Characterizations of Tight Frame Wavelets with Special Dilation Matrices

Abstract

We study all generalized low-pass filters and tight frame wavelets with special dilation matrix 𝑀 (M-TFW), where M satisfies 𝑀^𝑑=2𝐼_𝑑 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.

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+𝑒^3𝑖𝜉), 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 2^𝑑−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(ℝ^𝑑); 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(ℝ^𝑑). Therefore, we have to recur some special dilation matrices to solve the problem in 𝐿²(ℝ^𝑑). 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 𝑀^𝑑=2𝐼_𝑑 and generates the checkerboard lattice [14], that is, 𝑀ℤ^𝑑=𝑘∈ℤ^𝑑∶^𝑑_𝑖=1𝑘_𝑖∈2ℤ. (1.1) In this case, we only need one function 𝜓 such that the system {𝜓_𝑗,𝑘,𝑗∈ℤ,𝑘∈ℤ^𝑑} is a tight frame (with 𝐴=𝐵=1) for 𝐿²(ℝ^𝑑) (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. Section 2 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(ℝ^𝑑) by dimension function.

2. Preliminaries and Basic Definitions

In this paper, we denote by 𝐷 the transpose of 𝑀∶𝐷=𝑀^𝑇, and 𝕊=ℤ^𝑑⧵𝐷ℤ^𝑑, 𝜀₀ is a constant vector and 𝜀₀∈𝕊. For 𝜓∈𝐿²(ℝ^𝑑), and 𝑘∈ℤ^𝑑, we denote 𝜓_𝑗,𝑘∶=2^𝑗/2𝜓𝑀^𝑗𝑥−𝑘. (2.1)

Let us recall the concept of frame.

Given a countable index set 𝐼, a collection {𝑓_𝑖∶𝑖∈𝐼} in a Hilbert space ℍ is called a frame in ℍ if there exist two constants 𝐴,𝐵>0 such that 𝐴𝑓²_ℍ≤_𝑖∈𝐼𝑓,𝑓_𝑖²≤𝐵𝑓²_ℍ,∀𝑓∈ℍ. (2.2) If we can choose 𝐴=𝐵 in (2.2), then {𝑓_𝑖∶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 function 𝜓∈𝐿²(ℝ^𝑑) is a tight frame wavelet with dilation matrix 𝑀(briefly: M-TFW) if and only if the system {𝜓_𝑗,𝑘,𝑗∈ℤ,𝑘∈ℤ^𝑑} is a tight frame (with 𝐴=𝐵=1) for 𝐿²(ℝ^𝑑).

Definition 2.1 implies that 𝑓²₂=_{𝑗∈ℤ𝑘∈ℤ𝑑}𝑓,𝜓_𝑗,𝑘²,∀𝑓∈𝐿²ℝ^𝑑. (2.3) It is clearly that (2.3) is equivalent to 𝑓=_{𝑗∈ℤ𝑘∈ℤ𝑑}𝑓,𝜓_𝑗,𝑘𝜓_𝑗,𝑘 (2.4) for all f∈𝐿²(ℝ^𝑑), where the sum converges unconditionally in 𝐿²(ℝ^𝑑).

Bownik gave a deeper result as follows.

Proposition 2.2 (see [8, Theorem  4.2]). 𝜓∈𝐿²(ℝ^𝑑) is a M-TFW if and only if _𝑗∈ℤ𝜓𝐷^𝑗𝜉²=1,a.e.𝜉∈ℝ^𝑑,𝜚_𝑘(𝜉)=^+∞_𝑗=0𝜓𝐷^𝑗𝜉𝜓𝐷^𝑗(𝜉+2𝜋𝑘)=0,a.e.𝜉∈ℝ^𝑑,𝑘∈𝕊. (2.5)(2.6)

A M-TFW is said to be semi-orthogonal if and only if 𝜓_𝑗,𝑘 is orthogonal to 𝜓_ℓ,𝑚 whenever 𝑗≠ℓ (for all 𝑘,𝑚∈ℤ^𝑑). This is equivalent to the orthogonality of the subspaces 𝑊_𝑗 and 𝑊_ℓ if 𝑗≠ℓ, where 𝑊_𝑗=span𝜓_𝑗,𝑘∶𝑘∈ℤ^𝑑, (2.7) as 𝑗 ranges throughout ℤ.

Proposition 2.3 (see [3, Chapter  7]). Suppose 𝜓∈𝐿²(ℝ^𝑑) is a semi-orthogonal M-TFW, then, for each 𝑗⩾1, _{𝑘∈ℤ𝑑}𝜓𝐷^𝑗(𝜉+2𝜋𝑘)𝜓(𝜉+2𝜋𝑘)=0,a.e.𝜉∈ℝ^𝑑. (2.8)

Definition 2.4. A measurable 2𝜋-ℤ^𝑑 periodic function 𝐻 on ℝ^𝑑 is a generalized filter with dilation matrix 𝑀 (briefly: M-GF) if it satisfies 𝐻(𝜉)²+𝐻𝜉+2𝜋𝐷⁻¹𝜀₀²=1,a.e.𝜉∈ℝ^𝑑. (2.9)
As what was done in [20, 21], we will denote by 𝐹 the set of generalized filters with dilation matrix 𝑀 and let 𝐹⁺={𝐻∈𝐹∶𝐻⩾0}. Observe that 𝐻∈𝐹⇒|𝐻|∈𝐹⁺.

Definition 2.5. A function 𝜑∈𝐿²(ℝ^𝑑) is called a pseudoscaling function with dilation matrix 𝑀 (briefly: M-PSF) if there exists a generalized filter 𝐻∈𝐹 such that 𝜑(𝐷𝜉)=𝐻(𝜉)𝜑(𝜉),a.e.𝜉∈ℝ^𝑑. (2.10)

Remark 2.6. Notice that 𝐻 is not uniquely determined by the M-PSF 𝜑. Therefore, we will denote by 𝐹_𝜑 the set of all 𝐻∈𝐹 such that 𝐻 satisfies (2.10) for 𝜑. For example, if 𝜑=0, then 𝐹_𝜑=𝐹. If 𝜑 is a scaling function of MRA wavelet, then 𝐹_𝜑 is a singleton. On the other hand, if 𝜑 is an M-PSF, then |𝜑|^∨ is also an M-PSF, and if 𝐻∈𝐹, then |𝐻|∈𝐹_|𝜑^|∨.

Definition 2.7. For 𝐻∈𝐹, let 𝑁₀(|𝐻|)=𝜉∈ℝ^𝑑∶lim_𝑗→+∞𝜑_|𝐻|𝐷^−𝑗𝜉,𝜑=0_|𝐻|(𝜉)=^∞_𝑗=1𝐻𝐷^−𝑗𝜉. (2.11) We say that 𝐻∈𝐹 is a generalized low-pass filter with dilation matrix 𝑀(briefly: M-GLPF) if |𝑁₀(|𝐻|)|=0.

Now, we give the definition of MRA M-TFW.

Definition 2.8. A M-TFW 𝜓 is an MRA M-TFW if there exists a M-PSF 𝜑 and 𝐻∈𝐹_𝜑 such that 𝜓(𝐷𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝜌(𝐷𝜉)𝐻𝜉+2𝜋𝐷⁻¹𝜀₀𝜑(𝜉),a.e.𝜉∈ℝ^𝑑, (2.12) where 𝜌(𝜉) is a measurable 2𝜋-ℤ^𝑑 periodic function and |𝜌(𝜉)|=1.

In Section 3, 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 in 𝐿²(ℝ^𝑑). We will give some important characterizations about them.

Lemma 3.1. Suppose 𝜑 is a M-PSF and 𝐻∈𝐹_𝜑. If lim_𝑗→+∞𝜑𝐷^−𝑗𝜉=1,a.e.𝜉, (3.1) then, 𝜑(𝜉)=^∞_𝑗=1𝐻𝐷^−𝑗𝜉,a.e.𝜉, (3.2) and |𝑁₀(|𝐻|)|=0.

Proof. By (2.10), we have 𝜑(𝜉)=^𝑛_𝑗=1𝐻𝐷^−𝑗𝜉𝜑𝐷^−𝑛𝜉,a.e.𝜉∈ℝ^𝑑. (3.3) Using (3.1), we obtain that |𝜑(𝜉)|=𝜑_|𝐻| and, therefore (3.2) and |𝑁₀(|𝐻|)|=0 are clearly satisfied.

