Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article

Research Article | Open Access

Volume 2012 |Article ID 846852 | 18 pages | https://doi.org/10.1155/2012/846852

Orthogonal Multiwavelet Frames in ๐ฟ2(๐‘…๐‘‘)

Academic Editor: Yuesheng Xu
Received27 Jun 2011
Revised21 Sep 2011
Accepted25 Oct 2011
Published15 Dec 2011

Abstract

We characterize the orthogonal frames and orthogonal multiwavelet frames in ๐ฟ2(๐‘…๐‘‘) with matrix dilations of the form โˆš(๐ท๐‘“)(๐‘ฅ)=|det๐ด|๐‘“(๐ด๐‘ฅ), where ๐ด is an arbitrary expanding ๐‘‘ร—๐‘‘ 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 [1โ€“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 [5โ€“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 [11โ€“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 ๐ฟ2(๐‘…). 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 ๐ฟ2(๐‘…๐‘‘) 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 ๐‘™2(๐‘๐‘‘) 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 ๐ฟ2(๐‘…๐‘‘) with matrix dilation.

2. Preliminaries

Let us now establish some basic notations.

We denote by ๐‘‡๐‘‘ the d-dimensional torus. By ๐ฟ2(๐‘‡๐‘‘), we denote the space of all ๐‘๐‘‘-periodic functions ๐‘“ (i.e., ๐‘“ is 1-periodic in each variable) such that โˆซ๐‘‡๐‘‘|๐‘“(๐‘ฅ)|2๐‘‘๐‘ฅ<+โˆž. The subsets of ๐‘…๐‘‘ invariant under ๐‘๐‘‘ translations and the subsets of ๐‘‡๐‘‘ are often identified.

We use the Fourier transform in the form๎๎€œ๐‘“(๐œ”)=๐‘…๐‘‘๐‘“(๐‘ฅ)๐‘’โˆ’2๐œ‹๐‘–โŸจ๐‘ฅ,๐œ”โŸฉ๐‘‘๐‘ฅ,(2.1) where โŸจโ‹…,โ‹…โŸฉ denotes the standard inner product in ๐‘…๐‘‘. The Fourier inverse transform is defined by๎ƒถ๎๎€œ๐‘“(๐‘ฅ)=๐‘“(๐œ”)=๐‘…๐‘‘๎๐‘“(๐œ”)๐‘’2๐œ‹๐‘–โŸจ๐‘ฅ,๐œ”โŸฉ๐‘‘๐œ”.(2.2)

Let ๐ธ๐‘‘ denote the set of all expanding ๐‘‘ร—๐‘‘ matrices ๐ด with integer coefficients. The expanding matrices mean that all eigenvalues have magnitude greater than 1. For ๐ดโˆˆ๐ธ๐‘‘, we denote by ๐ต the transpose of ๐ด. It is obvious that ๐ตโˆˆ๐ธ๐‘‘.

A collection of elements {๐œ™๐‘—โˆถ๐‘—โˆˆ๐ฝ} in a Hilbert space ๐ป is called a frame if there exist constants ๐‘Ž and ๐‘, 0<๐‘Žโ‰ค๐‘<โˆž, such that๐‘Žโ€–๐‘“โ€–2โ‰ค๎“๐‘—โˆˆ๐ฝ||๎ซ๐‘“,๐œ™๐‘—๎ฌ||2โ‰ค๐‘โ€–๐‘“โ€–2,forall๐‘“โˆˆ๐ป.(2.3) If {๐œ™๐‘—โˆถ๐‘—โˆˆ๐ฝ} satisfies the second inequality, then {๐œ™๐‘—โˆถ๐‘—โˆˆ๐ฝ} is called a Bessel sequence. Let ๐‘Ž0 the supremum of all such numbers ๐‘Ž and ๐‘0 the infimum of all such numbers ๐‘, then ๐‘Ž0 and ๐‘0 are called the frame bounds of the frame {๐œ™๐‘—โˆถ๐‘—โˆˆ๐ฝ}. When ๐‘Ž0=๐‘0, we say that the frame is tight. When ๐‘Ž0=๐‘0=1, we say the frame is a Parseval frame.

In this paper, we will work with two families of unitary operators on ๐ฟ2(๐‘…๐‘‘). The first one consists of all translation operators ๐‘‡๐‘˜โˆถ๐ฟ2(๐‘…๐‘‘)โ†’๐ฟ2(๐‘…๐‘‘),๐‘˜โˆˆ๐‘๐‘‘, defined by (๐‘‡๐‘˜๐‘“)(๐‘ฅ)=๐‘“(๐‘ฅโˆ’๐‘˜). The second one consists of all integer powers of the dilation operator ๐ท๐ดโˆถ๐ฟ2(๐‘…๐‘‘)โ†’๐ฟ2(๐‘…๐‘‘) defined by โˆš(๐ท๐‘“)(๐‘ฅ)=|๐ด|๐‘“(๐ด๐‘ฅ) with ๐ดโˆˆ๐ธ๐‘‘.

Let us now fix an arbitrary matrix ๐ดโˆˆ๐ธ๐‘‘. For ฮจ={๐œ“1,โ€ฆ,๐œ“๐‘Ÿ}โŠ‚๐ฟ2(๐‘…๐‘‘), we will consider the affine system ๐‘‹(ฮจ) defined by๎‚†๐œ“๐‘‹(ฮจ)=๐‘™๐‘—,๐‘˜(๐‘ฅ)โˆฃ๐œ“๐‘™๐‘—,๐‘˜||||(๐‘ฅ)=det๐ด๐‘—/2๐œ“๐‘™๎€ท๐ด๐‘—๎€ธ๐‘ฅโˆ’๐‘˜โˆถ๐‘—โˆˆ๐‘;๐‘˜โˆˆ๐‘๐‘‘๎‚‡.;๐‘™=1,โ€ฆ,๐‘Ÿ(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 ๐‘‹(ฮจ)โŠ‚๐ฟ2(๐‘…๐‘‘) is a multiwavelet frame if the system (2.4) is a frame for ๐ฟ2(๐‘…๐‘‘).

Definition 2.2. We say that ๐‘‹(ฮจ)โŠ‚๐ฟ2(๐‘…๐‘‘) is a multiwavelet tight frame if the system (2.4) is a tight frame for ๐ฟ2(๐‘…๐‘‘).

Definition 2.3. We say that ๐‘‹(ฮจ)โŠ‚๐ฟ2(๐‘…๐‘‘) is a multiwavelet tight frame if the system (2.4) is a Parseval frame for ๐ฟ2(๐‘…๐‘‘).

We turn to the concept of multiresolution analysis (MRA) in ๐ฟ2(๐‘…๐‘‘) which is a useful tool in our study.

Definition 2.4. Let {๐‘‰๐‘š}๐‘šโˆˆ๐‘ be a sequence of closed subspaces of ๐ฟ2(๐‘…๐‘‘) satisfying:(1)โ€‰๐‘‰๐‘—โŠ‚๐‘‰๐‘—+1,(2)โ‹ƒ๐‘—โˆˆ๐‘๐‘‰๐‘—=๐ฟ2(๐‘…๐‘‘), (3)โ‹‚๐‘—โˆˆ๐‘๐‘‰๐‘—={0}, (4)๐‘“(๐‘ฅ)โˆˆ๐‘‰๐‘—โ‡”๐‘“(๐ด๐‘ฅ)โˆˆ๐‘‰๐‘—+1,๐‘—โˆˆ๐‘, where ๐ดโˆˆ๐ธ๐‘‘,(5)There exists a function ๐œ™(๐‘ฅ)โˆˆ๐‘‰0 such that {๐œ™(๐‘ฅโˆ’๐‘˜)}๐‘˜โˆˆ๐‘๐‘‘ is a frame of ๐‘‰0. Then, {๐‘‰๐‘—}๐‘—โˆˆ๐‘ is called an MRA and the function ๐œ™ in (5) a scaling function.

