤd periodic, measurable function H0:𝒵𝒞→ℂ such that
(4.36)|H0(ξ)|⩽1,a.e.ξ∈ZC,(4.37)φ0̂(Dξ)=H0(ξ)φ0̂(ξ),a.e.ξ∈ZC.Proof.
Again, by (4.33), the 2π-ℤd periodicity will be clear from the argument, and we need only to prove (4.36) and (4.37) on ⋃ℓ∈ℕ𝒫ℓ.
First, we consider𝒫ℓ,ℓ⩾2. For a.e. ξ∈𝒫ℓ, we obtain
(4.38)‖Ψl-1(Dξ)‖l2(Zd)2=∑k∈Zd|ψ̂(Dl-1(Dξ+2πk))|2=∑k∈Zd|ψ̂(Dl(ξ+2πD-1k))|2⩾∑k∈Zd|ψ̂(Dl(ξ+2πk))|2=‖Ψl(ξ)‖l2(Zd)2>0.
This and the definition (4.20) of 𝒫ℓ imply that for a.e. ξ∈𝒫ℓ, there exists k=k(ξ)∈{1,2,…,ℓ-1} such that Dξ∈𝒫k. It is easy to check, using (4.20), that ξ→k(ξ) is 2π-ℤd periodic and measurable (defined on 𝒫ℓ). Since, according to our assumption, dim𝔽ψ(Dξ)=1 for a.e. ξ∈𝒫ℓ, there exists a 2π-ℤd periodic, measurable function λ:𝒫ℓ→ℂ such that, for a.e. ξ∈𝒫ℓ,
(4.39)Ψl-1(Dξ)=λ(ξ)Ψk(ξ)(Dξ).
Notice that λ(ξ)≠0, since otherwise we would have
(4.40)0=‖Ψl-1(Dξ)‖l2(Zd)2⩾‖Ψl(ξ)‖l2(Zd)2⩾0
which would imply Ψℓ(ξ)=0; this is impossible for ξ∈𝒫ℓ. Hence, (4.39) implies that, for a.e. ξ∈𝒫ℓ,
(4.41)φ0̂(Dξ)=Dimψ(Dξ)∑j∈Zd|ψ̂(Dk(Dξ+2πj))|2⋅ψ̂(Dk⋅Dξ)=Dimψ(Dξ)∑j∈Zd|ψ̂(Dk(Dξ+2πj))|2⋅1λ(ξ)ψ̂(Dl-1⋅Dξ).
Notice that (4.41) shows that there exists a 2π-ℤd periodic, measurable function 𝒜:𝒫ℓ→ℂ such that, for a.e. ξ∈𝒫ℓ,
(4.42)φ0̂(Dξ)=A(ξ)ψ̂(Dlξ).
At the same time it follows directly from (4.21), that there exists a 2π-ℤd periodic, measurable function ℬ:𝒫ℓ→ℂ such that, for a.e. ξ∈𝒫ℓ,ℬ(ξ)≠0, and
(4.43)φ0̂(ξ)=B(ξ)ψ̂(Dlξ).
We defineH0(ξ) on 𝒫ℓ by
(4.44)H0(ξ):=A(ξ)B(ξ).
By (4.42) and (4.43), it is clear that H0(ξ) is a 2π-ℤd periodic, measurable function, and satisfies (4.37) on 𝒫ℓ. Therefore, it remains to define H0 on 𝒫1. Observe that for ξ∈𝒫1, either Dξ∈𝒵, or, otherwise, there is a 2π-ℤd periodic, measurable function ξ→ℓ(Dξ)∈ℕ, such that Dξ∈𝒫ℓ(Dξ). If Dξ∈𝒵, then we define H0(ξ) to be 0. Otherwise, we define, by (4.21), that there exists a 2π-ℤd periodic, measurable function 𝒜̃ such that
(4.45)φ0̂(Dξ)=Ã(Dξ)ψ̂(Dl(Dξ)⋅Dξ)=Ã(Dξ)ψ̂(Dl(Dξ)+1ξ).
Finally, we consider ξ∈𝒫1, we know that there exists a 2π-ℤd periodic, measurable function λ̃, such that
(4.46)Ψl(Dξ)+1(ξ)=λ̃(ξ)⋅Ψ1(ξ),
and a 2π-ℤd periodic, measurable function ℬ̃≠0, such that
(4.47)φ0̂(ξ)=B̃(ξ)⋅ψ̂(Dξ).
Hence, for ξ∈𝒫1 and Dξ∉𝒵, we define H0 by
(4.48)H0(ξ):=Ã(Dξ)λ̃(ξ)B̃(ξ).