Lemma 3.2 (see [25, Lemma  3.2]). If 𝑓∈𝐿¹(ℝ^𝑑), then, for a.e. 𝜔∈ℝ^𝑑, lim_𝑗→+∞|𝑓(𝐷^𝑗𝜔)|=0.

Proof. As the proof is simple, we omit it.

Theorem 3.3. Suppose that 𝜓 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 that 𝜓 is an MRA M-TFW, by (2.5), (2.9) and (2.12), we can obtain 1=_𝑗∈ℤ𝜓𝐷^𝑗𝜉²=_𝑗∈ℤ𝐻𝐷^𝑗−1𝜉+2𝜋𝐷⁻¹𝜀₀²𝜑𝐷^𝑗−1𝜉²=lim^𝑛_{𝑛→+∞𝑗=−𝑛}𝐻𝐷^𝑗−1𝜉+2𝜋𝐷⁻¹𝜀₀²𝜑𝐷^𝑗−1𝜉²=lim^𝑛_{𝑛→+∞𝑗=−𝑛}1−𝐻𝐷^𝑗−1𝜉²𝜑𝐷^𝑗−1𝜉²=lim_𝑛→+∞𝜑𝐷^−𝑛−1𝜉²−𝜑𝐷^𝑛−1𝜉². (3.4)
Since 𝜑∈𝐿²(ℝ^𝑑), Lemma 3.2 implies lim_𝑛→+∞|𝜑(𝐷^𝑛𝜉)|²=0 for a.e. 𝜉∈𝐿²(ℝ^𝑑). This shows that for a.e. 𝜉∈𝐿²(ℝ^𝑑), lim_𝑛→+∞|𝜑(𝐷^−𝑛𝜉)|²=1, thus, by Lemma 3.1, 𝐻 is a generalized low-pass filter.

Suppose that 𝜓 is a M-TFW, let 𝜎_𝜓(𝜉)=_{𝑘∈ℤ𝑑}𝜓(𝜉+2𝜋𝑘)²,Dim_𝜓(𝜉)=^∞_{𝑗=1𝑘∈ℤ𝑑}𝜓𝐷^𝑗(𝜉+2𝜋𝑘)². (3.5) We say that Dim_𝜓(𝜉) is the dimension function of M-TFW.

Lemma 3.4. Suppose that 𝜓 is an MRA M-TFW, the corresponding M-PSF of the 𝜓 is 𝜑, then 𝜑(𝜉)²=^∞_𝑗=1𝜓𝐷^𝑗𝜉²,a.e.𝜉∈ℝ^𝑑. (3.6)

Proof. By Definition 2.8, there exists a M-GF 𝐻∈𝐹 such that 𝜑(𝐷𝜉)=𝐻(𝜉)𝜑(𝜉),𝜓(𝐷𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝜌(𝐷𝜉)𝐻𝜉+2𝜋𝐷⁻¹𝜀₀𝜑(𝜉)=𝐻₁(𝜉)𝜑(𝜉) (3.7) we obtain that 𝜑(𝜉)²=𝐻(𝜉)²𝜑(𝜉)²+𝐻₁(𝜉)²𝜑(𝜉)²=𝜑(𝐷𝜉)²+𝜓(𝐷𝜉)². (3.8) Thus 𝜑(𝜉)²=𝜑(𝐷𝜉)²+𝜓(𝐷𝜉)²=𝜑𝐷²𝜉²+𝜓𝐷²𝜉²+𝜓(𝐷𝜉)²=⋯=𝜑𝐷^𝑁𝜉²+𝜓𝐷^𝑁𝜉²+⋯+𝜓(𝐷𝜉)²=𝜑𝐷^𝑁𝜉²+^𝑁_𝑗=1𝜓𝐷^𝑗𝜉². (3.9) Notice that |𝜑(𝜉)|⩽1, hence {∑^𝑁_𝑗=1|𝜓(𝐷^𝑗𝜉)|²}_𝑁 is an increased sequence and bounded with 1. Moreover, _ℝ𝑑𝜑𝐷^𝑁𝜉²1𝑑𝜉=2^𝑁_ℝ^𝑑𝜑(𝜉)²𝑑𝜉. (3.10) Therefore, by Fatou lemma, _ℝ𝑑lim_𝑁→∞𝜑𝐷^𝑁𝜉²𝑑𝜉⩽lim_{𝑁→∞ℝ𝑑}𝜑𝐷^𝑁𝜉²𝑑𝜉=lim_𝑁→∞12^𝑁_ℝ^𝑑𝜑(𝜉)²𝑑𝜉=0. (3.11) So lim_𝑁→∞|𝜑(𝐷^𝑁𝜉)|²=0, consequently, |𝜑(𝜉)|²=∑^∞_𝑗=1|𝜓(𝐷^𝑗𝜉)|²,a.e.𝜉∈ℝ^𝑑.

Lemma 3.5. Suppose that 𝜓 is an MRA M-TFW, then Dim_𝜓(𝐷𝜉)+𝜎_𝜓(𝐷𝜉)=Dim_𝜓(𝜉)+Dim_𝜓𝜉+2𝜋𝐷⁻¹𝜀₀. (3.12) Furthermore, if 𝜓 is a M-TFW, then (3.12) is also valid.

Proof. We only prove the case of 𝜓 is an MRA M-TFW. By Definition 2.8, there exists an M-GF 𝐻∈𝐹 such that 𝜑(𝐷𝜉)=𝐻(𝜉)𝜑(𝜉),𝜓(𝐷𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝜌(𝐷𝜉)𝐻𝜉+2𝜋𝐷⁻¹𝜀₀𝜑(𝜉)=𝐻₁(𝜉)𝜑(𝜉). (3.13) By Lemma 3.4, we can obtain Dim_𝜓(𝜉)=^∞_{𝑗=1𝑘∈ℤ𝑑}𝜓𝐷^𝑗(𝜉+2𝜋𝑘)²=_{𝑘∈ℤ𝑑}^∞_𝑗=1𝜓𝐷^𝑗(𝜉+2𝜋𝑘)²=_{𝑘∈ℤ𝑑}𝜑(𝜉+2𝜋𝑘)². (3.14) Thus, Dim_𝜓(𝐷𝜉)=_{𝑘∈ℤ𝑑}𝜑(𝐷𝜉+2𝜋𝑘)²=_{𝑘∈𝐷ℤ𝑑}𝜑(𝐷𝜉+2𝜋𝑘)²+_𝑘∈𝕊𝜑(𝐷𝜉+2𝜋𝑘)²=_{𝑘∈ℤ𝑑}𝜑(𝐷(𝜉+2𝜋𝑘))²+_{𝑘∈ℤ𝑑}𝜑𝐷𝜉+2𝜋𝑘+2𝜋𝐷⁻¹𝜀₀²=_{𝑘∈ℤ𝑑}𝐻(𝜉+2𝜋𝑘)²𝜑(𝜉+2𝜋𝑘)²+_{𝑘∈ℤ𝑑}𝐻𝜉+2𝜋𝑘+2𝜋𝐷⁻¹𝜀₀²𝜑𝜉+2𝜋𝑘+2𝜋𝐷⁻¹𝜀₀²=𝐻(𝜉)²_{𝑘∈ℤ𝑑}𝜑(𝜉+2𝜋𝑘)²+𝐻𝜉+2𝜋𝐷⁻¹𝜀₀²_{𝑘∈ℤ𝑑}𝜑𝜉+2𝜋𝑘+2𝜋𝐷⁻¹𝜀₀²=𝐻(𝜉)²Dim_𝜓(𝜉)+𝐻𝜉+2𝜋𝐷⁻¹𝜀₀²Dim_𝜓𝜉+2𝜋𝐷⁻¹𝜀₀. (3.15) Analogously, we have 𝜎_𝜓(𝐷𝜉)=𝐻₁(𝜉)²Dim_𝜓(𝜉)+𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀²Dim_𝜓𝜉+2𝜋𝐷⁻¹𝜀₀. (3.16) Thus, Dim_𝜓(𝐷𝜉)+𝜎_𝜓(𝐷𝜉)=Dim_𝜓(𝜉)+Dim_𝜓𝜉+2𝜋𝐷⁻¹𝜀₀. (3.17)