There is a standard procedure for constructing multiwavelets from a given MRA(๐‘‰๐‘—). Firstly, one defines ๐‘Š๐‘—=๐‘‰๐‘—+1โŠ–๐‘‰๐‘— for all ๐‘—โˆˆ๐‘. As an easy consequence of conditions (1)โ€“(4) from Definition 2.4, one obtains ๐ฟ2(๐‘…๐‘‘)=โŠ•๐‘—โˆˆ๐‘๐‘Š๐‘— and ๐‘Š๐‘—+1=๐ท๐‘Š๐‘—, for all ๐‘—โˆˆ๐‘. Suppose now that there exist functions ฮจโŠ‚๐‘Š0 such that the system ๐ธ(ฮจ)โˆถ={๐œ“(โ‹…โˆ’๐‘˜)โˆถ๐‘˜โˆˆ๐‘๐‘‘,๐œ“โˆˆฮจ} is a frame for ๐‘Š0. Then, {๐ท๐‘—๐‘‡๐‘˜๐œ“โˆถ๐‘˜โˆˆ๐‘๐‘‘,๐œ“โˆˆฮจ} is a frame for ๐‘Š๐‘—, for all ๐‘—โˆˆ๐‘, and, consequently, {๐ท๐‘—๐‘‡๐‘˜๐œ“โˆถ๐‘—โˆˆ๐‘,๐‘˜โˆˆ๐‘๐‘‘,๐œ“โˆˆฮจ} is a frame for ๐ฟ2(๐‘…๐‘‘).

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

Let ๐‘‹ be a (countable) Bessel system for a separable Hilbert space ๐ป over the complex field ๐ถ. The synthesis operator ๐‘‡๐‘‹โˆถ๐‘™2(๐‘‹)โ†’๐ป, which is well known to be bounded, is defined by ๐‘‡๐‘‹โˆ‘๐‘Žโˆถ=โ„Žโˆˆ๐‘‹๐‘Žโ„Žโ„Ž for ๐‘Ž={๐‘Žโ„Ž}โ„Žโˆˆ๐‘‹. The adjoint operator ๐‘‡โˆ—๐‘‹ of ๐‘‡๐‘‹, called the analysis operator, is ๐‘‡โˆ—๐‘‹โˆถ๐ปโ†’๐‘™2(๐‘‹);๐‘‡โˆ—๐‘‹๐‘“โˆถ={โŸจ๐‘“,โ„ŽโŸฉ}โ„Žโˆˆ๐‘‹. Recall that ๐‘‹ is a frame for ๐ป if and only ๐‘†๐‘‹โˆถ=๐‘‡๐‘‹๐‘‡โˆ—๐‘‹โˆถ๐ปโ†’๐ป, 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 ๐‘†๐‘‹ is the identity operator. The system ๐‘‹ is a Riesz system (for span๐‘‹) if and only its Gramian ๐บ๐‘‹โˆถ=๐‘‡โˆ—๐‘‹๐‘‡๐‘‹ is bounded and has a bounded inverse; it is an orthonormal system of ๐ป if and only if ๐บ๐‘‹ is the identity operator.

Definition 2.5. Let ๐‘‹ and ๐‘Œ=๐‘…๐‘‹, where ๐‘…โˆถโ„Žโ†’๐‘…โ„Ž is a bijection between ๐‘‹ and ๐‘Œ, be two frame for ๐ป. We call ๐‘‹ and ๐‘Œ a dual frames for ๐ป if ๐‘‡๐‘Œ๐‘‡โˆ—๐‘‹=๐ผ, that is, โˆ‘โ„Žโˆˆ๐‘‹โŸจ๐‘“,โ„ŽโŸฉ๐‘…โ„Ž=๐‘“ for all ๐‘“โˆˆ๐ป.

Definition 2.6. Let ๐‘‹ and ๐‘Œ=๐‘…๐‘‹, where ๐‘…โˆถโ„Žโ†’๐‘…โ„Ž is a bijection between ๐‘‹ and ๐‘Œ, be two frames for ๐ป. We call ๐‘‹ and ๐‘Œ a pair of orthogonal frames for ๐ป if ๐‘‡๐‘Œ๐‘‡โˆ—๐‘‹=0, that is, โˆ‘โ„Žโˆˆ๐‘‹โŸจ๐‘“,โ„ŽโŸฉ๐‘…โ„Ž=0 for all ๐‘“โˆˆ๐ป.

Definition 2.7. A closed subspace ๐‘‰โŠ‚๐ฟ2(๐‘…๐‘‘) is shift invariant if forall๐‘“โˆˆ๐‘‰ implies ๐‘‡๐‘˜๐‘“โˆˆ๐‘‰ for any ๐‘˜โˆˆ๐‘๐‘‘.

We consider orthogonal frames in a shift-invariant subspace of ๐ฟ2(๐‘…๐‘‘). Let ฮฆ be a countable subset of ๐ฟ2(๐‘…๐‘‘), and ๐ธ(ฮฆ)โˆถ={๐œ™(โ‹…โˆ’๐‘˜)โˆถ๐‘˜โˆˆ๐‘๐‘‘,๐œ™โˆˆฮฆ}. Define ๐‘†(ฮฆ)โˆถ=span๐ธ(ฮฆ), the smallest closed subspace that contains ๐ธ(ฮฆ). Throughout the rest of this paper, we assume that ๐ธ(ฮฆ) is a Bessel sequence for ๐‘†(ฮฆ). This assumption settles most of the convergence issues. The space ๐‘†(ฮฆ) is called the shift-invariant space generated by ฮฆ and ฮฆ a generating set for ๐‘†(ฮฆ). Shift-invariant spaces have been studied extensively in the literature, for example, [22, 23].

For ๐œ”โˆˆ๐‘…๐‘‘, we define the pre-Gramian by๐ฝฮฆ๎‚€๎๎‚(๐œ”)=๐œ™(๐œ”+๐›ผ)๐›ผโˆˆ๐‘๐‘‘,๐œ™โˆˆฮฆ,(2.5) where ๎๐œ™ is the Fourier transform of ๐œ™. Note that the domain of the pre-Gramian matrix as an operator is ๐‘™2(ฮฆ) and its codomain is ๐‘™2(๐‘๐‘‘). The pre-Gramian can be regarded as the synthesis operator represented in Fourier domain as it was extensively studied in [22].

Let ฮฆ and ฮจ=๐‘…ฮฆ, where ๐‘… is a bijection satisfying ๐‘…(๐œ™(โ‹…โˆ’๐‘˜))=(๐‘…๐œ™)(โ‹…โˆ’๐‘˜), be countable subsets of ๐ฟ2(๐‘…๐‘‘). Suppose that ๐‘†(ฮฆ)=๐‘†(ฮจ) and that both ๐ธ(ฮฆ) and ๐ธ(ฮจ) are frames for ๐‘†(ฮฆ). Then, by definition, ๐ธ(ฮฆ) and ๐ธ(ฮจ) are a pair of orthogonal frames for ๐‘†(ฮฆ) if and only if for all ๐‘“โˆˆ๐‘†(ฮฆ),๐‘†๐‘“โˆถ=๐‘‡๐ธ(ฮจ)๐‘‡โˆ—๐ธ(ฮฆ)=0.(2.6)

We define the mixed dual Gramian as ๎‚๐บ(๐œ”)=๐ฝฮจ(๐œ”)๐ฝโˆ—ฮฆ(๐œ”) and Gramians as๐บฮฆ(๐œ”)=๐ฝโˆ—ฮฆ(๐œ”)๐ฝฮฆ(๐œ”),๐บฮจ(๐œ”)=๐ฝโˆ—ฮจ(๐œ”)๐ฝฮจ(๐œ”).(2.7) Then, it is proven in [24] that, for any ๐‘“โˆˆ๐ฟ2(๐‘…๐‘‘),๎‚€๎‚Š๎‚๐‘†๐‘“|๐œ”+๐›ผ=๎‚๎๐‘“๐บ(๐œ”)|๐œ”+๐›ผ,(2.8) where ๎๐‘“|๐œ”+๐›ผ is the column vector (๎๐‘“(๐œ”+๐›ผ))๐‘‡๐›ผโˆˆ๐‘๐‘‘. By (2.8), one can prove easily that ๐‘†๐‘“=0 for all ๐‘“โˆˆ๐ฟ2(๐‘…๐‘‘) if and only if ๎‚๐บ(๐œ”)=0 for a.e. ๐œ”โˆˆ๐‘…๐‘‘.

