Orthogonal Multiwavelet Frames in 𝐿²(𝑅^𝑑)

Abstract

We characterize the orthogonal frames and orthogonal multiwavelet frames in 𝐿²(𝑅^𝑑) with matrix dilations of the form (𝐷𝑓)(𝑥)=√|det𝐴|𝑓(𝐴𝑥), where 𝐴 is an arbitrary expanding 𝑑×𝑑 matrix with integer coefficients. Firstly, through two arbitrarily multiwavelet frames, we give a simple construction of a pair of orthogonal multiwavelet frames. Then, by using the unitary extension principle, we present an algorithm for the construction of arbitrarily many orthogonal multiwavelet tight frames. Finally, we give a general construction algorithm for orthogonal multiwavelet tight frames from a scaling function.

1. Introduction

Wavelets are mathematical functions that take account into the resolutions and the frequencies simultaneously [1–4]. Moreover, wavelets could cut up data into different frequency components such that people can study each component with a resolution matched to its scale.

The classical MRA scaler wavelets are probably the most important class of orthonormal wavelets. However, the scalar wavelets cannot have the orthogonality, compact support, and symmetry at the same time (except the Haar wavelet). It is a disadvantage for signal processing. Multiwavelets have attracted much attention in the research community, since multiwavelets have more desired properties than any scalar wavelet function, such as orthogonality, short compact support, symmetry, and high approximation order [5–7]. It is natural, therefore, to develop the multiwavelets theory that can produce systems having these properties.

Although many compression applications of wavelets use wavelet or multiwavelet bases, the redundant representation offered by wavelet frames has already been put to good use for signal denoising and image compression. In fact, the concept of frame was introduced a long time ago [8] and has received much attention recently due to the development and study of wavelet theory [9, 10]. In particular, inspired by these and other applications, many people are interested in some types of frames, such as tight wavelet frames, dual wavelet frames, and orthogonal frames [11–19].

In [16], Weber proposed orthogonal wavelet frames, which are useful in multiple access communication systems and superframes. Later in [17], authors discussed a pair of orthogonal frames to be orthogonal in a shift-invariant space. In [18], authors presented sufficient conditions for the construction of orthogonal MRA wavelet frames in 𝐿²(𝑅). This led them to a vector-valued discrete wavelet transform. But all these results just base on 2 dilation wavelet transform. In this paper, we present the construction of orthogonal multiwavelet frames in 𝐿²(𝑅^𝑑) with matrix dilation, where the basic ingredients consists of two fixed multiwavelet basis and a paraunitary matrix of an appropriate size. Furthermore, by using the unitary extension principle, we present an algorithm for the construction of orthogonal multiwavelet tight frames from two suitable functions and give a general construction algorithm for orthogonal multiwavelet tight frames from a scaling function. These constructions lead to filter banks in 𝑙²(𝑍^𝑑) with similar orthogonality relations.

Let us now describe the organization of the material that follows. Section 2 contains some definitions in this paper. Also, we review some relative notations. In Section 3, we describe the construction of orthogonal multiwavelet frames and present different algorithms for the construction of orthogonal multiwavelet tight frames in 𝐿²(𝑅^𝑑) with matrix dilation.

2. Preliminaries

Let us now establish some basic notations.

We denote by 𝑇^𝑑 the d-dimensional torus. By 𝐿²(𝑇^𝑑), we denote the space of all 𝑍^𝑑-periodic functions 𝑓 (i.e., 𝑓 is 1-periodic in each variable) such that ∫_𝑇𝑑|𝑓(𝑥)|²𝑑𝑥<+∞. The subsets of 𝑅^𝑑 invariant under 𝑍^𝑑 translations and the subsets of 𝑇^𝑑 are often identified.

We use the Fourier transform in the form𝑓(𝜔)=_𝑅𝑑𝑓(𝑥)𝑒^{−2𝜋𝑖⟨𝑥,𝜔⟩}𝑑𝑥,(2.1) where ⟨⋅,⋅⟩ denotes the standard inner product in 𝑅^𝑑. The Fourier inverse transform is defined by𝑓(𝑥)=𝑓(𝜔)=_𝑅𝑑𝑓(𝜔)𝑒^{2𝜋𝑖⟨𝑥,𝜔⟩}𝑑𝜔.(2.2)

Let 𝐸_𝑑 denote the set of all expanding 𝑑×𝑑 matrices 𝐴 with integer coefficients. The expanding matrices mean that all eigenvalues have magnitude greater than 1. For 𝐴∈𝐸_𝑑, we denote by 𝐵 the transpose of 𝐴. It is obvious that 𝐵∈𝐸_𝑑.

A collection of elements {𝜙_𝑗∶𝑗∈𝐽} in a Hilbert space 𝐻 is called a frame if there exist constants 𝑎 and 𝑏, 0<𝑎≤𝑏<∞, such that𝑎‖𝑓‖²≤_𝑗∈𝐽||𝑓,𝜙_𝑗||²≤𝑏‖𝑓‖²,forall𝑓∈𝐻.(2.3) If {𝜙_𝑗∶𝑗∈𝐽} satisfies the second inequality, then {𝜙_𝑗∶𝑗∈𝐽} is called a Bessel sequence. Let 𝑎₀ the supremum of all such numbers 𝑎 and 𝑏₀ the infimum of all such numbers 𝑏, then 𝑎₀ and 𝑏₀ are called the frame bounds of the frame {𝜙_𝑗∶𝑗∈𝐽}. When 𝑎₀=𝑏₀, we say that the frame is tight. When 𝑎₀=𝑏₀=1, we say the frame is a Parseval frame.

In this paper, we will work with two families of unitary operators on 𝐿²(𝑅^𝑑). The first one consists of all translation operators 𝑇_𝑘∶𝐿²(𝑅^𝑑)→𝐿²(𝑅^𝑑),𝑘∈𝑍^𝑑, defined by (𝑇_𝑘𝑓)(𝑥)=𝑓(𝑥−𝑘). The second one consists of all integer powers of the dilation operator 𝐷_𝐴∶𝐿²(𝑅^𝑑)→𝐿²(𝑅^𝑑) defined by (𝐷𝑓)(𝑥)=√|𝐴|𝑓(𝐴𝑥) with 𝐴∈𝐸_𝑑.

Let us now fix an arbitrary matrix 𝐴∈𝐸_𝑑. For Ψ={𝜓₁,…,𝜓_𝑟}⊂𝐿²(𝑅^𝑑), we will consider the affine system 𝑋(Ψ) defined by𝑋(Ψ)=𝜓^{𝑙𝑗,𝑘}(𝑥)∣𝜓^{𝑙𝑗,𝑘}(𝑥)=||det𝐴||^𝑗/2𝜓^𝑙𝐴^𝑗𝑥−𝑘∶𝑗∈𝑍;𝑘∈𝑍^𝑑;𝑙=1,…,𝑟.(2.4)

Then, we define the multiwavelet frame, the multiwavelet tight frame, the multiwavelet tight frame, and the filter.