Theorem 3.6. Suppose that 𝜓 is an MRA M-TFW, the corresponding M-PSF of the 𝜓 is 𝜑, then 0⩽Dim_𝜓(𝜉)⩽1,a.e.𝜉∈ℝ^𝑑. (3.18)

Proof. Let 𝐻∈𝐹_𝜑, and it is the corresponding M-GLPF of the 𝜓. Let 𝜑_𝑛(𝜉)=^𝑛𝑗=1𝐻𝐷^−𝑗𝜉𝜒_{2𝑛/𝑑𝕋𝑑}(𝜉),𝑛=𝑞𝑑,𝑞∈ℤ,_𝑛+1𝑗=1𝐻𝐷^−𝑗𝜉𝜒_{2(𝑛+1)/𝑑𝕋𝑑}(𝜉)⋮,𝑛=𝑞𝑑−1,𝑞∈ℤ,_{𝑛+(𝑑−1)𝑗=1}𝐻𝐷^−𝑗𝜉𝜒_{2(𝑛+𝑑−1)/𝑑𝕋𝑑}(𝜉),𝑛=𝑞𝑑−(𝑑−1),𝑞∈ℤ, (3.19) where Γ denotes all the vertices of cube [0,1]^𝑑, 𝕋=[0,2𝜋]. It is clearly that lim_𝑛→∞|𝜑_𝑛(𝜉)|=|𝜑(𝜉)|. Now, we need to prove that {𝜑_𝑛(⋅−𝑘)}_{𝑘∈ℤ𝑑} is an orthonormal system for 𝐿²(ℝ^𝑑). Denote 𝐼_𝑛=_ℝ𝑑𝜑_𝑛(𝑥)𝜑_𝑛(𝑥−𝑘)𝑑𝑥. (3.20) So, if 𝑛=𝑞𝑑,𝑞∈ℤ, then 𝐼_𝑛=_ℝ𝑑𝜑_𝑛(𝑥)𝜑_𝑛()1𝑥−𝑘𝑑𝑥=(2𝜋)^{𝑑[0,2𝑞]2𝜋𝑑𝑛}_𝑗=1𝐻𝐷^−𝑗𝜉²𝑒^𝑖𝑘𝜉=1𝑑𝜉(2𝜋)^{𝑑[0,2𝑞]2𝜋𝑑(𝑞−1)𝑑}_𝑗=1𝐻𝐷^−𝑗𝜉²𝑒^{𝑖𝑘𝜉𝑑−1}_ℓ=0𝐻𝐷^{−𝑞𝑑+ℓ}𝜉²=1𝑑𝜉(2𝜋)^{𝑑[0,2𝑞−12𝜋]𝑑(𝑞−1)𝑑}_𝑗=1𝐻𝐷^−𝑗𝜉²𝑒^𝑖𝑘𝜉×_𝜔∈Γ^𝑑−1_ℓ=0𝐻𝐷^{−𝑞𝑑+ℓ}𝜉+2^𝑞𝜋𝜔²=1𝑑𝜉(2𝜋)^{𝑑[0,2𝑞−12𝜋]𝑑(𝑞−1)𝑑}_𝑗=1𝐻𝐷^−𝑗𝜉𝑒^𝑖𝑘𝜉=𝐼_{(𝑞−1)𝑑}=⋯=𝐼_𝑑. (3.21) However, 𝐼_𝑑=1(2𝜋)^{𝑑[]0,4𝜋𝑑𝑑}_𝑗=1𝐻𝐷^−𝑗𝜉²𝑒^𝑖𝑘𝜉=1𝑑𝜉(2𝜋)^{𝑑[]0,2𝜋𝑑}𝑒^𝑖𝑘𝜉×^𝑑_{𝜔∈Γℓ=1}𝐻𝐷^−ℓ(𝜉+2𝜋𝜔)²=1𝑑𝜉(2𝜋)^{𝑑[]0,2𝜋𝑑}𝑒^𝑖𝑘𝜉=𝛿_𝑘,0. (3.22) Thus we deduce from (3.19) and the above equation that if 𝑛=𝑞𝑑,𝑞∈ℤ, then 𝜑_𝑛(⋅),𝜑_𝑛(⋅−𝑘)=𝛿_𝑘,0. (3.23) 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∈ℝ^𝑑_{𝑘∈ℤ𝑑}𝜑_𝑛(𝜉+2𝜋𝑘)²=1,∀𝑛∈ℕ. (3.24) By Fatou lemma, Dim_𝜓(𝜉)=_{𝑘∈ℤ𝑑}𝜑(𝜉+2𝜋𝑘)²=_{𝑘∈ℤ𝑑}lim_𝑛→∞𝜑_𝑛(𝜉+2𝜋𝑘)²⩽lim_{𝑛→∞𝑘∈ℤ𝑑}𝜑_𝑛(𝜉+2𝜋𝑘)²=1. (3.25)

Recall that the MRA TFW are precisely those 𝜓∈𝐿²(ℝ) that can be constructed from a generalized low-pass filter as described in [20, 21]. What are the properties of 𝜎_𝜓 when 𝜓∈𝐿²(ℝ^𝑑) is a M-TFW? We will see that the answers to this question are important for determining the properties of Dim_𝜓.

For 𝜓∈𝐿²(ℝ^𝑑) (not necessarily a M-TFW), let us consider the principal shift-invariant space 𝑊(𝜓)≡span{𝜓(⋅−𝑘)∶𝑘∈ℤ^𝑑}. 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)}={𝜓_0𝑘},𝑘∈ℤ^𝑑. If 𝜓 is an orthonormal wavelet, then {𝜓(⋅−𝑘)} is an orthonormal basis for 𝑊₀. What will happen if 𝜓∈𝐿²(ℝ^𝑑) is a M-TFW?

It is not hard to see that we can find 𝜑∈𝑊(𝜓) such that {𝜑(⋅−𝑘),𝑘∈ℤ^𝑑} is a tight frame (with constant 1) for 𝑊(𝜓). Indeed, let 𝑠(𝜉)=1/𝜎_𝜓(𝜉) if 𝜉∈𝑈_𝜓 and 𝑠(𝜉)=0 if 𝜉∉𝑈_𝜓, where 𝑈_𝜓=𝑈={𝜉∈𝕋,𝜎_𝜓>0}. Straight forward calculations show that 𝜑 defined by 𝜑(𝜉)=𝑠(𝜉)𝜓(𝜉) provides us with the desired function. Moreover, 𝜎_𝜑=𝜒_𝑈𝜓(𝜉) (3.26) and 𝑈_𝜑=𝑈_𝜓. This result can be extended to the following known result.

Lemma 3.7 (see [5,Theorem  7.2.3]). {𝜓(⋅−𝑘)},𝑘∈ℤ^𝑑, is a tight frame with constant 𝐴,𝐵 for 𝑊(𝜓) if and only if 𝐴𝜒_𝑈⩽𝜎_𝜓⩽𝐵𝜒_𝑈. (3.27)

When 𝜓 is a M-TFW, for the space 𝑊₀, one has the following results.

Lemma 3.8 (see [21,Theorem  2.7]). Suppose that 𝜓 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,𝜓_0𝑘⟩|²,(c)𝜓 is semi-orthogonal,(d)𝜎_𝜓=𝜒_𝑈 a.e. on ℝ^𝑑.

Proof. It is clear that (a) implies (b), this is (2.3) with 𝑓=𝜓,𝐻=𝑊₀. Since 𝜓²₂=_{𝑘∈ℤ𝑑}𝜓,𝜓_0𝑘² (3.28) and {𝜓_𝑗𝑘} is a tight frame, (b) implies ⟨𝜓,𝜓_𝑗𝑘⟩=0 whenever 𝑗≠0. This clearly implies the semiorthogonality. Thus, (b) implies (c). If we assume (c), so that ⟨𝑓,𝜓_𝑗𝑘⟩=0 whenever 𝑓∈𝑊₀ and 𝑗≠0, an application of (2.3) gives us 𝑓²₂=_{𝑗∈ℤ𝑘∈ℤ𝑑}𝑓,𝜓_𝑗𝑘²=_{𝑘∈ℤ𝑑}𝑓,𝜓_0𝑘² (3.29) and we see that (c) implies (a).
Lemma 3.7 with 𝐴=𝐵=1 tells us that (a) and (d) are equivalent.