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 ฮจ1โˆถ={๐œ“11,๐œ“12,โ€ฆ,๐œ“1๐‘Ÿ} and ฮจ2โˆถ={๐œ“21,๐œ“22,โ€ฆ,๐œ“2๐‘Ÿ}. Suppose that ๐‘‹(ฮจ1) and ๐‘‹(ฮจ2) are multiwavelet frames for ๐ฟ2(๐‘…๐‘‘). ๐‘‹(ฮจ1) and ๐‘‹(ฮจ2) are a pair of orthogonal frames for ๐ฟ2(๐‘…๐‘‘) if and only if the following two equations are satisfied a.e.: ๐‘Ÿ๎“๐‘–=1๎“๐‘—โˆˆ๐‘๎‚Š๐œ“1๐‘–๎€ท๐ต๐‘—๐œ”๎€ธ๎‚Š๐œ“2๐‘–๎€ท๐ต๐‘—๐œ”๎€ธ=0,๐‘Ž.๐‘’.,๐‘Ÿ๎“๐‘–=1+โˆž๎“๐‘—=0๎‚Š๐œ“1๐‘–๎€ท๐ต๐‘—๐œ”๎€ธ๎‚Š๐œ“2๐‘–๎€ท๐ต๐‘—๎€ธ(๐œ”+๐‘ž)=0,๐‘Ž.๐‘’.,โˆ€๐‘˜โˆˆ๐‘๐‘‘,๐‘žโˆˆ๐‘๐‘‘โงต๐ต๐‘๐‘‘.(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 ฮจ1โˆถ={๐œ“11,๐œ“12,โ€ฆ,๐œ“1๐‘Ÿ}andฮจ2โˆถ={๐œ“21,๐œ“22,โ€ฆ,๐œ“2๐‘Ÿ}โˆˆ๐ฟ2(๐‘…๐‘‘) for some positive integer r. Suppose that ๐‘‹(ฮจ1) and ๐‘‹(ฮจ2) are multiwavelet frames for ๐ฟ2(๐‘…๐‘‘). Let ๐‘‰โˆถ=(๐‘‰1;๐‘‰2) be a 2๐‘Ÿร—2๐‘Ÿ constant unitary matrix, where ๐‘‰1 is the submatrix of the first r columns and ๐‘‰2 the remaining r columns. Then, ๐‘‹(ฮจ11) and ๐‘‹(ฮจ22) are a pair of orthogonal multiwavelet frames for ๐ฟ2(๐‘…๐‘‘), where ฮจ11โˆถ=๐‘‰1ฮจ1 and ฮจ22โˆถ=๐‘‰2ฮจ2.

Proof. Assume that ๐‘‰ is a constant matrix such that ๎ฮจ11โˆถ=๐‘‰1๎ฮจ1 and ๎ฮจ22โˆถ=๐‘‰2๎ฮจ2. Then, one can directly calculate the dual Gramians of ๐‘‹๐‘ž(ฮจ11) and ๐‘‹๐‘ž(ฮจ22). 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 ๐‘‰=(๐‘ฃ๐‘™๐‘š)1โ‰ค๐‘™,๐‘šโ‰ค2๐‘Ÿ. For a fixed ๐‘žโˆˆ๐‘๐‘‘โงต๐ต๐‘๐‘‘, ๐‘–โˆˆ1,2, we have 2๐‘Ÿ๎“๐‘™=1๎“๐‘šโ‰ฅ0๎‚Š๐œ“๐‘™๐‘–๐‘–(๐ต๐‘š๎‚Š๐œ“๐œ”)๐‘™๐‘–๐‘–(๐ต๐‘š(๐œ”+๐‘ž))=2๐‘Ÿ๎“๐‘™=1๎“๐‘Ÿ๐‘šโ‰ฅ0๎“๐‘›=1๐‘ฃ๐‘™,๐‘›๎‚Š๐œ“๐‘–๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๐‘ฃ๐‘™,๐‘›โ€ฒ๎‚Š๐œ“๐‘–๐‘›โ€ฒ(๐ต๐‘š=๎“(๐œ”+๐‘ž))๐‘Ÿ๐‘šโ‰ฅ0๎“๐‘›=1๎‚Š๐œ“๐‘–๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๎‚Š๐œ“๐‘–๐‘›โ€ฒ(๐ต๐‘š(๐œ”+๐‘ž))2๐‘Ÿ๎“๐‘™=1๐‘ฃ๐‘™,๐‘›๐‘ฃ๐‘™,๐‘›๎…ž=๎“๐‘Ÿ๐‘šโ‰ฅ0๎“๐‘›=1๎‚Š๐œ“๐‘–๐‘›(๐ต๐‘š๎‚Š๐œ“๐œ”)๐‘–๐‘›(๐ต๐‘š(๐œ”+๐‘ž)),(3.2) where we used the fact that the double sums converge absolutely a.e., ๐‘‰โˆ—๐‘‰=๐ผ2๐‘Ÿ, and that ๐‘‹(ฮจ1) and ๐‘‹(ฮจ2) are frames for ๐ฟ2(๐‘…๐‘‘). Moreover, 2๐‘Ÿ๎“๐‘™=1๎“๐‘šโˆˆ๐‘๎‚Š๐œ“๐‘™๐‘–๐‘–(๐ต๐‘š๎‚Š๐œ“๐œ”)๐‘™๐‘–๐‘–(๐ต๐‘š๐œ”)=2๐‘Ÿ๎“๐‘™=1๎“๐‘Ÿ๐‘šโˆˆ๐‘๎“๐‘›=1๐‘ฃ๐‘™,๐‘›๎‚Š๐œ“๐‘–๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๐‘ฃ๐‘™,๐‘›โ€ฒ๎‚Š๐œ“๐‘–๐‘›โ€ฒ(๐ต๐‘š=๎“๐œ”)๐‘Ÿ๐‘šโˆˆ๐‘๎“๐‘›=1๎‚Š๐œ“๐‘–๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๎‚Š๐œ“๐‘–๐‘›โ€ฒ(๐ต๐‘š๐œ”)2๐‘Ÿ๎“๐‘™=1๐‘ฃ๐‘™,๐‘›๐‘ฃ๐‘™,๐‘›๎…ž=๎“๐‘Ÿ๐‘šโˆˆ๐‘๎“๐‘›=1๎‚Š๐œ“๐‘–๐‘›(๐ต๐‘š๎‚Š๐œ“๐œ”)๐‘–๐‘›(๐ต๐‘š๐œ”).(3.3) From the above results, by using the dual Gramian characterization of frames in [25, Corollary 5.7], then ๐‘‹(ฮจ11) and ๐‘‹(ฮจ22) are frames for ๐ฟ2(๐‘…๐‘‘).
We now show that the multiwavelet systems generated by ฮจ11 and ฮจ22 are a pair of orthogonal frames for ๐ฟ2(๐‘…๐‘‘). We apply Lemma 3.1 to ฮจ11โˆถ={๐œ“111,๐œ“211,โ€ฆ,๐œ“112๐‘Ÿ} and ฮจ22โˆถ={๐œ“122,๐œ“222,โ€ฆ,๐œ“222๐‘Ÿ}. Let ๐‘‰=(๐‘ฃ๐‘™๐‘š)1โ‰ค๐‘™,๐‘šโ‰ค2๐‘Ÿ. For all ๐‘žโˆˆ๐‘๐‘‘โงต๐ต๐‘๐‘‘, we have 2๐‘Ÿ๎“๐‘™=1๎“๐‘šโ‰ฅ0๎‚ฟ๐œ“๐‘™11(๐ต๐‘š๎‚ฟ๐œ“๐œ”)๐‘™22(๐ต๐‘š(๐œ”+๐‘ž))=2๐‘Ÿ๎“๐‘™=1๎“๐‘Ÿ๐‘šโ‰ฅ0๎“๐‘›=1๐‘ฃ๐‘™,๐‘›๎‚Š๐œ“1๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๐‘ฃ๐‘™,๐‘Ÿ+๐‘›โ€ฒ๎‚Š๐œ“2๐‘›โ€ฒ(๐ต๐‘š=๎“(๐œ”+๐‘ž))๐‘Ÿ๐‘šโ‰ฅ0๎“๐‘›=1๎‚Š๐œ“1๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๎‚Š๐œ“2๐‘›โ€ฒ(๐ต๐‘š(๐œ”+๐‘ž))2๐‘Ÿ๎“๐‘™=1๐‘ฃ๐‘™,๐‘›๐‘ฃ๐‘™,๐‘Ÿ+๐‘›๎…ž=๎“๐‘Ÿ๐‘šโ‰ฅ0๎“๐‘›=1๎‚Š๐œ“1๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๎‚Š๐œ“2๐‘›โ€ฒ(๐ต๐‘š(๐œ”+๐‘ž))ร—0=0,(3.4) where we used the orthogonality of the columns of ๐‘‰.
Moreover, 2๐‘Ÿ๎“๐‘™=1๎“mโˆˆ๐‘๎‚ฟ๐œ“๐‘™11(๐ต๐‘š๎‚ฟ๐œ“๐œ”)๐‘™22(๐ต๐‘š(๐œ”))=2๐‘Ÿ๎“๐‘™=1๎“๐‘Ÿ๐‘šโˆˆ๐‘๎“๐‘›=1๐‘ฃ๐‘™,๐‘›๎‚Š๐œ“1๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๐‘ฃ๐‘™,๐‘Ÿ+๐‘›โ€ฒ๎‚Š๐œ“2๐‘›โ€ฒ(๐ต๐‘š=๎“(๐œ”))๐‘Ÿ๐‘šโˆˆ๐‘๎“๐‘›=1๎‚Š๐œ“1๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๎‚Š๐œ“2๐‘›โ€ฒ(๐ต๐‘š(๐œ”))2๐‘Ÿ๎“๐‘™=1๐‘ฃ๐‘™,๐‘š๐‘ฃ๐‘™,๐‘Ÿ+๐‘›๎…ž=๎“๐‘Ÿ๐‘šโˆˆ๐‘๎“๐‘›=1๎‚Š๐œ“1๐‘›(๐ต๐‘š๐œ”)๐‘Ÿ๎“๐‘›โ€ฒ=1๎‚Š๐œ“2๐‘›โ€ฒ(๐ต๐‘š(๐œ”))ร—0=0,(3.5) by Lemma 3.1, ฮจ11 and ฮจ22 generate a pair of orthogonal frames.

