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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 640789, 37 pages
http://dx.doi.org/10.1155/2012/640789
Research Article

Minimum-Energy Multiwavelet Frames with Arbitrary Integer Dilation Factor

1Institute of Information and System Science, Beifang University of Nationalities, Yinchuan 750021, China
2School of Information Science & Technology, East China Normal University, Shanghai 200241, China

Received 15 May 2012; Accepted 26 June 2012

Academic Editor: Carlo Cattani

Copyright © 2012 Yongdong Huang 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.

Abstract

In order to organically combine the minimum-energy frame with the significant properties of multiwavelets, minimum-energy multiwavelet frames with arbitrary integer dilation factor are studied. Firstly, we define the concept of minimum-energy multiwavelet frame with arbitrary dilation factor and present its equivalent characterizations. Secondly, some necessary conditions and sufficient conditions for minimum-energy multiwavelet frame are given. Thirdly, the decomposition and reconstruction formulas of minimum-energy multiwavelet frame with arbitrary integer dilation factor are deduced. Finally, we give several numerical examples based on B-spline functions.

1. Introduction

Wavelets transform has been widely applied to information processing, image processing, computer science, mathematical physics, engineering, and so on. As you all know, it is not possible for any orthogonal scaling wavelet function with compact support to be symmetric, except for the Haar wavelets. In 1993, Goodman and Lee [1] established the multiwavelet theory by introducing the multiresolution analysis (MRA) with multiplicity 𝑟, and gave the spline multiwavelet examples. Using the fractal interpolation technology, Geronimo et al. [2] constructed the GHM multiwavelet which have short support, (anti)symmetry, orthogonality and vanishing moment with order 2 in 1994. From then on, multiwavelet has been a hot research area. In 1996, Chui and Lian [3] reconstructed the GHM multiwavelet without using the fractal interpolation technology, and they gave the general method on constructing the multiwavelet with short support, (anti)symmetry, and orthogonality. After that, Plonka and Strela [4] used two-scale similarity transforms (TSTs) to raise the approximation order of multiwavelet and gave the important conclusions of the two-scale matrix symbol's factorizations and so on. And, by Lawton et al. [5], the construction of multiwavelet has been transformed into matrix extension problem in 1996. The construction theory of multiwavelet had a great development after Jiang [6, 7] putting forward a series of effective methods. Whether wavelets or multiwavelet, they require that the integer shifts of the scaling function form Riesz bases, orthogonal basis, or biorthogonal basis for its span space. And this will cause some defects: (1) the computational complexity can be increased during the course of decomposition and reconstruction; (2) the numerical instability can be caused during the procedure of reconstructing original signal (3) in the biorthogonal case, the analysis filter bank can not replaced by the synthetic filter bank, and vice verse.

Fortunately, besides orthogonal wavelets and multiwavelet minimum-energy frames can effectively avoid the difficulty which is caused by different bases functions during the course of decomposition and reconstruction, still use the same wavelets both for analysis and synthesis. The theory of frames comes from signal processing firstly. It was introduced by Duffin and Schaffer to deal with problems in nonharmonic Fourier series. But in a long time after that, people did not pay enough attention to it. After Daubechies et al. [8] defined affine frames (wavelets frames) by combining the theory of continuous wavelets transforms and frames while wavelets theory was booming, people start to research frames and its application again. Benedetto and Li [9] gave the definition of frame multiresolution analysis (FMRA), and their work laid the foundation for other people's further investigation. Frames cannot only overcome the disadvantages of wavelets and multiwavelet, but also increase redundancy properly, then the numerical computation become much more stable using frames to reconstruct signal. With well time-frequency localization and shift invariance, frames can be designed more easily than wavelets or multiwavelet. Nowadays frames have been used widely in theoretical and applied domain [1022], such as signal analysis, image processing, numerical calculation, Banach space theory, Besov space theory, and so on.

In 2000, Chui and He [11] proposed the concept of minimum-energy wavelets frames. The minimum-energy wavelets frames reduce the computational complexity, maintain the numerical stability, and do not need to search dual frames in the decomposition and reconstruction of functions (or signals). Therefore, many people pay a lot of attention to the study of minimum-energy wavelets frames. Huang and Cheng [15] studied the construction and characterizations of the minimum-energy with arbitrary integer dilation factor. Gao and Cao [18] researched the structure of the minimum-energy wavelets frames on the interval and its application on signal denoising systematically. Liang and Zhao [23] studied the minimum-energy multiwavelet frames with dilation factor 2 and multiplicity 2 and gave a characterization and a necessary condition of minimum-energy multiwavelet frames. Unfortunately, the authors did not give the sufficient conditions of minimum-energy multiwavelet frames. In fact, people need to pay close attention to the existence of sufficient conditions of minimum-energy wavelet frames in most cases. On the other hand B-spline functions which are the convolution of Shannon wavelets [2426]. It can be seen that also Shannon wavelets are minimum-energy wavelets. In this paper, in order to organically combine the minimum-energy frame with the significant properties of multiwavelet, minimum-energy multiwavelet frames with arbitrary integer dilation factor are studied. Firstly, we define the concept of minimum-energy multiwavelet frame with arbitrary dilation factor and present its equivalent characterizations. Secondly, some necessary conditions and sufficient conditions for minimum-energy multiwavelet frame are given; Thirdly, the decomposition and reconstruction formulas of minimum-energy multiwavelet frame with arbitrary integer dilation factor and the multiplicity 𝑟 are deduced. Finally, we give several numerical examples based on B-spline functions.