Although the proof of Lemma 3.8 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 𝑊_𝑗,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 Theorem 3.3, Lemma 3.4, and Lemma 3.8, the assumption of 𝑀^𝑑=2𝐼_𝑑 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 𝐷ℤ^𝑑⊂ℤ^𝑑 and ℤ^𝑑=𝐷ℤ^𝑑+ℤ^𝑑⧵𝐷ℤ^𝑑 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 wavelet 𝜓 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 Dim_𝜓, on sets of positive measure (see [8, 21]). We will investigate the properties of the dimension function for M-TFW.

Theorem 4.1. Suppose that 𝜓 is a M-TFW. Then 𝜓 is semi-orthogonal if and only if Dim_𝜓 is integer valued a.e.

Proof. First, let us prove the sufficiency. Suppose that Dim_𝜓 is integer valued a.e. From Lemma 3.5, we see that 𝜎_𝜓 must also be integer valued a.e. By (3.18) and 𝜎_𝜓(𝜉)⩽1 a.e., we have 𝜎_𝜓(𝜉)=𝜒_𝑈𝜓(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 that 𝜓 is semi-orthogonal, denote 𝐺_𝑛(𝜉)=^∞_𝑗=1𝜓𝐷^𝑗𝜉_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝑘)𝜓𝐷^𝑗(𝜉+2𝜋𝑘),𝑛⩾1. (4.1)
Apply Schwart's inequality to the sum in (4.1) with respect to 𝑘, we obtain ^∞_𝑗=1𝜓𝐷^𝑗𝜉_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝑘)⋅𝜓𝐷^𝑗(⩽𝜉+2𝜋𝑘)_∞𝑗=1𝜓𝐷^𝑗𝜉_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝑘)^21/2_{𝑘∈ℤ𝑑}𝜓𝐷^𝑗(𝜉+2𝜋𝑘)^21/2⩽^∞_𝑗=1𝜓𝐷^𝑗𝜉_{𝑘∈ℤ𝑑}𝜓𝐷^𝑗(𝜉+2𝜋𝑘)^21/2𝜎_𝜓𝐷^𝑛𝜉^1/2. (4.2) Using the fact that 𝜎_𝜓(𝐷^𝑛𝜉)⩽1 a.e. and appling Schwart's inequality to the sum over 𝑗, we obtain that ^∞_𝑗=1𝜓𝐷^𝑗𝜉^21/2∞_{𝑗=1𝑘∈ℤ𝑑}𝜓𝐷^𝑗(𝜉+2𝜋𝑘)^21/2⩽Dim_𝜓(𝜉). (4.3) Therefore, the sum in (4.1) is absolutely convergent. The fact allows us to interchange the sums in the expression 𝐺_𝑛(𝜉). By Proposition 2.3, we have 𝐺_𝑛(𝜉)=_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝑘)^∞_𝑗=1𝜓𝐷^𝑗𝜉𝜓𝐷^𝑗(+𝜓(𝜉)𝜉+2𝜋𝑘)_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝑘)𝜓()=𝜉+2𝜋𝑘_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝑘)^∞_𝑗=0𝜓𝐷^𝑗𝜉𝜓𝐷^𝑗(𝜉+2𝜋𝑘)=_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝑘)𝜚_𝑘(𝜉), (4.4) where 𝜚_𝑘(𝜉) is defined as in (2.6). Since 𝜚_𝑘(𝜉)=0, a.e. 𝜉∈ℝ^𝑑, 𝑘∈𝕊, we obtain 𝐺_𝑛(𝜉)=_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝐷𝑘)^∞_𝑗=0𝜓𝐷^𝑗𝜉𝜓𝐷^𝑗(𝜉+2𝜋𝐷𝑘). (4.5) On the other hand, 𝐺_𝑛+1𝐷⁻¹𝜉=_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛+1𝐷⁻¹𝜉+2𝜋𝑘^∞_𝑗=1𝜓𝐷^𝑗𝐷⁻¹𝜉𝜓𝐷^𝑗𝐷⁻¹𝜉+2𝜋𝑘=_{𝑘∈ℤ𝑑}𝜓𝐷^𝑛(𝜉+2𝜋𝐷𝑘)^∞_𝑗=0𝜓𝐷^𝑗𝜉𝜓𝐷^𝑗(𝜉+2𝜋𝐷𝑘)=𝐺_𝑛(𝜉). (4.6) Therefore, we have shown that 𝐺_𝑛(𝜉)=𝐺_𝑛−1(𝐷𝜉), and consequently, 𝐺_𝑛(𝜉)=𝐺₁(𝐷^𝑛−1𝜉) for 𝑛⩾1. By Propositions 2.2 and 2.3, we have 𝐺₁(𝜉)=_{𝑘∈ℤ𝑑}𝜓(𝐷(𝜉+2𝜋𝑘))^∞_𝑗=1𝜓𝐷^𝑗𝜉𝜓𝐷^𝑗(𝜉+2𝜋𝑘)=_{𝑘∈ℤ𝑑}𝜓(𝐷𝜉+2𝜋𝐷𝑘)^∞_𝑗=0𝜓𝐷^𝑗⋅𝐷𝜉𝜓𝐷^𝑗(𝐷𝜉+2𝜋𝐷𝑘)=_{𝑘∈ℤ𝑑}𝜓(𝐷𝜉+2𝜋𝐷𝑘)^∞_𝑗=0𝜓𝐷^𝑗⋅𝐷𝜉𝜓𝐷^𝑗(𝐷𝜉+2𝜋𝐷𝑘)+_𝑘∈𝕊𝜓(𝐷𝜉+2𝜋𝑘)^∞_𝑗=0𝜓𝐷^𝑗⋅𝐷𝜉𝜓𝐷^𝑗(𝐷𝜉+2𝜋𝑘)=_{𝑘∈ℤ𝑑}𝜓(𝐷𝜉+2𝜋𝑘)^∞_𝑗=0𝜓𝐷^𝑗⋅𝐷𝜉𝜓𝐷^𝑗()=𝐷𝜉+2𝜋𝑘_∞𝑗=0𝜓𝐷^𝑗⋅𝐷𝜉_{𝑘∈ℤ𝑑}𝜓(𝐷𝜉+2𝜋𝑘)𝜓𝐷^𝑗()=𝐷𝜉+2𝜋𝑘_∞𝑗=1𝜓𝐷^𝑗⋅𝐷𝜉_{𝑘∈ℤ𝑑}𝜓(𝐷𝜉+2𝜋𝑘)𝜓𝐷^𝑗(𝐷𝜉+2𝜋𝑘)+𝜓(𝐷𝜉)𝜎_𝜓()=𝜓()𝜎𝐷𝜉𝐷𝜉_𝜓(𝐷𝜉). (4.7)
This shows that 𝐺_𝑛(𝜉)=𝜓𝐷^𝑛𝜉𝜎_𝜓𝐷^𝑛𝜉. (4.8)
Finally, since 𝜓 is a semi-orthogonal M-TFW, 𝜎_𝜓(𝐷^𝑛𝜉) is either 0 or 1 a.e.; this and the last fact, which implies that 𝜓(𝐷^𝑛𝜉)=0 when 𝜎_𝜓(𝐷^𝑛𝜉)=0, give us 𝜓𝐷^𝑛𝜉=𝐺_𝑛(𝜉),a.e.,𝑛⩾1. (4.9)
Now, let Ψ_𝑗(𝜉)=𝜓𝐷^𝑗(𝜉+2𝑘𝜋)∶𝑘∈ℤ^𝑑,𝑗⩾1. (4.10) Using the fact that 𝜎_𝜓(𝜉)⩽1 once again, we see that Ψ_𝑗∈ℓ²(ℤ^𝑑) for a.e. 𝜉∈ℝ^𝑑. But, from (2.6), (2.7), and (2.8), we see that Ψ_𝑗(𝜉)=^∞_𝑛=1Ψ_𝑛(𝜉),Ψ_𝑗(𝜉)_{ℓ2(ℤ𝑑)}Ψ_𝑛(𝜉) (4.11) for 𝑗⩾1.
By the definition of Dim_𝜓, we have Dim_𝜓(𝜉)=^∞_𝑗=1Ψ_𝑗(𝜉)²_ℓ2ℤ𝑑. (4.12) Hence, we conclude that Dim_𝜓(𝜉)=dim𝔽_𝜓(𝜉),a.e.𝜉∈ℝ^𝑑, (4.13) where 𝔽_𝜓(𝜉)=span{Ψ_𝑗(𝜉)∶𝑗⩾1}; this is a well-defined subspace of ℓ²(ℤ^𝑑). Obviously, (4.13) implies that Dim_𝜓 is integer valued a.e. The proof of Theorem 4.1 is completed.