The following results give some properties of the orthogonal frames.

Proposition 3.3. Suppose that ๐ธ(๐œ“๐‘–) and ๐ธ(๐œ“๐‘—) are a pair of orthogonal affine Bessel sequences in ๐ฟ2(๐‘…๐‘‘). If ๐›ผโˆˆ๐ฟ2(๐‘…๐‘‘) is a ๐‘๐‘‘-periodic function, then ๐ธ(๐œ“๐‘–) and ๐ธ(๐›ผ๐œ“๐‘—) are a pair of orthogonal affine Bessel sequences.

Proof. Suppose that ๐ธ(๐œ“๐‘–) and ๐ธ(๐œ“๐‘—) are a pair of orthogonal affine Bessel sequences in ๐ฟ2(๐‘…๐‘‘). Then, for all ๐‘“โˆˆ๐ฟ2(๐‘…๐‘‘), we have ๎“๐‘†๐‘“(๐‘ฅ)=๐‘šโˆˆ๐‘๐‘‘โŸจ๐‘“(๐‘ฅ),๐œ“๐‘–(๐‘ฅ+๐‘š)โŸฉ๐œ“๐‘—(๐‘ฅ+๐‘š)=0.(3.6) Let ๐œ“๎…ž๐‘—โˆถ=๐›ผ๐œ“๐‘—. Since ๐›ผ is a ๐‘๐‘‘-periodic function, then ๐ธ(๐œ“๎…ž๐‘—) is an affine Bessel sequence for ๐ฟ2(๐‘…๐‘‘) from the fact that, for all ๐‘“โˆˆ๐ฟ2(๐‘…๐‘‘), ๎“๐‘˜โˆˆ๐‘๐‘‘||๎ซ๐‘“(๐‘ฅ),๐›ผ(๐‘ฅโˆ’๐‘˜)๐œ“๐‘—๎ฌ||(๐‘ฅโˆ’๐‘˜)2=๎“๐‘˜โˆˆ๐‘๐‘‘|||๎‚ฌ๐›ผ(๐‘ฅ)๐‘“(๐‘ฅ),๐œ“๐‘—๎‚ญ|||(๐‘ฅโˆ’๐‘˜)2โ€–โ€–โ‰ค๐ตโ€–โ€–๐›ผ๐‘“2โ€–โ€–โ‰ค๐ต๐›ผโ€–โ€–2โ€–๐‘“โ€–2=๐ต๎…žโ€–๐‘“โ€–2.(3.7)
Again by ๐›ผ being a ๐‘๐‘‘-periodic function, we have the following equation: ๎“๐‘†๐‘“(๐‘ฅ)=๐‘šโˆˆ๐‘๐‘‘โŸจ๐‘“(๐‘ฅ),๐œ“๐‘–(๐‘ฅ+๐‘š)โŸฉ๐œ“๐‘—(๐‘ฅ+๐‘š)๎…ž=๎“๐‘šโˆˆ๐‘๐‘‘โŸจ๐‘“(๐‘ฅ),๐œ“๐‘–(๐‘ฅ+๐‘š)โŸฉ๐œ“๐‘—(๎“๐‘ฅ+๐‘š)๐›ผ(๐‘ฅ+๐‘š)=๐›ผ(๐‘ฅ)๐‘šโˆˆ๐‘๐‘‘โŸจ๐‘“(๐‘ฅ),๐œ“๐‘–(๐‘ฅ+๐‘š)โŸฉ๐œ“๐‘—(๐‘ฅ+๐‘š)=0.(3.8) Hence, ๐ธ(๐œ“๐‘–) and ๐ธ(๐›ผ๐œ“๐‘—) are a pair of orthogonal affine Bessel sequences in ๐ฟ2(๐‘…๐‘‘).

Proposition 3.4. Suppose that ๐ธ(๐œ“๐‘–) and ๐ธ(๐œ“๐‘—) are a pair of orthogonal frames for ๐ปโŠ‚๐ฟ2(๐‘…๐‘‘). Let ๐›ผโˆˆ๐ฟ2(๐‘…๐‘‘) be a ๐‘๐‘‘-periodic function. If ๐ธ(๐›ผ๐œ“๐‘—) is a frame for ๐ป, then ๐ธ(๐œ“๐‘–) and ๐ธ(๐›ผ๐œ“๐‘—) are a pair of orthogonal frames for ๐ป.

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 ๐ฟ2(๐‘…๐‘‘).

Lemma 3.5. Suppose ฮฆ=(๐œ™๐‘—)๐‘—โˆˆ๐ฝ is a refinable vector with a mask ฮ“ such that ๎“๐‘—โˆˆ๐ฝโ€–โ€–๐œ‘๐‘—โ€–โ€–2=๎€œ๐‘…๐‘‘โ€–โ€–๎โ€–โ€–ฮฆ(๐œ‰)2๐‘™2(๐ฝ)๐‘‘๐œ‰<โˆž,lim๐‘—โ†’โˆžโ€–โ€–๎ฮฆ๎€ท๐ตโˆ’๐‘—๐œ‰๎€ธโ€–โ€–=1,fora.e.๐œ‰โˆˆ๐‘…๐‘‘.(3.9) Suppose also that ฮจ=(๐œ“๐‘—)๐‘—โˆˆ๎‚๐ฝ, where ๎‚๐ฝ={1,โ€ฆ,๐‘} is finite, is given by ๎๎ฮจ(๐ต๐œ‰)=๐ป(๐œ‰)ฮฆ(๐œ‰),(3.10) where ๐ป=(โ„Ž๐‘–,๐‘—)๐‘–โˆˆ๎‚๐ฝ,๐‘–โˆˆ๐ฝ is a ๐‘๐‘‘-periodic, measurable matrix function satisfying ฮ“โˆ—(๐œ‰)ฮ“(๐œ‰+๐‘‘)+๐ปโˆ—(๐œ‰)๐ป(๐œ‰+๐‘‘)=ฮฉ(๐œ‰)๐›ฟ0,๐‘‘,fora.e.๐œ‰,(3.11) and for any ๐‘‘โˆˆฮฅ, where ฮฅ consists of representatives of distinct cosets of ๐ตโˆ’1๐‘๐‘‘/๐‘๐‘‘, then ฮจโŠ‚๐ฟ2(๐‘…๐‘‘) is a multiwavelet tight frame.