Definition 2.1. We say that 𝑋(Ψ)⊂𝐿²(𝑅^𝑑) is a multiwavelet frame if the system (2.4) is a frame for 𝐿²(𝑅^𝑑).

Definition 2.2. We say that 𝑋(Ψ)⊂𝐿²(𝑅^𝑑) is a multiwavelet tight frame if the system (2.4) is a tight frame for 𝐿²(𝑅^𝑑).

Definition 2.3. We say that 𝑋(Ψ)⊂𝐿²(𝑅^𝑑) is a multiwavelet tight frame if the system (2.4) is a Parseval frame for 𝐿²(𝑅^𝑑).

We turn to the concept of multiresolution analysis (MRA) in 𝐿²(𝑅^𝑑) which is a useful tool in our study.

Definition 2.4. Let {𝑉_𝑚}_𝑚∈𝑍 be a sequence of closed subspaces of 𝐿²(𝑅^𝑑) satisfying:(1) 𝑉_𝑗⊂𝑉_𝑗+1,(2)⋃_𝑗∈𝑍𝑉_𝑗=𝐿²(𝑅^𝑑), (3)⋂_𝑗∈𝑍𝑉_𝑗={0}, (4)𝑓(𝑥)∈𝑉_𝑗⇔𝑓(𝐴𝑥)∈𝑉_𝑗+1,𝑗∈𝑍, where 𝐴∈𝐸_𝑑,(5)There exists a function 𝜙(𝑥)∈𝑉₀ such that {𝜙(𝑥−𝑘)}_𝑘∈𝑍^𝑑 is a frame of 𝑉₀. Then, {𝑉_𝑗}_𝑗∈𝑍 is called an MRA and the function 𝜙 in (5) a scaling function.

There is a standard procedure for constructing multiwavelets from a given MRA(𝑉_𝑗). Firstly, one defines 𝑊_𝑗=𝑉_𝑗+1⊖𝑉_𝑗 for all 𝑗∈𝑍. As an easy consequence of conditions (1)–(4) from Definition 2.4, one obtains 𝐿²(𝑅^𝑑)=⊕_𝑗∈𝑍𝑊_𝑗 and 𝑊_𝑗+1=𝐷𝑊_𝑗, for all 𝑗∈𝑍. Suppose now that there exist functions Ψ⊂𝑊₀ such that the system 𝐸(Ψ)∶={𝜓(⋅−𝑘)∶𝑘∈𝑍^𝑑,𝜓∈Ψ} is a frame for 𝑊₀. Then, {𝐷^𝑗𝑇_𝑘𝜓∶𝑘∈𝑍^𝑑,𝜓∈Ψ} is a frame for 𝑊_𝑗, for all 𝑗∈𝑍, and, consequently, {𝐷^𝑗𝑇_𝑘𝜓∶𝑗∈𝑍,𝑘∈𝑍^𝑑,𝜓∈Ψ} is a frame for 𝐿²(𝑅^𝑑).

In the following, we will borrow some notations from [17, 18] which will be used in this paper.

Let 𝑋 be a (countable) Bessel system for a separable Hilbert space 𝐻 over the complex field 𝐶. The synthesis operator 𝑇_𝑋∶𝑙²(𝑋)→𝐻, which is well known to be bounded, is defined by 𝑇_𝑋𝑎∶=∑_ℎ∈𝑋𝑎_ℎℎ for 𝑎={𝑎_ℎ}_ℎ∈𝑋. The adjoint operator 𝑇^∗𝑋 of 𝑇_𝑋, called the analysis operator, is 𝑇^∗𝑋∶𝐻→𝑙²(𝑋);𝑇^∗𝑋𝑓∶={⟨𝑓,ℎ⟩}_ℎ∈𝑋. Recall that 𝑋 is a frame for 𝐻 if and only 𝑆_𝑋∶=𝑇_𝑋𝑇^∗𝑋∶𝐻→𝐻, the frame operator or dual Gramian, is bounded and has a bounded inverse [20, 21], and it is a tight frame (with frame bound 1) if and only if 𝑆_𝑋 is the identity operator. The system 𝑋 is a Riesz system (for span𝑋) if and only its Gramian 𝐺_𝑋∶=𝑇^∗𝑋𝑇_𝑋 is bounded and has a bounded inverse; it is an orthonormal system of 𝐻 if and only if 𝐺_𝑋 is the identity operator.

Definition 2.5. Let 𝑋 and 𝑌=𝑅𝑋, where 𝑅∶ℎ→𝑅ℎ is a bijection between 𝑋 and 𝑌, be two frame for 𝐻. We call 𝑋 and 𝑌 a dual frames for 𝐻 if 𝑇_𝑌𝑇^∗𝑋=𝐼, that is, ∑_ℎ∈𝑋⟨𝑓,ℎ⟩𝑅ℎ=𝑓 for all 𝑓∈𝐻.

Definition 2.6. Let 𝑋 and 𝑌=𝑅𝑋, where 𝑅∶ℎ→𝑅ℎ is a bijection between 𝑋 and 𝑌, be two frames for 𝐻. We call 𝑋 and 𝑌 a pair of orthogonal frames for 𝐻 if 𝑇_𝑌𝑇^∗𝑋=0, that is, ∑_ℎ∈𝑋⟨𝑓,ℎ⟩𝑅ℎ=0 for all 𝑓∈𝐻.

Definition 2.7. A closed subspace 𝑉⊂𝐿²(𝑅^𝑑) is shift invariant if forall𝑓∈𝑉 implies 𝑇_𝑘𝑓∈𝑉 for any 𝑘∈𝑍^𝑑.

We consider orthogonal frames in a shift-invariant subspace of 𝐿²(𝑅^𝑑). Let Φ be a countable subset of 𝐿²(𝑅^𝑑), and 𝐸(Φ)∶={𝜙(⋅−𝑘)∶𝑘∈𝑍^𝑑,𝜙∈Φ}. Define 𝑆(Φ)∶=span𝐸(Φ), the smallest closed subspace that contains 𝐸(Φ). Throughout the rest of this paper, we assume that 𝐸(Φ) is a Bessel sequence for 𝑆(Φ). This assumption settles most of the convergence issues. The space 𝑆(Φ) is called the shift-invariant space generated by Φ and Φ a generating set for 𝑆(Φ). Shift-invariant spaces have been studied extensively in the literature, for example, [22, 23].

For 𝜔∈𝑅^𝑑, we define the pre-Gramian by𝐽_Φ(𝜔)=𝜙(𝜔+𝛼)_{𝛼∈𝑍𝑑,𝜙∈Φ},(2.5) where 𝜙 is the Fourier transform of 𝜙. Note that the domain of the pre-Gramian matrix as an operator is 𝑙²(Φ) and its codomain is 𝑙²(𝑍^𝑑). The pre-Gramian can be regarded as the synthesis operator represented in Fourier domain as it was extensively studied in [22].