Theorem 4.1 provides us with the following interesting statement.

Corollary 4.2. Suppose that 𝜓 is a M-TFW. Then Dim_𝜓(𝜉)=dim𝔽_𝜓(𝜉), for a.e. 𝜉∈ℝ^𝑑, if and only if 𝜓 is semi-orthogonal.

The dimension function of an orthonormal wavelet which attains value 𝑁>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 Dim_𝜓(𝐷𝜉)⩾Dim_𝜓(𝜉)−1,a.e.𝜉. (4.14) Fortunately, (4.14) is also valid for semi-orthogonal M-TFWs. More precisely, one has the following result.

Proposition 4.3. Suppose that 𝜓 is a semi-orthogonal M-TFW. Let 𝑁>1 be an integer. If there exits a set 𝐴⊆ℝ^𝑑 of positive Lebesgue measure, such that Dim_𝜓(𝜉)⩾𝑁 for all 𝜉∈𝐴, then there exits a set 𝐵⊆ℝ^𝑑 of positive Lebesgue measure such that Dim_𝜓(𝜉)=𝑁−1 for all 𝜉∈𝐵.

There is another interesting consequence of Theorem 4.1. Notice that a consequence of (2.5) is that for every M-TFW 𝜓, one has 𝜓(𝜉)⩽1,a.e. (4.15) An interesting class consists of those M-TFW for which |𝜓(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. If 𝜓 is an MSF M-TFW, then 𝜓 is semi-orthogonal.

Theorem 4.5. Suppose that 𝜓 is a M-TFW, then, 𝜓 is an MRA M-TFW if and only if dim𝔽_𝜓(𝜉)∈{0,1}, for a.e. 𝜉∈ℝ^𝑑.