We call ๐‘š a filter if ๐‘šโˆˆ๐ฟโˆž([0,1)๐‘‘). We shall call ๐‘š a low-pass filter if ๐‘š(0)=1, and we shall call ๐‘š a high-pass filter if ๐‘š(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 ๐‘€โ€ฒ={๐‘š0,๐‘š1,โ€ฆ,๐‘š๐‘Ÿ}โŠ‚๐ฟโˆž([0,1)๐‘‘), let ๐‘€โ€ฒ(๐œ‰) and ๎‚‹๐‘€โ€ฒ(๐œ‰) be the matrices๐‘€๎…žโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘š(๐œ‰)=0(๐œ‰)๐‘š0๐‘š(๐œ‰+๐›ฝ)1(๐œ‰)๐‘š1(๐‘š๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘Ÿ(๐œ‰)๐‘š๐‘ŸโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,๎‚‹๐‘€(๐œ‰+๐›ฝ)๎…žโŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘š(๐œ‰)=1(๐œ‰)๐‘š1๐‘š(๐œ‰+๐›ฝ)2(๐œ‰)๐‘š2(๐‘š๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘Ÿ(๐œ‰)๐‘š๐‘ŸโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(๐œ‰+๐›ฝ)(3.12) where ๐›ฝโˆˆฮฅ. 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 ๐œ™1,๐œ™2โˆˆ๐ฟ2(๐‘…๐‘‘) are refinable functions which satisfy the conditions of the unitary extension principe, and let ๐‘š1(๐œ‰),๐‘š2(๐œ‰) be the associated low-pass filter. Let ๐‘€={๐‘š0(๐œ‰),๐‘š1(๐œ‰),โ€ฆ,๐‘š๐‘Ÿ(๐œ‰)} and ๐‘={๐‘›0(๐œ‰),๐‘›1(๐œ‰),โ€ฆ,๐‘›๐‘Ÿ(๐œ‰)} be filter banks with ๐‘š0(๐œ‰)=๐‘š1(๐œ‰),๐‘›0(๐œ‰)=๐‘š2(๐œ‰). For all ๐›ฝโˆˆฮฅ, suppose that the following matrix equations hold: (a)โ€‰๐‘€โˆ—(๐œ‰)๐‘€(๐œ‰)=๐ผ2 for almost every ๐œ‰,(b)๐‘โˆ—(๐œ‰)๐‘(๐œ‰)=๐ผ2 for almost every ๐œ‰,(c)๎‚‹๐‘€โˆ—๎‚(๐œ‰)๐‘(๐œ‰)=0 for almost every ๐œ‰.Let ฬ‚๐œ‚๐‘˜(๐ต๐œ‰)=๐‘›๐‘˜๎๐œ™(๐œ‰)2(๐œ‰) and ๎๐œ“๐‘˜(๐ต๐œ‰)=๐‘š๐‘˜๎๐œ™(๐œ‰)1(๐œ‰),1โ‰ค๐‘˜โ‰ค๐‘Ÿ. Then, {๐œ“1,โ€ฆ,๐œ“๐‘Ÿ} and {๐œ‚1,โ€ฆ,๐œ‚๐‘Ÿ} generate orthogonal multiwavelet tight frames.

Proof. For Items (a) and (b), by Lemma 3.5, then {๐œ“1,โ€ฆ,๐œ“๐‘Ÿ} and {๐œ‚1,โ€ฆ,๐œ‚๐‘Ÿ} generate multiwavelet tight frames. We use the characterization equations of Lemma 3.1 to prove orthogonality.
Let us focus on โˆ‘๐‘Ÿ๐‘˜=1โˆ‘๐‘—โˆˆ๐‘๎๐œ“๐‘˜(๐ต๐‘—๐œ‰)ฬ‚๐œ‚๐‘˜(๐ต๐‘—๐œ‰). For each ๐‘˜, by Hรถlderโ€™s inequality and virtue of the fact that ๐œ“๐‘˜ and ๐œ‚๐‘˜ generate Bessel sequences [4, Theorem 8.3.2], we have ๎“๐‘—โˆˆ๐‘|||๎๐œ“๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธฬ‚๐œ‚๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธ|||โ‰ค๎“๐‘—โˆˆ๐‘||๎๐œ“๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธ||2๎“๐‘—โˆˆ๐‘||ฬ‚๐œ‚๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธ||2<โˆž,(3.13) then the order of summation can be reversed. With this, by Item (c), ๐‘Ÿ๎“๐‘˜=1๎“๐‘—โˆˆ๐‘๎๐œ“๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธฬ‚๐œ‚๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธ=๐‘Ÿ๎“๐‘˜=1๎“๐‘—โˆˆ๐‘๐‘š๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธ๎๐œ™1๎€ท๐ต๐‘—๐œ‰๎€ธ๐‘›๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธ๎๐œ™2๎€ท๐ต๐‘—๐œ‰๎€ธ=๎“๐‘—โˆˆ๐‘๎๐œ™1๎€ท๐ต๐‘—๐œ‰๎€ธ๎๐œ™2๎€ท๐ต๐‘—๐œ‰๎€ธ๐‘Ÿ๎“๐‘˜=1๐‘š๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธ๐‘›๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธ=0(3.14) holds for almost every ๐œ‰.
Likewise, for ๐‘žโˆˆ๐‘๐‘‘โงต๐ต๐‘๐‘‘, by item (c), ๐‘Ÿ๎“โˆž๐‘˜=1๎“๐‘—=0๎๐œ“๐‘˜๎€ท๐ต๐‘—๐œ‰๎€ธฬ‚๐œ‚๐‘˜๎€ท๐ต๐‘—๎€ธ=(๐œ‰+๐‘ž)๐‘Ÿ๎“โˆž๐‘˜=1๎“๐‘—=0๐‘š๐‘˜๎€ท๐ต๐‘—โˆ’1๐œ‰๎€ธ๎๐œ™1๎€ท๐ต๐‘—โˆ’1๐œ‰๎€ธ๐‘›๐‘˜๎€ท๐ต๐‘—โˆ’1๎€ธ๎๐œ™(๐œ‰+๐‘ž)2๎€ท๐ต๐‘—โˆ’1๎€ธ=(๐œ‰+๐‘ž)โˆž๎“๐‘—=0๎๐œ™1๎€ท๐ต๐‘—๐œ”๎€ธ๎๐œ™2๎€ท๐ต๐‘—๎€ท๐œ”+๐ตโˆ’1๐‘ž๎€ธ๎€ธ๐‘Ÿ๎“๐‘˜=1๐‘š๐‘˜๎€ท๐ต๐‘—๐œ”๎€ธ๐‘›๐‘˜๎€ท๐ต๐‘—๐œ”+๐ต๐‘—โˆ’1๐‘ž๎€ธ๎€ธ=0,(3.15) where ๐œ”=๐ตโˆ’1๐œ‰.

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

Theorem 3.7. Suppose that ๐‘‹(ฮจ๐‘–) and ๐‘‹(ฮจ๐‘—) are a pair of orthogonal MRA multiwavelet frames, where ฮจ๐‘–โˆถ={๐œ“๐‘–1,๐œ“๐‘–2,โ€ฆ,๐œ“๐‘–๐‘Ÿ}, ฮจ๐‘—โˆถ={๐œ“๐‘—1,๐œ“๐‘—2,โ€ฆ,๐œ“๐‘—๐‘Ÿ}. If ๐‘†(ฮจ๐‘–)=๐‘†(ฮจ๐‘—) and there exist functions ๐‘,๐‘คโˆˆ๐ฟ2(๐‘…๐‘‘) such that ฮจ๐‘ƒ๐‘–โˆถ={๐œ“1๐‘–๐‘,๐œ“2๐‘–๐‘,โ€ฆ,๐œ“๐‘Ÿ๐‘–๐‘} and ฮจ๐‘ƒ๐‘—โˆถ={๐œ“1๐‘—๐‘ค,๐œ“2๐‘—๐‘ค,โ€ฆ,๐œ“๐‘Ÿ๐‘—๐‘ค} are multiwavelet frames, where ๐œ“๐‘๐‘™ and ๐œ“๐‘ค๐‘™ defined by ๎„ž๐œ“๐‘™๐‘–๐‘(๐œ”)=๎๐œ“๐‘–๐‘™(๐œ”)ฬ‚๐‘(๐œ”), ๎„ž๐œ“๐‘™๐‘—๐‘ค(๐œ”)=๎๐œ“๐‘—๐‘™๎(๐œ”)๐‘ค(๐œ”),1โ‰ค๐‘™โ‰ค๐‘Ÿ respectively, then ๐‘‹(ฮจ๐‘๐‘–) and ๐‘‹(ฮจ๐‘ค๐‘—) are a pair of orthogonal multiwavelet frames for ๐ฟ2(๐‘…๐‘‘).