Let Φ and Ψ=𝑅Φ, where 𝑅 is a bijection satisfying 𝑅(𝜙(⋅−𝑘))=(𝑅𝜙)(⋅−𝑘), be countable subsets of 𝐿²(𝑅^𝑑). Suppose that 𝑆(Φ)=𝑆(Ψ) and that both 𝐸(Φ) and 𝐸(Ψ) are frames for 𝑆(Φ). Then, by definition, 𝐸(Φ) and 𝐸(Ψ) are a pair of orthogonal frames for 𝑆(Φ) if and only if for all 𝑓∈𝑆(Φ),𝑆𝑓∶=𝑇_𝐸(Ψ)𝑇^∗𝐸(Φ)=0.(2.6)

We define the mixed dual Gramian as 𝐺(𝜔)=𝐽_Ψ(𝜔)𝐽^∗Φ(𝜔) and Gramians as𝐺_Φ(𝜔)=𝐽^∗Φ(𝜔)𝐽_Φ(𝜔),𝐺_Ψ(𝜔)=𝐽^∗Ψ(𝜔)𝐽_Ψ(𝜔).(2.7) Then, it is proven in [24] that, for any 𝑓∈𝐿²(𝑅^𝑑),𝑆𝑓_|𝜔+𝛼=𝐺(𝜔)𝑓_|𝜔+𝛼,(2.8) where 𝑓_|𝜔+𝛼 is the column vector (𝑓(𝜔+𝛼))^{𝑇𝛼∈𝑍}_𝑑. By (2.8), one can prove easily that 𝑆𝑓=0 for all 𝑓∈𝐿²(𝑅^𝑑) if and only if 𝐺(𝜔)=0 for a.e. 𝜔∈𝑅^𝑑.

3. Orthogonal Multiwavelet Frames

In this section, we present a simple construction of a pair of orthogonal multiwavelet frames from two arbitrarily multiwavelet frames and get some interesting properties about the orthogonal multiwavelet frames. We also show different algorithms for the construction of arbitrarily many orthogonal multiwavelet tight frames.

Firstly, we give a lemma, which has been obtained by Weber in [16].

Lemma 3.1. Let Ψ₁∶={𝜓¹¹,𝜓¹²,…,𝜓^1𝑟} and Ψ₂∶={𝜓²¹,𝜓²²,…,𝜓^2𝑟}. Suppose that 𝑋(Ψ₁) and 𝑋(Ψ₂) are multiwavelet frames for 𝐿²(𝑅^𝑑). 𝑋(Ψ₁) and 𝑋(Ψ₂) are a pair of orthogonal frames for 𝐿²(𝑅^𝑑) if and only if the following two equations are satisfied a.e.: ^𝑟_𝑖=1_𝑗∈𝑍𝜓_1𝑖𝐵^𝑗𝜔𝜓_2𝑖𝐵^𝑗𝜔=0,𝑎.𝑒.,_𝑟_{𝑖=1+∞𝑗=0}𝜓_1𝑖𝐵^𝑗𝜔𝜓_2𝑖𝐵^𝑗(𝜔+𝑞)=0,𝑎.𝑒.,∀𝑘∈𝑍^𝑑,𝑞∈𝑍^𝑑⧵𝐵𝑍^𝑑.(3.1)

From Lemma 3.1, by Theorem 2.3 [17], we can construct a pair of orthogonal multiwavelet frames easily.

Theorem 3.2. Let Ψ¹∶={𝜓¹¹,𝜓¹²,…,𝜓^1𝑟}andΨ²∶={𝜓²¹,𝜓²²,…,𝜓^2𝑟}∈𝐿²(𝑅^𝑑) for some positive integer r. Suppose that 𝑋(Ψ¹) and 𝑋(Ψ²) are multiwavelet frames for 𝐿²(𝑅^𝑑). Let 𝑉∶=(𝑉₁;𝑉₂) be a 2𝑟×2𝑟 constant unitary matrix, where 𝑉₁ is the submatrix of the first r columns and 𝑉₂ the remaining r columns. Then, 𝑋(Ψ¹¹) and 𝑋(Ψ²²) are a pair of orthogonal multiwavelet frames for 𝐿²(𝑅^𝑑), where Ψ¹¹∶=𝑉₁Ψ¹ and Ψ²²∶=𝑉₂Ψ².

Proof. Assume that 𝑉 is a constant matrix such that Ψ¹¹∶=𝑉₁Ψ¹ and Ψ²²∶=𝑉₂Ψ². Then, one can directly calculate the dual Gramians of 𝑋^𝑞(Ψ¹¹) and 𝑋^𝑞(Ψ²²). It follows from the fact that the double sums in (3.1) are the entries of the dual Gramian of the affine systems [24].
Let 𝑉=(𝑣_𝑙𝑚)_{1≤𝑙,𝑚≤2𝑟}. For a fixed 𝑞∈𝑍^𝑑⧵𝐵𝑍^𝑑, 𝑖∈1,2, we have ^2𝑟_𝑙=1_𝑚≥0𝜓^𝑙𝑖𝑖(𝐵^𝑚𝜔)𝜓^𝑙𝑖𝑖(𝐵^𝑚(𝜔+𝑞))=^2𝑟_𝑙=1_𝑟𝑚≥0_𝑛=1𝑣_𝑙,𝑛𝜓^𝑖𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1=𝑣_𝑙,𝑛′𝜓^𝑖𝑛_′(𝐵^𝑚(𝜔+𝑞))_𝑟𝑚≥0_𝑛=1𝜓^𝑖𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1𝜓^𝑖𝑛_′(𝐵^𝑚(𝜔+𝑞))^2𝑟_𝑙=1𝑣_𝑙,𝑛𝑣_𝑙,𝑛=_𝑟𝑚≥0_𝑛=1𝜓^𝑖𝑛(𝐵^𝑚𝜔)𝜓^𝑖𝑛(𝐵^𝑚(𝜔+𝑞)),(3.2) where we used the fact that the double sums converge absolutely a.e., 𝑉^∗𝑉=𝐼_2𝑟, and that 𝑋(Ψ¹) and 𝑋(Ψ²) are frames for 𝐿²(𝑅^𝑑). Moreover, ^2𝑟_𝑙=1_𝑚∈𝑍𝜓^𝑙𝑖𝑖(𝐵^𝑚𝜔)𝜓^𝑙𝑖𝑖(𝐵^𝑚𝜔)=^2𝑟_𝑙=1_{𝑟𝑚∈𝑍}_𝑛=1𝑣_𝑙,𝑛𝜓^𝑖𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1=𝑣_𝑙,𝑛′𝜓^𝑖𝑛_′(𝐵^𝑚𝜔)_{𝑟𝑚∈𝑍}_𝑛=1𝜓^𝑖𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1𝜓^𝑖𝑛_′(𝐵^𝑚𝜔)^2𝑟_𝑙=1𝑣_𝑙,𝑛𝑣_𝑙,𝑛=_{𝑟𝑚∈𝑍}_𝑛=1𝜓^𝑖𝑛(𝐵^𝑚𝜔)𝜓^𝑖𝑛(𝐵^𝑚𝜔).(3.3) From the above results, by using the dual Gramian characterization of frames in [25, Corollary 5.7], then 𝑋(Ψ¹¹) and 𝑋(Ψ²²) are frames for 𝐿²(𝑅^𝑑).
We now show that the multiwavelet systems generated by Ψ¹¹ and Ψ²² are a pair of orthogonal frames for 𝐿²(𝑅^𝑑). We apply Lemma 3.1 to Ψ¹¹∶={𝜓¹¹¹,𝜓²¹¹,…,𝜓^112𝑟} and Ψ²²∶={𝜓¹²²,𝜓²²²,…,𝜓^222𝑟}. Let 𝑉=(𝑣_𝑙𝑚)_{1≤𝑙,𝑚≤2𝑟}. For all 𝑞∈𝑍^𝑑⧵𝐵𝑍^𝑑, we have ^2𝑟_𝑙=1_𝑚≥0𝜓_𝑙11(𝐵^𝑚𝜔)𝜓^𝑙22(𝐵^𝑚(𝜔+𝑞))=^2𝑟_𝑙=1_𝑟𝑚≥0_𝑛=1𝑣_𝑙,𝑛𝜓_1𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1=𝑣_{𝑙,𝑟+𝑛′}𝜓_2𝑛′(𝐵^𝑚(𝜔+𝑞))_𝑟𝑚≥0_𝑛=1𝜓_1𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1𝜓_2𝑛′(𝐵^𝑚(𝜔+𝑞))^2𝑟_𝑙=1𝑣_𝑙,𝑛𝑣_{𝑙,𝑟+𝑛}=_𝑟𝑚≥0_𝑛=1𝜓_1𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1=0,𝜓_2𝑛′(𝐵^𝑚(𝜔+𝑞))×0(3.4) where we used the orthogonality of the columns of 𝑉.
Moreover, ^2𝑟_𝑙=1_m∈𝑍𝜓_𝑙11(𝐵^𝑚𝜔)𝜓^𝑙22(𝐵^𝑚(𝜔))=^2𝑟_𝑙=1_{𝑟𝑚∈𝑍}_𝑛=1𝑣_𝑙,𝑛𝜓_1𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1=𝑣_{𝑙,𝑟+𝑛′}𝜓_2𝑛′(𝐵^𝑚(𝜔))_{𝑟𝑚∈𝑍}_𝑛=1𝜓_1𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1𝜓_2𝑛′(𝐵^𝑚(𝜔))^2𝑟_𝑙=1𝑣_𝑙,𝑚𝑣_{𝑙,𝑟+𝑛}=_{𝑟𝑚∈𝑍}_𝑛=1𝜓_1𝑛(𝐵^𝑚𝜔)^𝑟_𝑛_′=1=0,𝜓_2𝑛′(𝐵^𝑚(𝜔))×0(3.5) by Lemma 3.1, Ψ¹¹ and Ψ²² generate a pair of orthogonal frames.