Proof of Theorem 4.5. First, we prove the necessity. Suppose that 𝜓 is an MRA M-TFW, thus, it satisfies (2.12) for an appropriate 𝐻,𝜌 and 𝜑. If dim𝔽_𝜓(𝜉)⩾2, for some 𝜉∈ℝ^𝑑, then there exist 𝑘,ℓ such that Ψ_𝑘(𝜉) and Ψ_ℓ(𝜉) are linearly independent in ℓ²(ℤ^𝑑). It is then impossible to have a vector 𝜈∈ℓ²(ℤ^𝑑) such that both Ψ_𝑘(𝜉) and Ψ_ℓ(𝜉) are in the one-dimensional subspace span{𝜈}. However, by (2.10) and (2.12), for every 𝑗⩾1, we have 𝜓𝐷^𝑗(𝜉+2𝜋𝑘)=𝑒^{𝑖𝜀0⋅𝐷𝑗−1(𝜉+2𝜋𝑘)}𝜌𝐷^𝑗(𝜉+2𝜋𝑘)𝐻𝐷^𝑗−1(𝜉+2𝜋𝑘)+2𝜋𝐷⁻¹𝜀₀𝜑𝐷^𝑗−1(𝜉+2𝜋𝑘)=𝑒^{𝑖𝜀0⋅𝐷𝑗−1𝜉}𝜌𝐷^𝑗𝜉𝐻𝐷^𝑗−1𝜉+2𝜋𝐷⁻¹𝜀₀^𝑗−2_𝑛=0𝐻𝐷^𝑛𝜉𝜑(𝜉+2𝜋𝑘),𝑘∈ℤ^𝑑. (4.16) Since {𝜑(𝜉+2𝜋𝑘)∶𝑘∈ℤ^𝑑} is in ℓ²(ℤ^𝑑), for a.e. 𝜉∈ℝ^𝑑, by Lemma 3.5 and Theorem 3.6, we conclude that dim𝔽_𝜓(𝜉) is at most 1, for a.e. 𝜉∈ℝ^𝑑.
Now, we will prove the sufficiency. In other words, we have to prove that 𝜓 is an MRA M-TFW, assuming that 𝜓 is an M-TFW such that dim𝔽_𝜓(𝜉) is either 0 or 1, for a.e. 𝜉∈ℝ^𝑑. We noticed that 𝔽_𝜓(𝜉) is always defined for a M-TFW 𝜓 and is a closed subspace of ℓ²(ℤ^𝑑). Theorem 4.1 implies that, for 𝑗⩾1,Ψ_𝑗(𝜉)∈ℓ²(ℤ^𝑑), for a.e. 𝜉∈ℝ^𝑑.
Therefore, we need to prove that there exist a generalized filter 𝐻 and a corresponding pseudoscaling function 𝜑 such that (2.12) is satisfied, that is, a.e. 𝜉∈ℝ^𝑑𝜓(𝐷𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝜌(𝐷𝜉)𝐻𝜉+2𝜋𝐷⁻¹𝜀₀𝜑(𝜉). (4.17)
Thus, we begin to prove that it is enough to show that there exist a generalized filter 𝐻₀, a corresponding pseudoscaling function 𝜑₀, and a 2𝜋-ℤ^𝑑 periodic, unimodular function 𝜌 such that 𝜓(𝐷𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝜌(𝐷𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀𝜑₀(𝜉). (4.18)
Let, 𝒵∶=𝜉∈ℝ^𝑑∶Dim_𝜓(𝜉)=0 (4.19) and, for 𝑗∈ℕ, we define the set 𝒫_𝑗 to be 𝒫_𝑗∶=𝜉∈ℝ^𝑑∶Ψ_𝑗(𝜉)_{ℓ2(ℤ𝑑)}≠0,Ψ_ℓ(𝜉)₂=0for1⩽ℓ⩽𝑗−1. (4.20)
It is clearly that 𝒵 is the set where all vectors Ψ_𝑗(𝜉),𝑗⩾1, are zero (or, equivalently, where dim𝔽_𝜓(𝜉)=0). Moreover, the set 𝒵 and the 𝒫_𝑗,𝑗⩾1, are 2𝜋-ℤ^𝑑 periodic, measurable and they form a partition of ℝ^𝑑.
We define 𝜑₀ as follows: 𝜑₀(𝜉)=0,𝜉∈𝒵,Dim_𝜓(𝜉)∑_{𝑘∈ℤ𝑑}𝜓𝐷^𝑗(𝜉+2𝜋𝑘)²⋅𝜓𝐷^𝑗𝜉,𝜉∈𝒫_𝑗. (4.21)
Observe that (4.21) makes sense and defines a measurable function 𝜑₀∶ℝ^𝑑→ℂ. Furthermore, the function clearly satisfies _{𝑘∈ℤ𝑑}𝜑₀(𝜉+2𝜋𝑘)²=Dim_𝜓(𝜉),a.e. (4.22) Using (4.22), we obtain that ‖𝜑₀‖²₂=∫_𝕋𝑑Dim_𝜓(𝜉)𝑑𝜉. It follows that 𝜑₀∈𝐿²(ℝ^𝑑). Thus, 𝜑₀∈𝐿²(ℝ^𝑑).
Next, we divide the argument into six steps (six lemmas).
Lemma 4.6. For a.e. 𝜉∈ℝ^𝑑, one has 𝜑₀(𝜉)²=^∞_𝑗=1𝜓𝐷^𝑗𝜉². (4.23)Proof. If 𝜉∈𝒵, then, (4.23) is trivially true. We only need to consider 𝜉∈𝒫_ℓ,ℓ∈ℕ. Thus, Ψ_ℓ(𝜉)≠0 and, by assumption, dim𝔽_𝜓(𝜉)=1. This implies that for every 𝑗⩾1 there exists a 2𝜋-ℤ^𝑑 periodic, measurable function 𝜆^ℓ_𝑗∶𝒫_ℓ→ℂ, such that, for a.e. 𝜉∈𝒫_ℓ, Ψ_𝑗(𝜉)=𝜆^ℓ_𝑗(𝜉)Ψ_𝑙(𝜉). (4.24) Coordinatewise, this means that for every 𝑘∈ℤ^𝑑, a.e. 𝜉∈𝒫_ℓ𝜓𝐷^𝑗(𝜉+2𝜋𝑘)=𝜆^ℓ_𝑗(𝜉)𝜓𝐷^ℓ(𝜉+2𝜋𝑘). (4.25)
It follows that the right hand side of (4.23) satisfies ^∞_𝑗=1𝜓𝐷^𝑗𝜉²=^∞_𝑗=1𝜆^ℓ_𝑗(𝜉)²⋅𝜓𝐷^ℓ𝜉² (4.26) for a.e. 𝜉∈𝒫_ℓ. Keeping in mind formula (4.21) for 𝜉∈𝒫_ℓ, (4.26) implies that ^∞_𝑗=1𝜆^ℓ_𝑗(𝜉)²=∑^∞_𝑗=1𝜓𝐷^𝑗𝜉²𝜓𝐷^ℓ𝜉², (4.27) for a.e. 𝜉∈𝒫_ℓ. On the other hand, we obtain Dim_𝜓(𝜉)=^∞_{𝑗=1𝑘∈ℤ𝑑}𝜓𝐷^𝑗(𝜉+2𝜋𝑘)²=^∞_{𝑗=1𝑘∈ℤ𝑑}𝜆^ℓ_𝑗(𝜉)²𝜓𝐷^ℓ(𝜉+2𝜋𝑘)²=^∞_𝑗=1𝜆^ℓ_𝑗(𝜉)²_{𝑘∈ℤ𝑑}𝜓𝐷^ℓ(𝜉+2𝜋𝑘)². (4.28)
Using (4.26), (4.27), and the above equation, we obtain ^∞_𝑗=1𝜓𝐷^𝑗𝜉²=Dim_𝜓(𝜉)∑_{𝑘∈ℤ𝑑}𝜓𝐷^ℓ(𝜉+2𝜋𝑘)²𝜓𝐷^𝑙𝜉². (4.29) Combine (4.29) with definition of 𝜑₀, for a.e. 𝜉∈𝒫_ℓ, we obtain 𝜑₀(𝜉)²=^∞_𝑗=1𝜓𝐷^𝑗𝜉². (4.30) This completes the proof of Lemma 4.6 since 𝒵 and 𝒫_𝑗,𝑗⩾1 cover all of ℝ^𝑑.Lemma 4.7. There exists a 2𝜋-ℤ^𝑑 periodic, measurable function 𝐻₁∶𝒵^𝒞→ℂ such that 𝐻₁(𝜉)⩽1,a.e.𝜉∈𝒵^𝒞,𝜓(𝐷𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝐻₁(𝜉)𝜑₀(𝜉),a.e.𝜉∈𝒵^𝒞. (4.31)(4.32)Proof. Notice that the 2𝜋-ℤ^𝑑 periodicity of 𝐻₁ is consistent with the 2𝜋-ℤ^𝑑 periodicity of the measurable set 𝒵^𝒞=_ℓ∈ℕ𝒫_ℓ. (4.33)
Define 𝐻₁ as the following formula 𝐻₁(𝜉)=0,𝜉∈𝒫_ℓ𝑒,ℓ⩾2,_{𝑖𝜀0⋅𝜉}∑_{𝑘∈ℤ𝑑}𝜓(𝐷(𝜉+2𝜋𝑘))²Dim_𝜓(𝜉),𝜉∈𝒫₁. (4.34)
Since all the functions and sets involved in (4.34) are measurable and 2𝜋-ℤ^𝑑 periodic, it is obvious that (4.34) defines a 2𝜋-ℤ^𝑑 periodic, measurable function 𝐻₁∶𝒵^𝒞→ℂ. Furthermore, _{𝑘∈ℤ𝑑}𝜓(𝐷(𝜉+2𝜋𝑘))²⩽Dim_𝜓(𝜉) (4.35) implies that (4.31) is valid. For 𝜉∈𝒫₁, (4.32) is the direct consequence of (4.21) and (4.34). For 𝜉∈𝒫_ℓ,ℓ⩾2,Ψ₁(𝜉)=0. In particular, 𝜓(𝐷𝜉)=0; thus (4.32) is trivially satisfied. The proof of Lemma 4.7 is completed.Lemma 4.8. There exists a 2𝜋-ℤ^𝑑 periodic, measurable function 𝐻₀∶𝒵^𝒞→ℂ such that 𝐻₀(𝜉)⩽1,a.e.𝜉∈𝒵^𝒞,𝜑₀(𝐷𝜉)=𝐻₀(𝜉)𝜑₀(𝜉),a.e.𝜉∈𝒵^𝒞. (4.36)(4.37)Proof. Again, by (4.33), the 2𝜋-ℤ^𝑑 periodicity will be clear from the argument, and we need only to prove (4.36) and (4.37) on ⋃_ℓ∈ℕ𝒫_ℓ.