Proof. Suppose that ๐‘‹(ฮจ๐‘–), ๐‘‹(ฮจ๐‘—) are a pair of orthogonal MRA multiwavelet frames and ๐‘†(ฮจ๐‘–)=๐‘†(ฮจ๐‘—), then, by the property of MRA multiwavelet frames, for any ๐‘›โ‰ ๐‘šโˆˆ๐‘, we have ๐‘†(๐ด๐‘šฮจ๐‘–)โŸ‚๐‘†(๐ด๐‘›ฮจ๐‘–). Hence, for all ๐‘“1โˆˆ๐‘†(ฮจ๐‘–)0=๐‘†๐‘“1=(๐‘ฅ)๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎“๐‘ โˆˆ๐‘๎ซ๐‘“1(๐‘ฅ),๐œ“๐‘–๐‘™(๐ด๐‘ )๎ฌ๐œ“๐‘ฅโˆ’๐‘˜๐‘—๐‘™(๐ด๐‘ )=๐‘ฅโˆ’๐‘˜๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“1(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐‘ฅโˆ’๐‘˜).(3.16)
For any ๐‘“โˆˆ๐ฟ2(๐‘…๐‘‘), define ๐‘“=๐‘“1+๐‘“2, where ๐‘“1โˆˆ๐‘†(ฮจ๐‘–),๐‘“2โˆˆ(๐ฟ2(๐‘…๐‘‘)โงต๐‘†(ฮจ๐‘–)), then, โŸจ๐‘“1,๐‘“2โŸฉ=0. With this, we get ๐‘†๐‘“2(๐‘ฅ)=๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“2(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐‘ฅโˆ’๐‘˜)=0.(3.17) Hence, for all ๐‘“โˆˆ๐ฟ2(๐‘…๐‘‘), the following equation holds: ๐‘†๐‘“(๐‘ฅ)=๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™=(๐‘ฅโˆ’๐‘˜)๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“1(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐‘ฅโˆ’๐‘˜)+๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“2(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐‘ฅโˆ’๐‘˜)=0.(3.18)
Notice that ฮจ๐‘—โˆถ={๐œ“๐‘—1,โ€ฆ,๐œ“๐‘—๐‘Ÿ}, since ๎„ž๐œ“๐‘™๐‘—๐‘ค(๐œ‰)=๎๐œ“๐‘—๎(๐œ‰)๐‘ค(๐œ‰),1โ‰ค๐‘™โ‰ค๐‘Ÿ, by ๐‘†๐‘“(๐‘ฅ)=0, then ๎„ž=0=๐‘†๐‘“(๐‘ฅ)๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎‚Š๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”=๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“1(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎‚Š๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”+๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“2(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎‚Š๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”๎ƒฉ=๐‘ค(๐œ”)๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“1(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎‚Š๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”+๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“2(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎‚Š๐œ“(๐‘ฅโˆ’๐‘˜)๐‘—๐‘™(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”๎ƒช=๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“1(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎‚ฟ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘™๐‘—๐‘ค(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”+๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“2(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎‚ฟ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘™๐‘—๐‘ค(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”=๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎‚ฟ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘™๐‘—๐‘ค(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”.(3.19)
Applying Fourier inverse transform on (3.19), we have ๎„ก๎„ž=0=๐‘†๐‘“(๐‘ฅ)๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๎ƒ‚๎‚ฟ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘™๐‘—๐‘ค(๐œ”)๐‘’โˆ’2๐œ‹๐‘–๐‘˜๐œ”=๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘–๐‘™(๎ฌ๐œ“๐‘ฅโˆ’๐‘˜)๐‘™๐‘—๐‘ค(๐‘ฅโˆ’๐‘˜).(3.20)
From the above result, we get the following equation: ๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘–๐‘™(๐‘ฅโˆ’๐‘˜)๎ฌ๎ซ๐‘“(๐‘ฅ),๐œ“๐‘™๐‘—๐‘ค๎ฌ=๎„”(๐‘ฅโˆ’๐‘˜)๐‘“(๐‘ฅ),๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘–๐‘™๎ฌ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘™๐‘—๐‘ค๎„•=๎„”(๐‘ฅโˆ’๐‘˜)=โŸจ๐‘“(๐‘ฅ),0โŸฉ๐‘“(๐‘ฅ),๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘™๐‘—๐‘ค(๎ฌ๐œ“๐‘ฅโˆ’๐‘˜)๐‘–๐‘™(๎„•,๐‘ฅโˆ’๐‘˜)(3.21) hence, ๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘™๐‘—๐‘ค๎ฌ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘–๐‘™(๐‘ฅโˆ’๐‘˜)=0.(3.22) Similar to the calculation of (3.19), clearly ๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘™๐‘—๐‘ค๎ฌ๐œ“(๐‘ฅโˆ’๐‘˜)๐‘™๐‘–๐‘(๐‘ฅโˆ’๐‘˜)=0.(3.23)
For any ๐‘ โˆˆ๐‘, ๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘™๐‘—๐‘ค(๐ด๐‘ ๎ฌ๐œ“๐‘ฅโˆ’๐‘˜)๐‘™๐‘–๐‘(๐ด๐‘ ๐‘ฅโˆ’๐‘˜)=๐ด๐‘Ÿโˆ’๐‘ ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“๎€ท๐ดโˆ’๐‘ ๐‘ฅ๎…ž๎€ธ,๐œ“๐‘™๐‘—๐‘ค๎€ท๐‘ฅ๎…ž๐œ“โˆ’๐‘˜๎€ธ๎ฌ๐‘™๐‘–๐‘๎€ท๐‘ฅ๎…ž๎€ธ.โˆ’๐‘˜(3.24) Let ๐‘”(๐‘ฅ)โˆถ=๐‘“(๐ดโˆ’๐‘ ๐‘ฅ). Define operator ๐‘‡โˆถ๐ฟ2(๐‘…๐‘‘)โ†’๐ฟ2(๐‘…๐‘‘);๐‘‡๐‘“(๐‘ฅ)=๐‘”(๐‘ฅ), obviously ๐‘‡ is a surjection operator. If ๐‘  is fixed, for all ๐‘”โˆˆ๐ฟ2(๐‘…๐‘‘), we get ๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘™๐‘—๐‘ค(๐ด๐‘ ๎ฌ๐œ“๐‘ฅโˆ’๐‘˜)๐‘™๐‘–๐‘(๐ด๐‘ ๐‘ฅโˆ’๐‘˜)=๐ดโˆ’๐‘ ๎ƒฉ๐‘Ÿ๎“๐‘™=1๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘”๎€ท๐‘ฅ๎…ž๎€ธ,๐œ“๐‘™๐‘—๐‘ค๎€ท๐‘ฅ๎…ž๐œ“โˆ’๐‘˜๎€ธ๎ฌ๐‘™๐‘–๐‘๎€ท๐‘ฅ๎…ž๎€ธ๎ƒชโˆ’๐‘˜=0.(3.25)
Putting everything together, we have ๐‘Ÿ๎“๐‘™=1๎“๐‘ โˆˆ๐‘๎“๐‘˜โˆˆ๐‘๐‘‘๎ซ๐‘“(๐‘ฅ),๐œ“๐‘™๐‘—๐‘ค(๐ด๐‘ ๎ฌ๐œ“๐‘ฅโˆ’๐‘˜)๐‘™๐‘–๐‘(๐ด๐‘ ๐‘ฅโˆ’๐‘˜)=0,(3.26) then, ๐‘‹(ฮจ๐‘๐‘–) and ๐‘‹(ฮจ๐‘ค๐‘—) are a pair of orthogonal multiwavelet frames.