It is now clear that H0 is a 2π-ℤd periodic, measurable function on 𝒫1 and satisfies (4.37) on 𝒫1. It follows that we have a 2π-ℤd periodic, measurable function H0:𝒵𝒞→ℂ, such that (4.37) is satisfied.
Let us prove (4.36). For ξ∈𝒵𝒞, we have, by (4.22), that there exists k∈ℤd such that φ̂0(ξ+2πk)≠0. Since H0 is a 2π-ℤd periodic we obtain, by (4.37),
(4.49)φ0̂(D(ξ+2πk))=H0(ξ+2πk)φ0̂(ξ+2πk)=H0(ξ)φ0̂(ξ+2πk).
Hence
(4.50)|H0(ξ)|=|φ0̂(D(ξ+2πk))||φ0̂(ξ+2πk)|.
However, Lemma 4.6 implies that for a.e. μ∈ℝd,|φ0̂(Dμ)|2⩽|φ̂0(μ)|2. Thus, |H0(ξ)|⩽1, and this completes the proof of Lemma 4.8.
Now, let us extend the definition ofH0 and H1 to 𝒵, as well. Since H0 is already defined on 𝒵𝒞, the following definition of H1 on 𝒵 makes sense:
(4.51)H1(ξ)={12,ξ∈Z,ξ+2πD-1ε0∈Z,H0(ξ+2πD-1ε0),ξ∈Z,ξ+2πD-1ε0∉Z.
Using Lemmas 4.7 and 4.8 and (4.51), we obtain that H1:ℝd→ℂ is a 2π-ℤd periodic, measurable function, such that
(4.52)|H1(ξ)|⩽1,a.e.ξ∈Rd.
Moreover, we conclude that (4.32) is now satisfied for a.e. ξ∈ℝd. By Lemma 4.7, we need to check (4.32) only on 𝒵. But, for ξ∈𝒵, we have that φ0̂(ξ)=0, by (4.21), and ψ̂(Dξ)=0, since Dimψ(ξ)=0. Hence, on 𝒵 (4.32) is satisfied irrespective of the value on H1(ξ). We have established, that
(4.53)ψ̂(Dξ)=eiε0⋅ξH1(ξ)¯φ0̂(ξ),a.e.ξ∈ZC.
Let us turn our attention toH0. Now that H1 is defined on all of ℝd, we define H0 on 𝒵 by
(4.54)H0(ξ):=H1(ξ+2πD-1ε0),a.e.ξ∈Rd.
Obviously, H0:ℝ→ℂ is a 2π-ℤd periodic, measurable function, such that
(4.55)|H0(ξ)|⩽1,a.e.ξ∈Rd.
Again, we obtain that such H0 satisfies (4.37) on ℝd. And, again, because of Lemma 4.8, it is enough to check (4.37) on 𝒵. Since φ̂0(ξ)=0, for ξ∈𝒵, it is enough to show that φ̂0(Dξ)=0, for ξ∈𝒵. But this is an immediate consequence of Lemma 4.6. We conclude that, for a.e. ξ∈ℝ,
(4.56)φ̂0(Dξ)=H0(ξ)φ̂0(ξ).
The following lemma connectsH0 and H1.
Lemma 4.9.
For a.e.ξ∈ℝd(4.57)|H0(ξ)|2+|H1(ξ)|2=1.Proof.
(1) Ifξ∈𝒵 and ξ+2πD-1ε0∈𝒵, then, by (4.51) and (4.37), H1(ξ)=1/2, and, since ξ+4πD-1ε0∈𝒵, so
(4.58)H0(ξ)=H1(ξ+2πD-1ε0)=12.
Obviously, (4.57) is satisfied.
(2) Ifξ∈𝒵, and ξ+2πD-1ε0∉𝒵, then by (4.51) and (4.54),
(4.59)|H0(ξ)|2+|H1(ξ)|2=|H0(ξ+2πD-1ε0)|2+|H1(ξ+2πD-1ε0)|2.
Hence, (4.57) is going to be satisfied if we can prove it on 𝒵𝒞. We present this argument.
Lemma4.6 implies that, for a.e. ξ∈ℝd(4.60)|φ̂0(ξ)|2=|ψ̂(Dξ)|2+|φ̂0(Dξ)|2.
From (4.53) and (4.56) we obtain that, for a.e. ξ∈ℝd(4.61)|φ̂0(ξ)|2=[|H0(ξ)|2+|H1(ξ)|2]⋅|φ̂0(ξ)|2.