First, we consider 𝒫_ℓ,ℓ⩾2. For a.e. 𝜉∈𝒫_ℓ, we obtain Ψ_ℓ−1(𝐷𝜉)²_ℓ2ℤ𝑑=_{𝑘∈ℤ𝑑}𝜓𝐷^ℓ−1(𝐷𝜉+2𝜋𝑘)²=_{𝑘∈ℤ𝑑}𝜓𝐷^ℓ𝜉+2𝜋𝐷⁻¹𝑘²⩾_{𝑘∈ℤ𝑑}𝜓𝐷^ℓ(𝜉+2𝜋𝑘)²=Ψ_ℓ(𝜉)²_ℓ2ℤ𝑑>0. (4.38) This and the definition (4.20) of 𝒫_ℓ imply that for a.e. 𝜉∈𝒫_ℓ, there exists 𝑘=𝑘(𝜉)∈{1,2,…,ℓ−1} such that 𝐷𝜉∈𝒫_𝑘. It is easy to check, using (4.20), that 𝜉→𝑘(𝜉) is 2𝜋-ℤ^𝑑 periodic and measurable (defined on 𝒫_ℓ). Since, according to our assumption, dim𝔽_𝜓(𝐷𝜉)=1 for a.e. 𝜉∈𝒫_ℓ, there exists a 2𝜋-ℤ^𝑑 periodic, measurable function 𝜆∶𝒫_ℓ→ℂ such that, for a.e. 𝜉∈𝒫_ℓ, Ψ_ℓ−1(𝐷𝜉)=𝜆(𝜉)Ψ_𝑘(𝜉)(𝐷𝜉). (4.39) Notice that 𝜆(𝜉)≠0, since otherwise we would have 0=Ψ_ℓ−1(𝐷𝜉)²_{ℓ2(ℤ𝑑)}⩾Ψ_ℓ(𝜉)²_{ℓ2(ℤ𝑑)}⩾0 (4.40) which would imply Ψ_ℓ(𝜉)=0; this is impossible for 𝜉∈𝒫_ℓ. Hence, (4.39) implies that, for a.e. 𝜉∈𝒫_ℓ, 𝜑₀(𝐷𝜉)=Dim_𝜓(𝐷𝜉)∑_{𝑗∈ℤ𝑑}𝜓𝐷^𝑘𝐷𝜉+2𝜋𝑗²⋅𝜓𝐷^𝑘=⋅𝐷𝜉Dim_𝜓(𝐷𝜉)∑_{𝑗∈ℤ𝑑}𝜓𝐷^𝑘𝐷𝜉+2𝜋𝑗²⋅1𝜆(𝜉)𝜓𝐷^ℓ−1⋅𝐷𝜉. (4.41) Notice that (4.41) shows that there exists a 2𝜋-ℤ^𝑑 periodic, measurable function 𝒜∶𝒫_ℓ→ℂ such that, for a.e. 𝜉∈𝒫_ℓ, 𝜑₀(𝐷𝜉)=𝒜(𝜉)𝜓𝐷^ℓ𝜉. (4.42) At the same time it follows directly from (4.21), that there exists a 2𝜋-ℤ^𝑑 periodic, measurable function ℬ∶𝒫_ℓ→ℂ such that, for a.e. 𝜉∈𝒫_ℓ,ℬ(𝜉)≠0, and 𝜑₀(𝜉)=ℬ(𝜉)𝜓𝐷^ℓ𝜉. (4.43)
We define 𝐻₀(𝜉) on 𝒫_ℓ by 𝐻₀(𝜉)𝒜(𝜉)∶=ℬ(𝜉). (4.44) By (4.42) and (4.43), it is clear that 𝐻₀(𝜉) is a 2𝜋-ℤ^𝑑 periodic, measurable function, and satisfies (4.37) on 𝒫_ℓ. Therefore, it remains to define 𝐻₀ on 𝒫₁. Observe that for 𝜉∈𝒫₁, either 𝐷𝜉∈𝒵, or, otherwise, there is a 2𝜋-ℤ^𝑑 periodic, measurable function 𝜉→ℓ(𝐷𝜉)∈ℕ, such that 𝐷𝜉∈𝒫_{ℓ(𝐷𝜉)}. If 𝐷𝜉∈𝒵, then we define 𝐻₀(𝜉) to be 0. Otherwise, we define, by (4.21), that there exists a 2𝜋-ℤ^𝑑 periodic, measurable function 𝒜 such that 𝜑₀(𝐷𝜉)=𝒜(𝐷𝜉)𝜓𝐷^{ℓ(𝐷𝜉)}⋅𝐷𝜉=𝒜(𝐷𝜉)𝜓𝐷^{ℓ(𝐷𝜉)+1}𝜉. (4.45) Finally, we consider 𝜉∈𝒫₁, we know that there exists a 2𝜋-ℤ^𝑑 periodic, measurable function 𝜆, such that Ψ_{ℓ(𝐷𝜉)+1}(𝜉)=𝜆(𝜉)⋅Ψ₁(𝜉), (4.46) and a 2𝜋-ℤ^𝑑 periodic, measurable function ℬ≠0, such that 𝜑₀(𝜉)=ℬ(𝜉)⋅𝜓(𝐷𝜉). (4.47) Hence, for 𝜉∈𝒫₁ and 𝐷𝜉∉𝒵, we define 𝐻₀ by 𝐻₀(𝜉)𝒜()𝜆(𝜉)∶=𝐷𝜉ℬ(𝜉). (4.48) It is now clear that 𝐻₀ is a 2𝜋-ℤ^𝑑 periodic, measurable function on 𝒫₁ and satisfies (4.37) on 𝒫₁. It follows that we have a 2𝜋-ℤ^𝑑 periodic, measurable function 𝐻₀∶𝒵^𝒞→ℂ, such that (4.37) is satisfied.
Let us prove (4.36). For 𝜉∈𝒵^𝒞, we have, by (4.22), that there exists 𝑘∈ℤ^𝑑 such that 𝜑₀(𝜉+2𝜋𝑘)≠0. Since 𝐻₀ is a 2𝜋-ℤ^𝑑 periodic we obtain, by (4.37), 𝜑₀(𝐷(𝜉+2𝜋𝑘))=𝐻₀(𝜉+2𝜋𝑘)𝜑₀(𝜉+2𝜋𝑘)=𝐻₀(𝜉)𝜑₀(𝜉+2𝜋𝑘). (4.49) Hence 𝐻₀(𝜉=𝜑)⁰(𝐷(𝜉+2𝜋𝑘))𝜑₀(.𝜉+2𝜋𝑘) (4.50) However, Lemma 4.6 implies that for a.e. 𝜇∈ℝ^𝑑,|𝜑₀(𝐷𝜇)|²⩽|𝜑₀(𝜇)|². Thus, |𝐻₀(𝜉)|⩽1, and this completes the proof of Lemma 4.8.
Now, let us extend the definition of 𝐻₀ and 𝐻₁ to 𝒵, as well. Since 𝐻₀ is already defined on 𝒵^𝒞, the following definition of 𝐻₁ on 𝒵 makes sense: 𝐻₁(𝜉)=1√2,𝜉∈𝒵,𝜉+2𝜋𝐷⁻¹𝜀₀𝐻∈𝒵,₀𝜉+2𝜋𝐷⁻¹𝜀₀,𝜉∈𝒵,𝜉+2𝜋𝐷⁻¹𝜀₀∉𝒵. (4.51) Using Lemmas 4.7 and 4.8 and (4.51), we obtain that 𝐻₁∶ℝ^𝑑→ℂ is a 2𝜋-ℤ^𝑑 periodic, measurable function, such that 𝐻₁(𝜉)⩽1,a.e.𝜉∈ℝ^𝑑. (4.52) Moreover, we conclude that (4.32) is now satisfied for a.e. 𝜉∈ℝ^𝑑. By Lemma 4.7, we need to check (4.32) only on 𝒵. But, for 𝜉∈𝒵, we have that 𝜑₀(𝜉)=0, by (4.21), and 𝜓(𝐷𝜉)=0, since Dim_𝜓(𝜉)=0. Hence, on 𝒵 (4.32) is satisfied irrespective of the value on 𝐻₁(𝜉). We have established, that 𝜓(𝐷𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝐻₁(𝜉)𝜑₀(𝜉),a.e.𝜉∈𝒵^𝒞. (4.53)
Let us turn our attention to 𝐻₀. Now that 𝐻₁ is defined on all of ℝ^𝑑, we define 𝐻₀ on 𝒵 by 𝐻₀(𝜉)∶=𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀,a.e.𝜉∈ℝ^𝑑. (4.54) Obviously, 𝐻₀∶ℝ→ℂ is a 2𝜋-ℤ^𝑑 periodic, measurable function, such that 𝐻₀(𝜉)⩽1,a.e.𝜉∈ℝ^𝑑. (4.55) Again, we obtain that such 𝐻₀ satisfies (4.37) on ℝ^𝑑. And, again, because of Lemma 4.8, it is enough to check (4.37) on 𝒵. Since 𝜑₀(𝜉)=0, for 𝜉∈𝒵, it is enough to show that 𝜑₀(𝐷𝜉)=0, for 𝜉∈𝒵. But this is an immediate consequence of Lemma 4.6. We conclude that, for a.e. 𝜉∈ℝ, 𝜑₀(𝐷𝜉)=𝐻₀(𝜉)𝜑₀(𝜉). (4.56)
The following lemma connects 𝐻₀ and 𝐻₁.
Lemma 4.9. For a.e. 𝜉∈ℝ^𝑑𝐻₀(𝜉)²+𝐻₁(𝜉)²=1. (4.57)Proof. (1) If 𝜉∈𝒵 and 𝜉+2𝜋𝐷⁻¹𝜀₀∈𝒵, then, by (4.51) and (4.37), 𝐻₁(𝜉)=1/√2, and, since 𝜉+4𝜋𝐷⁻¹𝜀₀∈𝒵, so 𝐻₀(𝜉)=𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀=1√2. (4.58) Obviously, (4.57) is satisfied.
(2) If 𝜉∈𝒵, and 𝜉+2𝜋𝐷⁻¹𝜀₀∉𝒵, then by (4.51) and (4.54), 𝐻₀(𝜉)²+𝐻₁(𝜉)²=𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀²+𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀². (4.59) Hence, (4.57) is going to be satisfied if we can prove it on 𝒵^𝒞. We present this argument.
Lemma 4.6 implies that, for a.e. 𝜉∈ℝ^𝑑𝜑₀(𝜉)²=𝜓(𝐷𝜉)²+𝜑₀(𝐷𝜉)². (4.60) From (4.53) and (4.56) we obtain that, for a.e. 𝜉∈ℝ^𝑑𝜑₀(𝜉)²=𝐻₀(𝜉)²+𝐻₁(𝜉)²⋅𝜑₀(𝜉)². (4.61) Using the fact that 𝐻₀(𝜉) and 𝐻₁(𝜉) are 2𝜋-ℤ^𝑑 periodic, we can periodize (4.61). This periodization and (4.22) provides us with Dim_𝜓(𝜉)=𝐻₀(𝜉)²+𝐻₁(𝜉)²⋅Dim_𝜓(𝜉). (4.62) Recall that on 𝒵^𝒞,Dim_𝜓(𝜉)≠0; thus (4.62) provides us with (4.57) on 𝒵^𝒞.
The following lemma establishes that 𝐻₀(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 𝜚_𝑞(𝜉)=0,𝑞∈𝕊.