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

Theorem 3.8. Suppose ๐พ(๐œ‰) is an ๐‘Ÿร—๐‘Ÿ paraunitary matrix with ๐ตโˆ’1๐‘๐‘‘-periodic entries ๐‘Ž๐‘˜,๐‘ (๐œ‰); let ๐พ๐‘—(๐œ‰) denote the jth column. For all ๐›ฝโˆˆฮฅ, suppose ๐‘€={๐‘š0(๐œ‰),๐‘š1(๐œ‰),โ€ฆ,๐‘š๐‘Ÿ(๐œ‰)} and ๐‘€โˆ—(๐œ‰)๐‘€(๐œ‰)=๐ผ2 hold for almost every ๐œ‰, where ๐‘š0 and {๐‘š1,โ€ฆ,๐‘š๐‘™} are low- and high-pass filters, respectively, for a multiwavelet tight frame with scaling function ๐œ™. For ๐‘—=1,โ€ฆ,๐‘Ÿ, define new filters via โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘›๐‘—1,1โ‹ฎ๐‘›(๐œ‰)๐‘—1,๐‘Ÿโ‹ฎ๐‘›(๐œ‰)๐‘—๐‘™,1โ‹ฎ๐‘›(๐œ‰)๐‘—๐‘™,๐‘Ÿ(โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽ๐พ๐œ‰)๐‘—(๐œ‰)๐‘š1โ‹ฎ๐พ(๐œ‰)๐‘—(๐œ‰)๐‘š๐‘™โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(๐œ‰)(3.27) Then, for ๐‘—=1,โ€ฆ,๐‘Ÿ, the affine systems generated by ฮจ๐‘—={๐œ“๐‘—๐‘–,๐‘กโˆถ๐‘–=1,โ€ฆ,๐‘™,๐‘ก=1,โ€ฆ,๐‘Ÿ} obtained via ๎๐œ“๐‘—๐‘–,๐‘ก(๐ต๐œ‰)=๐‘›๐‘—๐‘–,๐‘ก๎๐œ™(๐œ‰)(๐œ‰)(3.28) are multiwavelet tight frames and are pairwise orthogonal.

Proof. Firstly, we prove that ๐‘‹(ฮจ๐‘—),1โ‰ค๐‘—โ‰ค๐‘Ÿ, are multiwavelet tight frames. Assume ๐‘€๐‘—={๐‘š0(๐œ‰),๐‘›๐‘—1,1(๐œ‰),โ€ฆ,๐‘›๐‘—1,๐‘Ÿ(๐œ‰),โ€ฆ,๐‘›๐‘—๐‘™,1(๐œ‰),โ€ฆ,๐‘›๐‘—๐‘™,๐‘Ÿ(๐œ‰)}. Define ๐‘€๐‘—(๐œ‰) according to (3.12): ๐‘€๐‘—(โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘š๐œ‰)=0(๐œ‰)๐‘š0๐‘›(๐œ‰+๐›ฝ)๐‘—1,1(๐œ‰)๐‘›๐‘—1,1(๐‘›๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—1,๐‘Ÿ(๐œ‰)๐‘›๐‘—1,๐‘Ÿ๐‘›(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—๐‘™,1(๐œ‰)๐‘›๐‘—๐‘™,1๐‘›(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—๐‘™,๐‘Ÿ(๐œ‰)๐‘›๐‘—๐‘™,๐‘ŸโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ,(๐œ‰+๐›ฝ)(3.29) where ๐›ฝโˆˆฮฅ. Then, ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—(๐œ‰) is a 2ร—2 matrix. Next, we examine the entries of ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—(๐œ‰) individually. Note that the columns of ๐พ(๐œ‰) have length 1, by ๐‘€โˆ—(๐œ‰)๐‘€(๐œ‰)=๐ผ2, it follows that ๎€บ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—๎€ป(๐œ‰)1,1=||๐‘š0||(๐œ‰)2+๐‘Ÿ๎“๐‘™๐‘˜=1๎“๐‘ก=1||๐‘Ž๐‘˜,๐‘—(๐œ‰)๐‘š๐‘ก||(๐œ‰)2=||๐‘š0||(๐œ‰)2+๐‘Ÿ๎“๐‘˜=1||๐‘Ž๐‘˜,๐‘—||(๐œ‰)2๐‘™๎“๐‘ก=1||๐‘š๐‘ก||(๐œ‰)2=||๐‘š0||(๐œ‰)2+๐‘™๎“๐‘ก=1||๐‘š๐‘ก||(๐œ‰)2=1,(3.30) where [๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—(๐œ‰)]1,1 means the (1,1) entry of the matrix ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—(๐œ‰).
Similarly, ๎€บ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—๎€ป(๐œ‰)2,2=||๐‘š0||(๐œ‰+๐›ฝ)2+๐‘Ÿ๎“๐‘™๐‘˜=1๎“๐‘ก=1||๐‘Ž๐‘˜,๐‘—(๐œ‰+๐›ฝ)๐‘š๐‘ก||(๐œ‰+๐›ฝ)2=||๐‘š0||(๐œ‰+๐›ฝ)2+๐‘Ÿ๎“๐‘˜=1||๐‘Ž๐‘˜,๐‘—||(๐œ‰+๐›ฝ)2๐‘™๎“๐‘ก=1||๐‘š๐‘ก||(๐œ‰+๐›ฝ)2=||๐‘š0||(๐œ‰+๐›ฝ)2+๐‘™๎“๐‘ก=1||๐‘š๐‘ก||(๐œ‰+๐›ฝ)2=1.(3.31)
Now, since the entries of ๐พ(๐œ‰) are ๐ตโˆ’1๐‘๐‘‘-periodic, again by ๐‘€โˆ—(๐œ‰)๐‘€(๐œ‰)=๐ผ2, ๎€บ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—๎€ป(๐œ‰)1,2=๐‘š0(๐œ‰+๐›ฝ)๐‘š0(๐œ‰)+๐‘Ÿ๎“๐‘™๐‘˜=1๎“๐‘ก=1๐‘Ž๐‘˜,๐‘—(๐œ‰)๐‘š๐‘ก(๐œ‰)๐‘Ž๐‘˜,๐‘—(๐œ‰+๐›ฝ)๐‘š๐‘ก(๐œ‰+๐›ฝ)=๐‘š0(๐œ‰+๐›ฝ)๐‘š0(๐œ‰)+๐‘Ÿ๎“๐‘˜=1||๐‘Ž๐‘˜,๐‘—||(๐œ‰)2๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ)=๐‘š0(๐œ‰+๐›ฝ)๐‘š0(๐œ‰)+๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ)=0.(3.32) Finally, the (2,1)-entry must be zero by conjugate symmetry of ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—(๐œ‰). Hence, ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—(๐œ‰)=๐ผ2,1โ‰ค๐‘—โ‰ค๐‘Ÿ.(3.33) Putting everything together, from Theorem 3.6, the affine systems generated by {๐œ“๐‘—๐‘–,๐‘กโˆถ๐‘–=1,โ€ฆ,๐‘™,๐‘ก=1,โ€ฆ,๐‘Ÿ} obtained via ๎๐œ“๐‘—๐‘–,๐‘ก(๐ต๐œ‰)=๐‘›๐‘—๐‘–,๐‘ก๎๐œ™(๐œ‰)(๐œ‰)(3.34) are multiwavelet tight frames.
For orthogonality, according to (3.12), for ๐‘—=1,โ€ฆ,๐‘Ÿ, we have ๎‚‹๐‘€๐‘—โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘›(๐œ‰)=๐‘—1,1(๐œ‰)๐‘›๐‘—1,1๐‘›(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—1,๐‘Ÿ(๐œ‰)๐‘›๐‘—1,๐‘Ÿ๐‘›(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—๐‘™,1(๐œ‰)๐‘›๐‘—๐‘™,1๐‘›(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—๐‘™,๐‘Ÿ(๐œ‰)๐‘›๐‘—๐‘™,๐‘Ÿ(โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽ๐พ๐œ‰+๐›ฝ)๐‘—(๐œ‰)๐‘š1(๐œ‰)๐พ๐‘—(๐œ‰+๐›ฝ)๐‘š1๐พ(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—(๐œ‰)๐‘š๐‘™(๐œ‰)๐พ๐‘—(๐œ‰+๐›ฝ)๐‘š๐‘™โŽžโŽŸโŽŸโŽŸโŽŸโŽ .(๐œ‰+๐›ฝ)(3.35) If 1โ‰ค๐‘—โ‰ ๐‘—โ€ฒโ‰ค๐‘Ÿ, then ๎‚‹๐‘€โˆ—๐‘—(๎‚‹๐‘€๐œ‰)๐‘—โ€ฒ(=โŽ›โŽœโŽœโŽœโŽœโŽ๐พ๐œ‰)๐‘—(๐œ‰)๐‘š1(๐œ‰)๐พ๐‘—(๐œ‰+๐›ฝ)๐‘š1๐พ(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—(๐œ‰)๐‘š๐‘™(๐œ‰)๐พ๐‘—(๐œ‰+๐›ฝ)๐‘š๐‘™โŽžโŽŸโŽŸโŽŸโŽŸโŽ (๐œ‰+๐›ฝ)โˆ—โŽ›โŽœโŽœโŽœโŽœโŽ๐พ๐‘—โ€ฒ(๐œ‰)๐‘š1(๐œ‰)๐พ๐‘—โ€ฒ(๐œ‰+๐›ฝ)๐‘š1๐พ(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘—โ€ฒ(๐œ‰)๐‘š๐‘™(๐œ‰)๐พ๐‘—โ€ฒ(๐œ‰+๐›ฝ)๐‘š๐‘™โŽžโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐พ(๐œ‰+๐›ฝ)โˆ—๐‘—(๐œ‰)๐พ๐‘—โ€ฒ(๐œ‰)๐‘™๎“๐‘ก=1||๐‘š๐‘ก||(๐œ‰)2๐พโˆ—๐‘—(๐œ‰)๐พ๐‘—โ€ฒ(๐œ‰+๐›ฝ)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก๐พ(๐œ‰+๐›ฝ)โˆ—๐‘—(๐œ‰+๐›ฝ)๐พ๐‘—โ€ฒ(๐œ‰)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰+๐›ฝ)๐‘š๐‘ก(๐œ‰)๐พโˆ—๐‘—(๐œ‰+๐›ฝ)๐พ๐‘—โ€ฒ(๐œ‰+๐›ฝ)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰+๐›ฝ)๐‘š๐‘กโŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ =โŽ›โŽœโŽœโŽœโŽœโŽœโŽ๐พ(๐œ‰+๐›ฝ)โˆ—๐‘—(๐œ‰)๐พ๐‘—โ€ฒ(๐œ‰)๐‘™๎“๐‘ก=1||๐‘š๐‘ก||(๐œ‰)2๐พโˆ—๐‘—(๐œ‰)๐พ๐‘—โ€ฒ(๐œ‰)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก๐พ(๐œ‰+๐›ฝ)โˆ—๐‘—(๐œ‰)๐พ๐‘—โ€ฒ(๐œ‰)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰+๐›ฝ)๐‘š๐‘ก(๐œ‰)๐พโˆ—๐‘—(๐œ‰)๐พ๐‘—โ€ฒ(๐œ‰)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰+๐›ฝ)๐‘š๐‘ก(โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽ ๐œ‰+๐›ฝ)=0,(3.36) where we use the fact that the product of the two matrices ๐พโˆ—๐‘—(๐œ‰)๐พ๐‘—๎…ž(๐œ‰) is 0 by the orthogonality of the columns of ๐พ(๐œ‰). 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 ๐œ™ is compactly supported, the paraunitary matrix K in Theorem 3.8 must have entries which are ๐ตโˆ’1๐‘๐‘‘-periodic.