The following results give some properties of the orthogonal frames.

Proposition 3.3. Suppose that 𝐸(𝜓_𝑖) and 𝐸(𝜓_𝑗) are a pair of orthogonal affine Bessel sequences in 𝐿²(𝑅^𝑑). If 𝛼∈𝐿²(𝑅^𝑑) is a 𝑍^𝑑-periodic function, then 𝐸(𝜓_𝑖) and 𝐸(𝛼𝜓_𝑗) are a pair of orthogonal affine Bessel sequences.

Proof. Suppose that 𝐸(𝜓_𝑖) and 𝐸(𝜓_𝑗) are a pair of orthogonal affine Bessel sequences in 𝐿²(𝑅^𝑑). Then, for all 𝑓∈𝐿²(𝑅^𝑑), we have 𝑆𝑓(𝑥)=_{𝑚∈𝑍𝑑}⟨𝑓(𝑥),𝜓_𝑖(𝑥+𝑚)⟩𝜓_𝑗(𝑥+𝑚)=0.(3.6) Let 𝜓^𝑗∶=𝛼𝜓_𝑗. Since 𝛼 is a 𝑍^𝑑-periodic function, then 𝐸(𝜓^𝑗) is an affine Bessel sequence for 𝐿²(𝑅^𝑑) from the fact that, for all 𝑓∈𝐿²(𝑅^𝑑), _𝑘_∈𝑍𝑑||𝑓(𝑥),𝛼(𝑥−𝑘)𝜓_𝑗(𝑥−𝑘)||²=_{𝑘∈𝑍𝑑}|||𝛼(𝑥)𝑓(𝑥),𝜓_𝑗(𝑥−𝑘)|||²≤𝐵‖‖𝛼𝑓‖‖²≤𝐵‖‖𝛼‖‖²‖𝑓‖²=𝐵^‖𝑓‖².(3.7)
Again by 𝛼 being a 𝑍^𝑑-periodic function, we have the following equation: 𝑆𝑓(𝑥)=_{𝑚∈𝑍𝑑}⟨𝑓(𝑥),𝜓_𝑖(𝑥+𝑚)⟩𝜓_𝑗(𝑥+𝑚)^==𝛼(𝑥_{𝑚∈𝑍𝑑})⟨𝑓(𝑥_{𝑚∈𝑍𝑑}),𝜓_𝑖(𝑥+𝑚)⟩𝜓_𝑗(𝑥+𝑚)𝛼(𝑥+𝑚)=0.⟨𝑓(𝑥),𝜓_𝑖(𝑥+𝑚)⟩𝜓_𝑗(𝑥+𝑚)(3.8) Hence, 𝐸(𝜓_𝑖) and 𝐸(𝛼𝜓_𝑗) are a pair of orthogonal affine Bessel sequences in 𝐿²(𝑅^𝑑).

Proposition 3.4. Suppose that 𝐸(𝜓_𝑖) and 𝐸(𝜓_𝑗) are a pair of orthogonal frames for 𝐻⊂𝐿²(𝑅^𝑑). Let 𝛼∈𝐿²(𝑅^𝑑) be a 𝑍^𝑑-periodic function. If 𝐸(𝛼𝜓_𝑗) is a frame for 𝐻, then 𝐸(𝜓_𝑖) and 𝐸(𝛼𝜓_𝑗) are a pair of orthogonal frames for 𝐻.

Proof. Similar to the proof in Proposition 3.3, we have the desired result.

Then, we recall a result from [26] that characterizes unitary extension principle (UEP) associated with more general matrix dilations in 𝐿²(𝑅^𝑑).

Lemma 3.5. Suppose Φ=(𝜙_𝑗)_𝑗∈𝐽 is a refinable vector with a mask Γ such that _𝑗∈𝐽lim_𝑗→∞‖‖𝜑_𝑗‖‖²=_𝑅𝑑‖‖Φ(𝜉)‖‖^2𝑙_2(𝐽)𝑑𝜉<∞,‖‖Φ𝐵^−𝑗𝜉‖‖=1,fora.e.𝜉∈𝑅^𝑑.(3.9) Suppose also that Ψ=(𝜓^𝑗)_𝑗∈𝐽, where 𝐽={1,…,𝑁} is finite, is given by Ψ(𝐵𝜉)=𝐻(𝜉)Φ(𝜉),(3.10) where 𝐻=(ℎ_𝑖,𝑗)_𝑖∈𝐽_,𝑖∈𝐽 is a 𝑍^𝑑-periodic, measurable matrix function satisfying Γ^∗(𝜉)Γ(𝜉+𝑑)+𝐻^∗(𝜉)𝐻(𝜉+𝑑)=Ω(𝜉)𝛿_0,𝑑,fora.e.𝜉,(3.11) and for any 𝑑∈Υ, where Υ consists of representatives of distinct cosets of 𝐵⁻¹𝑍^𝑑/𝑍^𝑑, then Ψ⊂𝐿²(𝑅^𝑑) is a multiwavelet tight frame.