Using the fact that H0(ξ) and H1(ξ) are 2π-ℤd periodic, we can periodize (4.61). This periodization and (4.22) provides us with
(4.62)Dimψ(ξ)=[|H0(ξ)|2+|H1(ξ)|2]⋅Dimψ(ξ).
Recall that on 𝒵𝒞,Dimψ(ξ)≠0; thus (4.62) provides us with (4.57) on 𝒵𝒞.
The following lemma establishes thatH0(ξ) and H1(ξ) are generalized filters, and provides the crucial step to find ρ such that (4.18) is satisfied. It is interesting to observe that the proof of this lemma relies on the fact that ϱq(ξ)=0,q∈𝕊.
Lemma 4.10.
For a.e.ξ∈ℝd(4.63)H0(ξ)H0(ξ+2πD-1ε0)¯=H1(ξ+2πD-1ε0)H1(ξ)¯,(4.64)|H0(ξ)|=|H1(ξ+2πD-1ε0)|,(4.65)|H0(ξ)|2+|H0(ξ+2πD-1ε0)|2=|H1(ξ)|2+|H1(ξ+2πD-1ε0)|2=1.Proof.
We only need to prove (4.63). Indeed, (4.63) and Lemma 4.9 imply that
(4.66)|H0(ξ)|2=|H0(ξ)|2⋅|H0(ξ+2πD-1ε0)|2+|H0(ξ)|2⋅|H1(ξ+2πD-1ε0)|2=|H1(ξ)|2⋅|H1(ξ+2πD-1ε0)|2+|H0(ξ)|2⋅|H1(ξ+2πD-1ε0)|2=|H1(ξ+2πD-1ε0)|2.
This clearly implies (4.64). Lemma 4.9 and (4.64) imply (4.65) straightforwardly.
Let us prove (4.63). First, we consider ξ∈𝒵⋃(𝒵+2πD-1ε0). We have three cases.
(1) Ifξ∈𝒵 and ξ+2πD-1ε0∉𝒵. By (4.51) and (4.54), we have
(4.67)H0(ξ)=H1(ξ+2πD-1ε0),H1(ξ)=H0(ξ+2πD-1ε0)
which clearly establishes (4.63).
(2) Ifξ∈𝒵 and ξ+2πD-1ε0∈𝒵. By (4.51) and (4.54), we have
(4.68)H0(ξ)=H1(ξ+2πD-1ε0),H1(ξ)=H0(ξ+2πD-1ε0)=12.
Again, (4.63) follows.
(3) Ifξ∉𝒵 and ξ+2πD-1ε0∈𝒵, then ξ+4πD-1ε0∉𝒵, thus, by (4.51) and (4.54), we have,
(4.69)H1(ξ+2πD-1ε0)=H0(ξ+4πD-1ε0)=H0(ξ),H0(ξ+2πD-1ε0)=H1(ξ).
Obviously, (4.63) follows.
Finally, let us considerξ∈[𝒵⋃(𝒵+2πD-1ε0)]𝒞. Using (4.53), (4.56), and (4.57), we obtain, for j⩾1,q∈𝕊, and a.e. ζ∈ℝd,
(4.70)ψ̂(Djζ)ψ̂(Dj(ζ+2πq))¯=φ0̂(Dj-1ζ)φ0̂(Dj-1(ζ+2πq))¯-φ0̂(Djζ)φ0̂(Dj(ζ+2πq))¯.
By Proposition2.2, for q∈𝕊, and a.e. ζ∈ℝd, we obtain
(4.71)0=ϱq(ζ)=∑j=0+∞ψ̂(Djζ)ψ̂(Dj(ζ+2πq))¯=ψ̂(ζ)ψ̂(ζ+2πq)¯+φ0̂(ζ)φ0̂(ζ+2πq)¯=[H0(D-1ζ)H0(D-1ζ+2πD-1ε0)¯-H1(D-1ζ)¯H1(D-1ζ+2πD-1ε0)]⋅φ0̂(D-1ζ)φ0̂(D-1ζ+2πD-1q)¯.
Now supposeξ∉𝒵 and ξ+2πD-1ε0∉𝒵. By (4.22) there exist k,ℓ∈ℤd such that
(4.72)φ0̂(ξ+2πk)≠0,φ0̂(ξ+2πl+2πD-1ε0)≠0.
Let(4.73)D-1ζ=ξ+2πk,q=D(l-k)+ε0
and we conclude
(4.74)0=H0(ξ+2πk)H0(ξ+2πD-1ε0)¯-H1(ξ+2πk)¯H1(ξ+2πD-1ε0)=H0(ξ)H0(ξ+2πD-1ε0)¯-H1(ξ)¯H1(ξ+2πD-1ε0).