Proof. The proof will follow the notation of Theorem 3.8. For 1โ‰ค๐‘—โ‰ค๐‘Ÿ, for all ๐œ‰โˆˆ๐ตโˆ’1๐‘๐‘‘/๐‘๐‘‘, the matrix ๐‘€๐‘—(โŽ›โŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽœโŽ๐‘š๐œ‰)=0(๐œ‰)๐‘š0๐‘Ž(๐œ‰+๐›ฝ)1,๐‘—(๐œ‰)๐‘š1(๐œ‰)๐‘Ž1,๐‘—(๐œ‰+๐›ฝ)๐‘š1(๐‘Ž๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘Ÿ,๐‘—(๐œ‰)๐‘š1(๐œ‰)๐‘Ž๐‘Ÿ,๐‘—(๐œ‰+๐›ฝ)๐‘š1๐‘Ž(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ1,๐‘—(๐œ‰)๐‘š๐‘™(๐œ‰)๐‘Ž1,๐‘—(๐œ‰+๐›ฝ)๐‘š๐‘™๐‘Ž(๐œ‰+๐›ฝ)โ‹ฎโ‹ฎ๐‘Ÿ,๐‘—(๐œ‰)๐‘š๐‘™(๐œ‰)๐‘Ž๐‘Ÿ,๐‘—(๐œ‰+๐›ฝ)๐‘š๐‘™โŽžโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽŸโŽ (๐œ‰+๐›ฝ)(3.37) satisfies the equation ๐‘€โˆ—๐‘—(๐œ‰)๐‘€๐‘—(๐œ‰)=๐ผ2a.e.๐œ‰.(3.38) Then, for almost every ๐œ‰, the following equation ๐‘š0(๐œ‰+๐›ฝ)๐‘š0(๐œ‰)+๐‘Ÿ๎“๐‘˜=1๐‘Ž๐‘˜,๐‘—(๐œ‰)๐‘Ž๐‘˜,๐‘—(๐œ‰+๐›ฝ)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ)=0(3.39) must hold. Notice that ๐‘š0 and {๐‘š1,โ€ฆ,๐‘š๐‘™} are low- and high-pass filters, respectively, which meet Theorem 3.8. Then, ๐‘š0(๐œ‰+๐›ฝ)๐‘š0(๐œ‰)+๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ)=0.(3.40) Thus, we have ๐‘š0(๐œ‰+๐›ฝ)๐‘š0(๐œ‰)=โˆ’๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ).(3.41)
From the above results, we get the following equation: 0=๐‘š0(๐œ‰+๐›ฝ)๐‘š0(๐œ‰)+๐‘Ÿ๎“๐‘˜=1๐‘Ž๐‘˜,๐‘—(๐œ‰)๐‘Ž๐‘˜,๐‘—(๐œ‰+๐›ฝ)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ)=โˆ’๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ)+๐‘Ÿ๎“๐‘˜=1๐‘Ž๐‘˜,๐‘—(๐œ‰)๐‘Ž๐‘˜,๐‘—(๐œ‰+๐›ฝ)๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก=๎ƒฉ(๐œ‰+๐›ฝ)๐‘Ÿ๎“๐‘˜=1๐‘Ž๐‘˜,๐‘—(๐œ‰)๐‘Ž๐‘˜,๐‘—๎ƒช(๐œ‰+๐›ฝ)โˆ’1๐‘™๎“๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ).(3.42)
Hence, โˆ‘๐‘™๐‘ก=1๐‘š๐‘ก(๐œ‰)๐‘š๐‘ก(๐œ‰+๐›ฝ)=0 or โˆ‘๐‘Ÿ๐‘˜=1๐‘Ž๐‘˜,๐‘—(๐œ‰)๐‘Ž๐‘˜,๐‘—(๐œ‰+๐›ฝ)=1. If ๐œ™ 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 ๐‘Ž๐‘˜,๐‘—(๐œ‰) and ๐‘Ž๐‘˜,๐‘—(๐œ‰+๐›ฝ), each of which has length 1. Applying Cauchy-Schwarz inequation yields that the two vectors must be identical for almost every ๐œ‰.

4. Conclusion

In this paper, motivated by the notion of orthogonal frames, we present the construction of orthogonal multiwavelet frames in ๐ฟ2(๐‘…๐‘‘) 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).

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Copyright ยฉ 2012 Liu Zhanwei et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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