We call 𝑚 a filter if 𝑚∈𝐿^∞([0,1)^𝑑). We shall call 𝑚 a low-pass filter if 𝑚(0)=1, and we shall call 𝑚 a high-pass filter if 𝑚(0)=0. Though not necessary, we will assume that every filter is continuous on a neighborhood of 0, so there will be no ambiguity in these definitions. Given a collection of filter 𝑀′={𝑚₀,𝑚₁,…,𝑚_𝑟}⊂𝐿^∞([0,1)^𝑑), let 𝑀′(𝜉) and 𝑀′(𝜉) be the matrices𝑀^(𝜉)=⎛⎜⎜⎜⎜⎜⎜⎝𝑚⁰𝑚(𝜉)𝑚₀(𝜉+𝛽)₁𝑚⋮(𝜉)𝑚₁(𝜉⋮_𝑟+𝛽)(𝜉)𝑚_𝑟(𝜉+𝛽)⎞⎟⎟⎟⎟⎟⎟⎠,𝑀^(𝜉)=⎛⎜⎜⎜⎜⎜⎜⎝𝑚¹𝑚(𝜉)𝑚₁(𝜉+𝛽)₂𝑚⋮(𝜉)𝑚₂(𝜉⋮_𝑟+𝛽)(𝜉)𝑚_𝑟(𝜉+𝛽)⎞⎟⎟⎟⎟⎟⎟⎠,(3.12) where 𝛽∈Υ. In the remainder of the paper, the filter banks will be composed of a single low-pass filter (with index 0) and a number of high-pass filters.

With the above definitions, we present an algorithm for the construction of arbitrarily many orthogonal multiwavelet tight frames.

Theorem 3.6. Suppose that 𝜙₁,𝜙₂∈𝐿²(𝑅^𝑑) are refinable functions which satisfy the conditions of the unitary extension principe, and let 𝑚¹(𝜉),𝑚²(𝜉) be the associated low-pass filter. Let 𝑀={𝑚₀(𝜉),𝑚₁(𝜉),…,𝑚_𝑟(𝜉)} and 𝑁={𝑛₀(𝜉),𝑛₁(𝜉),…,𝑛_𝑟(𝜉)} be filter banks with 𝑚₀(𝜉)=𝑚¹(𝜉),𝑛₀(𝜉)=𝑚²(𝜉). For all 𝛽∈Υ, suppose that the following matrix equations hold: (a) 𝑀^∗(𝜉)𝑀(𝜉)=𝐼₂ for almost every 𝜉,(b)𝑁^∗(𝜉)𝑁(𝜉)=𝐼₂ for almost every 𝜉,(c)𝑀^∗(𝜉)𝑁(𝜉)=0 for almost every 𝜉.Let 𝜂̂_𝑘(𝐵𝜉)=𝑛_𝑘(𝜉)𝜙₂(𝜉) and 𝜓_𝑘(𝐵𝜉)=𝑚_𝑘(𝜉)𝜙₁(𝜉),1≤𝑘≤𝑟. Then, {𝜓₁,…,𝜓_𝑟} and {𝜂₁,…,𝜂_𝑟} generate orthogonal multiwavelet tight frames.

Proof. For Items (a) and (b), by Lemma 3.5, then {𝜓₁,…,𝜓_𝑟} and {𝜂₁,…,𝜂_𝑟} generate multiwavelet tight frames. We use the characterization equations of Lemma 3.1 to prove orthogonality.
Let us focus on ∑^𝑟𝑘=1∑_𝑗∈𝑍𝜓_𝑘(𝐵^𝑗𝜉)𝜂̂_𝑘(𝐵^𝑗𝜉). For each 𝑘, by Hölder’s inequality and virtue of the fact that 𝜓_𝑘 and 𝜂_𝑘 generate Bessel sequences [4, Theorem 8.3.2], we have _𝑗_∈𝑍|||𝜓_𝑘𝐵^𝑗𝜉𝜂̂_𝑘𝐵^𝑗𝜉|||≤_𝑗∈𝑍||𝜓_𝑘𝐵^𝑗𝜉||²_𝑗∈𝑍||𝜂̂_𝑘𝐵^𝑗𝜉||²<∞,(3.13) then the order of summation can be reversed. With this, by Item (c), ^𝑟_𝑘=1_𝑗∈𝑍𝜓_𝑘𝐵^𝑗𝜉𝜂̂_𝑘𝐵^𝑗𝜉=^𝑟_𝑘=1_𝑗∈𝑍=_𝑗∈𝑍𝑚_𝑘𝐵^𝑗𝜉𝜙₁𝐵^𝑗𝜉𝑛_𝑘𝐵^𝑗𝜉𝜙₂𝐵^𝑗𝜉𝜙₁𝐵^𝑗𝜉𝜙₂𝐵^𝑗𝜉^𝑟_𝑘=1=0𝑚_𝑘𝐵^𝑗𝜉𝑛_𝑘𝐵^𝑗𝜉(3.14) holds for almost every 𝜉.
Likewise, for 𝑞∈𝑍^𝑑⧵𝐵𝑍^𝑑, by item (c), ^𝑟_∞𝑘=1_𝑗=0𝜓_𝑘𝐵^𝑗𝜉𝜂̂_𝑘𝐵^𝑗(𝜉+𝑞)=^𝑟_∞𝑘=1_𝑗=0=𝑚_𝑘𝐵^𝑗−1𝜉𝜙₁𝐵^𝑗−1𝜉𝑛_𝑘𝐵^𝑗−1(𝜉+𝑞)𝜙₂𝐵^𝑗−1(𝜉+𝑞)_∞_𝑗=0𝜙₁𝐵^𝑗𝜔𝜙₂𝐵^𝑗𝜔+𝐵⁻¹𝑞^𝑟_𝑘=1=0,𝑚_𝑘𝐵^𝑗𝜔𝑛_𝑘𝐵^𝑗𝜔+𝐵^𝑗−1𝑞(3.15) where 𝜔=𝐵⁻¹𝜉.

The following results show the relationship between a pair of orthogonal MRA multiwavelet frames.