This proves the case ξ∈[𝒵⋃(𝒵+2πD-1ε0)]𝒞 and completes the proof of Lemma 4.10.
We have established so far thatH0 and H1 are generalized filters, and that φ0 is a pseudoscaling function and that H0 is its corresponding filter, and that ψ̂ satisfies (4.53). We will now prove that (4.53) provides us with (4.21), which is going to complete the proof of the Theorem 4.5.
Let(4.75)Ω={ξ∈Rd:H0(ξ+2πD-1ε0)=0}.
It is clear that both Ω and Ω𝒞 are measurable and 2π-ℤd periodic. Furthermore, by (4.65), we have
(4.76)ξ∈Ω⟶ξ+2πD-1ε0∈ΩC,a.e.
Forξ∈Ω𝒞, define a function 𝒮 as follows:
(4.77)S(ξ):=H1(ξ)H0(ξ+2πD-1ε0).
By (4.64), it follows that 𝒮:Ω𝒞→ℂ is a 2π-ℤd periodic, unimodular function. For ξ∈Ω, we use (4.76) to conclude that ξ+2πD-1ε0∈Ω𝒞, and we define 𝒮 by
(4.78)S(ξ):=S(ξ+2πD-1ε0).
Hence, 𝒮(ξ) is an unimodular, 2π-ℤd periodic function on ℝd, which satisfies
(4.79)H1(ξ)=S(ξ)H0(ξ+2πD-1ε0),a.e..
Lemma 4.11.
𝒮(ξ) is a π-ℤd periodic function.Proof.
Firstly, ifξ∈ℝd such that H0(ξ)H0(ξ+2πD-1ε0)¯=0. By (4.76), we have two possibilities: ξ∈Ω and ξ+2πD-1ε0∉Ω, or ξ∉Ω and ξ+2πD-1ε0∈Ω. In both cases (4.78) and 2π-ℤd periodic 𝒮(ξ) provide us with the conclusion that 𝒮(ξ)=𝒮(ξ+2πD-1ε0).
Secondly, ifξ∈ℝd such that H0(ξ)H0(ξ+2πD-1ε0)¯≠0. In this case, we apply (4.63), (4.79), and the unimodularity of 𝒮(ξ) to obtain
(4.80)H0(ξ)H0(ξ+2πD-1ε0)¯=H1(ξ)¯H1(ξ+2πD-1ε0)=S(ξ+2πD-1ε0)H0(ξ)S(ξ)H0(ξ+2πD-1ε0)¯={S(ξ+2πD-1ε0)S(ξ)¯}H0(ξ)H0(ξ+2πD-1ε0)¯.
Therefore,
(4.81)S(ξ)=S(ξ+2πD-1ε0).
The proof of Lemma 4.11 is completed.
Forξ∈ℝd, let
(4.82)ρ(ξ)=S(D-1ξ)¯.
Since 𝒮(ξ) is a π-ℤd periodic, unimodular function, so ρ(ξ) is a 2π-ℤd periodic, unimodular function. Hence, H0,φ0 and ρ(ξ) satisfy all the necessary requirements. It remains to check that they satisfy (4.18), as well. Indeed, by (4.53), (4.79), and (4.82), we obtain
(4.83)ψ̂(Dξ)=eiε0⋅ξH1(ξ)¯φ0̂(ξ)=eiε0⋅ξρ(Dξ)H0(ξ+2πD-1ε0)¯φ0̂(ξ).
The proof of Theorem 4.5 is completed.Theorem 4.12.
Suppose thatψ is a M-TFW. Then, ψ is a semi-orthogonal MRA M-TFW if and only if
(4.84)Dimψ(ξ)∈{0,1},fora.e.ξ∈Rd.Proof.
Suppose thatψ is a semi-orthogonal MRA M-TFW, using Theorems 4.1 and 4.5, then (4.84) is valid.
Suppose thatψ is a M-TFW and (4.84) is valid. By Theorem 4.1, we conclude that ψ is semi-orthogonal, and, thus, by (4.13), Dimψ(ξ)∈{0,1}, for a.e. ξ∈ℝd. By Theorem 4.5, we conclude that ψ is an MRA M-TFW.


## 5. Conclusions

In this paper, we study all generalized low-pass filters and M-TFW. Firstly, we study the pseudoscaling function, generalized low-pass filters, and MRA M-TFW and give some important characterizations about them. Then, we characterize all M-TFW by showing that they correspond precisely to those of which the dimension function is nonnegative integer valued. Finally, we also show that an M-TFW arises from our MRA construction if and only if the dimension of a particular linear space is either zero or one.



---

*Source: 128294-2010-12-29.xml*