Lemma 4.10. For a.e. 𝜉∈ℝ^𝑑𝐻₀(𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀=𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀𝐻₁(𝜉),𝐻₀(𝜉)=𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀𝐻₀,(𝜉)²+𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀²=𝐻₁(𝜉)²+𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀²=1. (4.63)(4.64)(4.65)Proof. We only need to prove (4.63). Indeed, (4.63) and Lemma 4.9 imply that 𝐻₀(𝜉)²=𝐻₀(𝜉)²⋅𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀²+𝐻₀(𝜉)²⋅𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀²=𝐻₁(𝜉)²⋅𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀²+𝐻₀(𝜉)²⋅𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀²=𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀². (4.66) 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) If 𝜉∈𝒵 and 𝜉+2𝜋𝐷⁻¹𝜀₀∉𝒵. By (4.51) and (4.54), we have 𝐻₀(𝜉)=𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀,𝐻₁(𝜉)=𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀ (4.67) which clearly establishes (4.63).
(2) If 𝜉∈𝒵 and 𝜉+2𝜋𝐷⁻¹𝜀₀∈𝒵. By (4.51) and (4.54), we have 𝐻₀(𝜉)=𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀,𝐻₁(𝜉)=𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀=1√2. (4.68) Again, (4.63) follows.
(3) If 𝜉∉𝒵 and 𝜉+2𝜋𝐷⁻¹𝜀₀∈𝒵, then 𝜉+4𝜋𝐷⁻¹𝜀₀∉𝒵, thus, by (4.51) and (4.54), we have, 𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀=𝐻₀𝜉+4𝜋𝐷⁻¹𝜀₀=𝐻₀(𝜉),𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀=𝐻₁(𝜉). (4.69) Obviously, (4.63) follows.
Finally, let us consider 𝜉∈[𝒵⋃(𝒵+2𝜋𝐷⁻¹𝜀₀)]^𝒞. Using (4.53), (4.56), and (4.57), we obtain, for 𝑗⩾1,𝑞∈𝕊, and a.e. 𝜁∈ℝ^𝑑, 𝜓𝐷^𝑗𝜁𝜓𝐷^𝑗𝜁+2𝜋𝑞=𝜑₀𝐷^𝑗−1𝜁𝜑₀𝐷^𝑗−1𝜁+2𝜋𝑞−𝜑₀𝐷^𝑗𝜁𝜑₀𝐷^𝑗𝜁+2𝜋𝑞. (4.70)
By Proposition 2.2, for 𝑞∈𝕊, and a.e. 𝜁∈ℝ^𝑑, we obtain 0=𝜚_𝑞(𝜁)=^+∞_𝑗=0𝜓𝐷^𝑗𝜁𝜓𝐷^𝑗=𝜓(𝜁)𝜁+2𝜋𝑞𝜓𝜁+2𝜋𝑞+𝜑₀(𝜁)𝜑₀=𝐻𝜁+2𝜋𝑞₀𝐷⁻¹𝜁𝐻₀𝐷⁻¹𝜁+2𝜋𝐷⁻¹𝜀₀−𝐻₁𝐷⁻¹𝜁𝐻₁𝐷⁻¹𝜁+2𝜋𝐷⁻¹𝜀₀⋅𝜑₀𝐷⁻¹𝜁𝜑₀𝐷⁻¹𝜁+2𝜋𝐷⁻¹𝑞. (4.71)
Now suppose 𝜉∉𝒵 and 𝜉+2𝜋𝐷⁻¹𝜀₀∉𝒵. By (4.22) there exist 𝑘,ℓ∈ℤ^𝑑 such that 𝜑₀(𝜉+2𝜋𝑘)≠0,𝜑₀𝜉+2𝜋ℓ+2𝜋𝐷⁻¹𝜀₀≠0. (4.72)
Let 𝐷⁻¹𝜁=𝜉+2𝜋𝑘,𝑞=𝐷(ℓ−𝑘)+𝜀₀ (4.73) and we conclude 0=𝐻₀(𝜉+2𝜋𝑘)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀−𝐻₁(𝜉+2𝜋𝑘)𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀=𝐻₀(𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀−𝐻₁(𝜉)𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀. (4.74) This proves the case 𝜉∈[𝒵⋃(𝒵+2𝜋𝐷⁻¹𝜀₀)]^𝒞 and completes the proof of Lemma 4.10.
We have established so far that 𝐻₀ and 𝐻₁ are generalized filters, and that 𝜑₀ is a pseudoscaling function and that 𝐻₀ is its corresponding filter, and that 𝜓 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 Ω=𝜉∈ℝ^𝑑∶𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀=0. (4.75) It is clear that both Ω and Ω^𝒞 are measurable and 2𝜋-ℤ^𝑑 periodic. Furthermore, by (4.65), we have 𝜉∈Ω⟶𝜉+2𝜋𝐷⁻¹𝜀₀∈Ω^𝒞,a.e. (4.76)
For 𝜉∈Ω^𝒞, define a function 𝒮 as follows: 𝒮(𝜉)𝐻∶=¹(𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀. (4.77) By (4.64), it follows that 𝒮∶Ω^𝒞→ℂ is a 2𝜋-ℤ^𝑑 periodic, unimodular function. For 𝜉∈Ω, we use (4.76) to conclude that 𝜉+2𝜋𝐷⁻¹𝜀₀∈Ω^𝒞, and we define 𝒮 by 𝒮(𝜉)∶=𝒮𝜉+2𝜋𝐷⁻¹𝜀₀. (4.78) Hence, 𝒮(𝜉) is an unimodular, 2𝜋-ℤ^𝑑 periodic function on ℝ^𝑑, which satisfies 𝐻₁(𝜉)=𝒮(𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀,a.e. (4.79).
Lemma 4.11. 𝒮(𝜉) is a 𝜋-ℤ^𝑑 periodic function.Proof. Firstly, if 𝜉∈ℝ^𝑑 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-ℤ^𝑑 periodic 𝒮(𝜉) provide us with the conclusion that 𝒮(𝜉)=𝒮(𝜉+2𝜋𝐷⁻¹𝜀₀).
Secondly, if 𝜉∈ℝ^𝑑 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 𝐻₀(𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀=𝐻₁(𝜉)𝐻₁𝜉+2𝜋𝐷⁻¹𝜀₀=𝒮𝜉+2𝜋𝐷⁻¹𝜀₀𝐻₀(𝜉)𝒮(𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀=𝒮𝜉+2𝜋𝐷⁻¹𝜀₀𝒮(𝜉)𝐻₀(𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀. (4.80) Therefore, 𝒮(𝜉)=𝒮𝜉+2𝜋𝐷⁻¹𝜀₀. (4.81) The proof of Lemma 4.11 is completed.
For 𝜉∈ℝ^𝑑, let 𝜌(𝜉)=𝒮𝐷⁻¹𝜉. (4.82) Since 𝒮(𝜉) is a 𝜋-ℤ^𝑑 periodic, unimodular function, so 𝜌(𝜉) is a 2𝜋-ℤ^𝑑 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 𝜓(𝐷𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝐻₁(𝜉)𝜑₀(𝜉)=𝑒^{𝑖𝜀0⋅𝜉}𝜌(𝐷𝜉)𝐻₀𝜉+2𝜋𝐷⁻¹𝜀₀𝜑₀(𝜉). (4.83) The proof of Theorem 4.5 is completed.

Theorem 4.12. Suppose that 𝜓 is a M-TFW. Then, 𝜓 is a semi-orthogonal MRA M-TFW if and only if Dim_𝜓(𝜉)∈{0,1},fora.e.𝜉∈ℝ^𝑑. (4.84)

Proof. Suppose that 𝜓 is a semi-orthogonal MRA M-TFW, using Theorems 4.1 and 4.5, then (4.84) is valid.
Suppose that 𝜓 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), Dim_𝜓(𝜉)∈{0,1}, for a.e. 𝜉∈ℝ^𝑑. 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.

Acknowledgments

The authors would like to express their gratitude to the referee for his (or her) valuable comments and suggestions. This work was supported by the National Natural Science Foundation of China (Grant no. 10961001), the Key Project of Chinese Ministry of Eduction (Grant no. 209152), and the Science Foundation of Ningxia higher School (no. 2010JY003).