Theorem 3.7. Suppose that 𝑋(Ψ_𝑖) and 𝑋(Ψ_𝑗) are a pair of orthogonal MRA multiwavelet frames, where Ψ_𝑖∶={𝜓^𝑖1,𝜓^𝑖2,…,𝜓^𝑖𝑟}, Ψ_𝑗∶={𝜓^𝑗1,𝜓^𝑗2,…,𝜓^𝑗𝑟}. If 𝑆(Ψ_𝑖)=𝑆(Ψ_𝑗) and there exist functions 𝑝,𝑤∈𝐿²(𝑅^𝑑) such that Ψ^𝑃𝑖∶={𝜓^1𝑖𝑝,𝜓^2𝑖𝑝,…,𝜓^𝑟𝑖𝑝} and Ψ^𝑃𝑗∶={𝜓^1𝑗𝑤,𝜓^2𝑗𝑤,…,𝜓^𝑟𝑗𝑤} are multiwavelet frames, where 𝜓^𝑝𝑙 and 𝜓^𝑤𝑙 defined by 𝜓^𝑙𝑖𝑝(𝜔)=𝜓^𝑖𝑙(𝜔)𝑝(̂𝜔), 𝜓^𝑙𝑗𝑤(𝜔)=𝜓^𝑗𝑙(𝜔)𝑤(𝜔),1≤𝑙≤𝑟 respectively, then 𝑋(Ψ^𝑝𝑖) and 𝑋(Ψ^𝑤𝑗) are a pair of orthogonal multiwavelet frames for 𝐿²(𝑅^𝑑).

Proof. Suppose that 𝑋(Ψ_𝑖), 𝑋(Ψ_𝑗) are a pair of orthogonal MRA multiwavelet frames and 𝑆(Ψ_𝑖)=𝑆(Ψ_𝑗), then, by the property of MRA multiwavelet frames, for any 𝑛≠𝑚∈𝑍, we have 𝑆(𝐴^𝑚Ψ_𝑖)⟂𝑆(𝐴^𝑛Ψ_𝑖). Hence, for all 𝑓₁∈𝑆(Ψ_𝑖)0==𝑆𝑓₁(𝑥)_𝑟_𝑙=1_{𝑘∈𝑍𝑑}_𝑠∈𝑍𝑓₁=(𝑥),𝜓^𝑖𝑙(𝐴^𝑠𝑥−𝑘)𝜓^𝑗𝑙(𝐴^𝑠𝑥−𝑘_𝑟)_𝑙=1_{𝑘∈𝑍𝑑}𝑓₁(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝑥−𝑘).(3.16)
For any 𝑓∈𝐿²(𝑅^𝑑), define 𝑓=𝑓₁+𝑓₂, where 𝑓₁∈𝑆(Ψ_𝑖),𝑓₂∈(𝐿²(𝑅^𝑑)⧵𝑆(Ψ_𝑖)), then, ⟨𝑓₁,𝑓₂⟩=0. With this, we get 𝑆𝑓₂(𝑥)=^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₂(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝑥−𝑘)=0.(3.17) Hence, for all 𝑓∈𝐿²(𝑅^𝑑), the following equation holds: 𝑆𝑓(𝑥)=^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑖𝑙=(𝑥−𝑘)𝜓^𝑗𝑙(𝑥−𝑘)_𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₁(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝑥−𝑘)+^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₂=0.(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝑥−𝑘)(3.18)
Notice that Ψ_𝑗∶={𝜓^𝑗1,…,𝜓^𝑗𝑟}, since 𝜓^𝑙𝑗𝑤(𝜉)=𝜓_𝑗(𝜉)𝑤(𝜉),1≤𝑙≤𝑟, by 𝑆𝑓(𝑥)=0, then 0==𝑆𝑓(𝑥)_𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}=^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₁(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}+^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₂(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}=𝑤(𝜔)^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₁(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}+^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₂(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑗𝑙(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}=_𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₁(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑙𝑗𝑤(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}+^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓₂(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑙𝑗𝑤(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}=^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑙𝑗𝑤(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}.(3.19)
Applying Fourier inverse transform on (3.19), we have 0==𝑆𝑓(𝑥)_𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑙𝑗𝑤(𝜔)𝑒^{−2𝜋𝑖𝑘𝜔}=^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝜓^𝑙𝑗𝑤(𝑥−𝑘).(3.20)
From the above result, we get the following equation: ^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑖𝑙(𝑥−𝑘)𝑓(𝑥),𝜓^𝑙𝑗𝑤(𝑥−𝑘)=𝑓(𝑥),^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑖𝑙==⟨𝑓𝑓(𝑥(𝑥),)0,⟩_𝑟(𝑥−𝑘)𝜓^𝑙𝑗𝑤(𝑥−𝑘)_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑙𝑗𝑤(𝑥−𝑘)𝜓^𝑖𝑙(𝑥−𝑘),(3.21) hence, ^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑙𝑗𝑤(𝑥−𝑘)𝜓^𝑖𝑙(𝑥−𝑘)=0.(3.22) Similar to the calculation of (3.19), clearly ^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑙𝑗𝑤(𝑥−𝑘)𝜓^𝑙𝑖𝑝(𝑥−𝑘)=0.(3.23)
For any 𝑠∈𝑍, ^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑙𝑗𝑤(𝐴^𝑠𝑥−𝑘)𝜓^𝑙𝑖𝑝(𝐴^𝑠𝑥−𝑘)=𝐴^𝑟−𝑠_𝑙=1_{𝑘∈𝑍𝑑}𝑓𝐴^−𝑠𝑥^,𝜓^𝑙𝑗𝑤𝑥^−𝑘𝜓^𝑙𝑖𝑝𝑥^−𝑘.(3.24) Let 𝑔(𝑥)∶=𝑓(𝐴^−𝑠𝑥). Define operator 𝑇∶𝐿²(𝑅^𝑑)→𝐿²(𝑅^𝑑);𝑇𝑓(𝑥)=𝑔(𝑥), obviously 𝑇 is a surjection operator. If 𝑠 is fixed, for all 𝑔∈𝐿²(𝑅^𝑑), we get ^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑙𝑗𝑤(𝐴^𝑠𝑥−𝑘)𝜓^𝑙𝑖𝑝(𝐴^𝑠𝑥−𝑘)=𝐴^−𝑠^𝑟_𝑙=1_{𝑘∈𝑍𝑑}𝑔𝑥^,𝜓^𝑙𝑗𝑤𝑥^−𝑘𝜓^𝑙𝑖𝑝𝑥^−𝑘=0.(3.25)
Putting everything together, we have ^𝑟_𝑙=1_𝑠∈𝑍_{𝑘∈𝑍𝑑}𝑓(𝑥),𝜓^𝑙𝑗𝑤(𝐴^𝑠𝑥−𝑘)𝜓^𝑙𝑖𝑝(𝐴^𝑠𝑥−𝑘)=0,(3.26) then, 𝑋(Ψ^𝑝𝑖) and 𝑋(Ψ^𝑤𝑗) are a pair of orthogonal multiwavelet frames.

The following theorem describes a general construction algorithm for orthogonal multiwavelet tight frames.

Theorem 3.8. Suppose 𝐾(𝜉) is an 𝑟×𝑟 paraunitary matrix with 𝐵⁻¹𝑍^𝑑-periodic entries 𝑎_𝑘,𝑠(𝜉); let 𝐾_𝑗(𝜉) denote the jth column. For all 𝛽∈Υ, suppose 𝑀={𝑚₀(𝜉),𝑚₁(𝜉),…,𝑚_𝑟(𝜉)} and 𝑀^∗(𝜉)𝑀(𝜉)=𝐼₂ hold for almost every 𝜉, where 𝑚₀ and {𝑚₁,…,𝑚_𝑙} are low- and high-pass filters, respectively, for a multiwavelet tight frame with scaling function 𝜙. For 𝑗=1,…,𝑟, define new filters via ⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑛𝑛⋮^𝑗1,1(𝜉)_𝑗1,𝑟𝑛⋮(𝜉)_𝑗𝑙,1𝑛⋮(𝜉)_{𝑗𝑙,𝑟}(𝜉)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎝𝐾^𝑗𝐾(𝜉)⋮𝑚₁(𝜉)_𝑗(𝜉)𝑚_𝑙(𝜉)⎞⎟⎟⎟⎟⎠.(3.27) Then, for 𝑗=1,…,𝑟, the affine systems generated by Ψ^𝑗={𝜓^{𝑗𝑖,𝑡}∶𝑖=1,…,𝑙,𝑡=1,…,𝑟} obtained via 𝜓^{𝑗𝑖,𝑡}(𝐵𝜉)=𝑛^{𝑗𝑖,𝑡}(𝜉)𝜙(𝜉)(3.28) are multiwavelet tight frames and are pairwise orthogonal.

Proof. Firstly, we prove that 𝑋(Ψ^𝑗),1≤𝑗≤𝑟, are multiwavelet tight frames. Assume 𝑀_𝑗={𝑚₀(𝜉),𝑛^𝑗1,1(𝜉),…,𝑛^𝑗1,𝑟(𝜉),…,𝑛^𝑗𝑙,1(𝜉),…,𝑛^{𝑗𝑙,𝑟}(𝜉)}. Define 𝑀_𝑗(𝜉) according to (3.12): 𝑀_𝑗(𝜉)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑚⁰𝑛(𝜉)𝑚₀(𝜉+𝛽)_𝑗1,1𝑛⋮(𝜉)𝑛^𝑗1,1(⋮_𝑗1,𝑟𝜉+𝛽)𝑛⋮(𝜉)𝑛^𝑗1,𝑟(⋮_𝑗𝑙,1𝜉+𝛽)𝑛⋮(𝜉)𝑛^𝑗𝑙,1(⋮_{𝑗𝑙,𝑟}𝜉+𝛽)(𝜉)𝑛^{𝑗𝑙,𝑟}(𝜉+𝛽)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(3.29) where 𝛽∈Υ. Then, 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉) is a 2×2 matrix. Next, we examine the entries of 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉) individually. Note that the columns of 𝐾(𝜉) have length 1, by 𝑀^∗(𝜉)𝑀(𝜉)=𝐼₂, it follows that 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉)_1,1=||𝑚₀(𝜉)||²+^𝑟_𝑙𝑘=1_𝑡=1||𝑎_𝑘,𝑗(𝜉)𝑚_𝑡(𝜉)||²=||𝑚₀(𝜉)||²+^𝑟_𝑘=1||𝑎_𝑘,𝑗(𝜉)||^2𝑙_𝑡=1||𝑚_𝑡(𝜉)||²=||𝑚₀(𝜉)||²+^𝑙_𝑡=1||𝑚_𝑡(𝜉)||²=1,(3.30) where [𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉)]_1,1 means the (1,1) entry of the matrix 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉).
Similarly, 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉)_2,2=||𝑚₀(𝜉+𝛽)||²+^𝑟_𝑙𝑘=1_𝑡=1||𝑎_𝑘,𝑗(𝜉+𝛽)𝑚_𝑡(𝜉+𝛽)||²=||𝑚₀(𝜉+𝛽)||²+^𝑟_𝑘=1||𝑎_𝑘,𝑗(𝜉+𝛽)||^2𝑙_𝑡=1||𝑚_𝑡(𝜉+𝛽)||²=||𝑚₀(𝜉+𝛽)||²+^𝑙_𝑡=1||𝑚_𝑡(𝜉+𝛽)||²=1.(3.31)
Now, since the entries of 𝐾(𝜉) are 𝐵⁻¹𝑍^𝑑-periodic, again by 𝑀^∗(𝜉)𝑀(𝜉)=𝐼₂, 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉)_1,2==𝑚𝑚₀₀(𝜉+𝛽)𝑚₀(𝜉)+^𝑟_𝑙𝑘=1_𝑡=1𝑎_𝑘,𝑗(𝜉)𝑚_𝑡(𝜉)𝑎_𝑘,𝑗(𝜉+𝛽)𝑚_𝑡(𝜉+𝛽)(𝜉+𝛽)𝑚₀(𝜉)+^𝑟_𝑘=1||𝑎_𝑘,𝑗(𝜉)||^2𝑙_𝑡=1=𝑚₀𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)(𝜉+𝛽)𝑚₀(𝜉)+^𝑙_𝑡=1=0.𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)(3.32) Finally, the (2,1)-entry must be zero by conjugate symmetry of 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉). Hence, 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉)=𝐼₂,1≤𝑗≤𝑟.(3.33) Putting everything together, from Theorem 3.6, the affine systems generated by {𝜓^{𝑗𝑖,𝑡}∶𝑖=1,…,𝑙,𝑡=1,…,𝑟} obtained via 𝜓^{𝑗𝑖,𝑡}(𝐵𝜉)=𝑛^{𝑗𝑖,𝑡}(𝜉)𝜙(𝜉)(3.34) are multiwavelet tight frames.
For orthogonality, according to (3.12), for 𝑗=1,…,𝑟, we have 𝑀_𝑗(𝜉)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑛^𝑗1,1𝑛⋮(𝜉)𝑛^𝑗1,1(⋮_𝑗1,𝑟𝜉+𝛽)𝑛⋮(𝜉)𝑛^𝑗1,𝑟(⋮_𝑗𝑙,1𝜉+𝛽)𝑛⋮(𝜉)𝑛^𝑗𝑙,1(⋮_{𝑗𝑙,𝑟}𝜉+𝛽)(𝜉)𝑛^{𝑗𝑙,𝑟}(𝜉+𝛽)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎝𝐾^𝑗𝐾(𝜉)⋮𝑚₁(𝜉)𝐾_𝑗(𝜉+𝛽)⋮_𝑗𝑚₁(𝜉+𝛽)(𝜉)𝑚_𝑙(𝜉)𝐾_𝑗(𝜉+𝛽)𝑚_𝑙(𝜉+𝛽)⎞⎟⎟⎟⎟⎠.(3.35) If 1≤𝑗≠𝑗′≤𝑟, then 𝑀^∗𝑗(=𝜉)⎛⎜⎜⎜⎜⎝𝑀𝐾_𝑗′(𝜉)_𝑗𝐾(𝜉)⋮𝑚₁(𝜉)𝐾_𝑗(𝜉+𝛽)⋮_𝑗𝑚₁(𝜉+𝛽)(𝜉)𝑚_𝑙(𝜉)𝐾_𝑗(𝜉+𝛽)𝑚_𝑙(𝜉+𝛽)⎞⎟⎟⎟⎟⎠^∗⎛⎜⎜⎜⎜⎝𝐾𝐾_𝑗′(𝜉⋮)𝑚₁(𝜉)𝐾_𝑗′(𝜉+𝛽⋮_𝑗′)𝑚₁(𝜉+𝛽)=⎛⎜⎜⎜⎜⎜⎝𝐾(𝜉)𝑚_𝑙(𝜉)𝐾_𝑗′(𝜉+𝛽)𝑚_𝑙(𝜉+𝛽)_∗𝑗⎞⎟⎟⎟⎟⎠(𝜉)𝐾_𝑗′(𝜉)^𝑙_𝑡=1||𝑚_𝑡(𝜉)||²𝐾^∗𝑗(𝜉)𝐾_𝑗′(𝜉+𝛽)^𝑙_𝑡=1𝐾𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)_∗𝑗(𝜉+𝛽)𝐾_𝑗′(𝜉)^𝑙_𝑡=1𝑚_𝑡(𝜉+𝛽)𝑚_𝑡(𝜉)𝐾^∗𝑗(𝜉+𝛽)𝐾_𝑗′(𝜉+𝛽)^𝑙_𝑡=1=⎛⎜⎜⎜⎜⎜⎝𝐾𝑚_𝑡(𝜉+𝛽)𝑚_𝑡(𝜉+𝛽)_∗𝑗⎞⎟⎟⎟⎟⎟⎠(𝜉)𝐾_𝑗′(𝜉)^𝑙_𝑡=1||𝑚_𝑡(𝜉)||²𝐾^∗𝑗(𝜉)𝐾_𝑗′(𝜉)^𝑙_𝑡=1𝐾𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)_∗𝑗(𝜉)𝐾_𝑗′(𝜉)^𝑙_𝑡=1𝑚_𝑡(𝜉+𝛽)𝑚_𝑡(𝜉)𝐾^∗𝑗(𝜉)𝐾_𝑗′(𝜉)^𝑙_𝑡=1=0,𝑚_𝑡(𝜉+𝛽)𝑚_𝑡(𝜉+𝛽)⎞⎟⎟⎟⎟⎟⎠(3.36) where we use the fact that the product of the two matrices 𝐾^∗𝑗(𝜉)𝐾_𝑗(𝜉) is 0 by the orthogonality of the columns of 𝐾(𝜉). By Theorem 3.6, we have the desired result.

The following proposition is directly related to the construction algorithm in Theorem 3.8.

Proposition 3.9. If 𝜙 is compactly supported, the paraunitary matrix K in Theorem 3.8 must have entries which are 𝐵⁻¹𝑍^𝑑-periodic.

Proof. The proof will follow the notation of Theorem 3.8. For 1≤𝑗≤𝑟, for all 𝜉∈𝐵⁻¹𝑍^𝑑/𝑍^𝑑, the matrix 𝑀_𝑗(𝜉)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝𝑚⁰𝑎(𝜉)𝑚₀(𝜉+𝛽)_1,𝑗𝑎(𝜉⋮)𝑚₁(𝜉)𝑎_1,𝑗(𝜉+𝛽⋮_𝑟,𝑗)𝑚₁(𝜉+𝛽)𝑎(𝜉⋮)𝑚₁(𝜉)𝑎_𝑟,𝑗(𝜉+𝛽⋮_1,𝑗)𝑚₁(𝜉+𝛽)𝑎(𝜉⋮)𝑚_𝑙(𝜉)𝑎_1,𝑗(𝜉+𝛽⋮_𝑟,𝑗)𝑚_𝑙(𝜉+𝛽)(𝜉)𝑚_𝑙(𝜉)𝑎_𝑟,𝑗(𝜉+𝛽)𝑚_𝑙(𝜉+𝛽)⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠(3.37) satisfies the equation 𝑀^∗𝑗(𝜉)𝑀_𝑗(𝜉)=𝐼₂a.e.𝜉.(3.38) Then, for almost every 𝜉, the following equation 𝑚₀(𝜉+𝛽)𝑚₀(𝜉)+^𝑟_𝑘=1𝑎_𝑘,𝑗(𝜉)𝑎_𝑘,𝑗(𝜉+𝛽)^𝑙_𝑡=1𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)=0(3.39) must hold. Notice that 𝑚₀ and {𝑚₁,…,𝑚_𝑙} are low- and high-pass filters, respectively, which meet Theorem 3.8. Then, 𝑚₀(𝜉+𝛽)𝑚₀(𝜉)+^𝑙_𝑡=1𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)=0.(3.40) Thus, we have 𝑚₀(𝜉+𝛽)𝑚₀(𝜉)=−^𝑙_𝑡=1𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽).(3.41)
From the above results, we get the following equation: 0=𝑚₀(𝜉+𝛽)𝑚₀(𝜉)+^𝑟_𝑘=1𝑎_𝑘,𝑗(𝜉)𝑎_𝑘,𝑗(𝜉+𝛽)^𝑙_𝑡=1=−_𝑙𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)_𝑡=1𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)+^𝑟_𝑘=1𝑎_𝑘,𝑗(𝜉)𝑎_𝑘,𝑗(𝜉+𝛽)^𝑙_𝑡=1=𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)_𝑟_𝑘=1𝑎_𝑘,𝑗(𝜉)𝑎_𝑘,𝑗(𝜉+𝛽)−1^𝑙_𝑡=1𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽).(3.42)
Hence, ∑^𝑙𝑡=1𝑚_𝑡(𝜉)𝑚_𝑡(𝜉+𝛽)=0 or ∑^𝑟𝑘=1𝑎_𝑘,𝑗(𝜉)𝑎_𝑘,𝑗(𝜉+𝛽)=1. If 𝜙 is compactly supported, then the first possibility is eliminated except possibly on a set of measure 0, whence the second must hold almost everywhere. Now, the sum is precisely the inner product of the two vectors 𝑎_𝑘,𝑗(𝜉) and 𝑎_𝑘,𝑗(𝜉+𝛽), each of which has length 1. Applying Cauchy-Schwarz inequation yields that the two vectors must be identical for almost every 𝜉.

4. Conclusion

In this paper, motivated by the notion of orthogonal frames, we present the construction of orthogonal multiwavelet frames in 𝐿²(𝑅^𝑑) with matrix dilation, where the basic ingredients consist of two fixed multiwavelet basis and a paraunitary matrix of an appropriate size. The number of orthogonal multiwavelet frames that can be constructed is arbitrary, and is determined by the size of the paraunitary matrix. Moreover, by using the unitary extension principle, we present an algorithm for the construction of orthogonal multiwavelet tight frames and give a general construction algorithm for orthogonal multiwavelet tight frames from a scaling function.

Acknowledgment

L. Zhanwei was supported by the Henan Provincial Natural Science Foundation of China (Grant no. 102300410205). H. Guoen was supported by the National Natural Science Foundation of China (Grant no. 10971228).