Let us now describe the organization of the material that as follows. Section 2 is preliminaries and basic definitions. Section 3 is main result. In Section 4, we give the decomposition and reconstruction formulas of minimum-energy multiwavelet frame. Section 5 is numerical examples.

2. Preliminaries and Basic Definitions

Throughout this paper, let , , and denote the set of integers, real numbers, and complex numbers respectively; 𝑎 with 𝑎2, 𝜔𝑗=cos(2𝑗𝜋/𝑎)+𝑖sin(2𝑗𝜋/𝑎), 𝑗=0,1,,𝑎1.

A multiscaling function vector (refinable function vector) is a vector-valued function: 𝜙Φ=1(𝑥),,𝜙𝑟(𝑥)𝑇,𝜙𝑙(𝑥)𝐿2(),𝑙=1,,𝑟,(2.1) which satisfies a two-scale matrix refinement equation of the form: Φ(𝑥)=𝑘𝑃𝑘Φ(𝑎𝑥𝑘),𝑥,(2.2)𝑟 is called the multiplicity of Φ, the integer 𝑎 is said to be dilation factor. The recursion coefficients {𝑃𝑘}𝑘 are 𝑟×𝑟 matrices.

The Fourier transform of the formula (2.2) is Φ𝜔Φ(𝜔)=𝑃(𝑧)𝑎,𝑧=𝑒𝑖𝜔/𝑎,(2.3) where 1𝑃(𝑧)=𝑎𝑘𝑃𝑘𝑧𝑘.(2.4)𝑃(𝑧) is the symbol of the matrix sequence {𝑃𝑘}𝑘.

The multiresolution analysis (MRA) with multiplicity 𝑟 and dilation factor 𝑎 generated by Φ(𝑥) is defined as 𝑉𝑗=𝜙span𝜏,𝑗,𝑘,1𝜏𝑟,𝑘,𝑗(2.5) where 𝜙𝜏,𝑗,𝑘=𝑎𝑗/2𝜙𝜏(𝑎𝑗𝑥𝑘), and the sequence of closed subspace of 𝐿2() has the following properties:(1) 𝑉𝑗𝑉𝑗+1, 𝑗.(2) 𝑗𝑉𝑗=𝐿2(), 𝑗𝑉𝑗={0};(3) 𝑓(𝑥)𝑉𝑗𝑓(𝑎𝑥)𝑉𝑗+1, for all 𝑗;(4) 𝑓(𝑥)𝑉j𝑓(𝑥𝑎𝑗𝑘)𝑉𝑗, for all 𝑘,𝑗;(5) {𝜙𝜏,0,𝑘1𝜏𝑟,𝑘𝐙} forms a Riesz basis of 𝑉0;

Definition 2.1. A finite family vector-valued function Ψ𝑖=(𝜓𝑖1,,𝜓𝑖𝑟)𝑇, 𝑖=1,,𝑁 generates a multiwavelet frames for 𝐿2(), if there exist constants 0<𝐴𝐵< such that for any 𝑓(𝑥)𝐿2()𝐴𝑓2𝑁𝑟𝑖=1𝜏=1𝑗,𝑘|||𝑓,𝜓𝑖𝜏,𝑗,𝑘|||2𝐵𝑓2,(2.6) where 𝜓𝑖𝜏,𝑗,𝑘=𝑎𝑗/2𝜓𝑖𝜏(𝑎𝑗𝑥𝑘).

Definition 2.2. A nested subspace generated by a multiscaling vector-valued function Φ(𝑥) satisfies formula (2.5) and its additional conditions, then finite family vector-valued function {Ψ1,,Ψ𝑁} generates a frame multiresolution analysis associated the vector-valued function Φ(𝑥), if the finite family Ψ𝑖=(𝜓𝑖1,,𝜓𝑖𝑟)𝑇, 𝑖=1,,𝑁 satisfies the formulation (2.6) with 𝜓𝑖𝜏𝑉1, 𝑖=1,,𝑁; 𝜏=1,,𝑟.

Definition 2.3. Let Φ(𝑥)=(𝜙1(𝑥),,𝜙𝑟(𝑥))𝑇, with 𝜙𝜏𝐿()𝐿2(), 𝜏=1,,𝑟, Φ continuous at 0 and Φ(0)0, be a multiscaling vector-valued function that generates the nested subspace {𝑉𝑗}𝑗𝐙 in the sense of (2.5). Then a finite family vector-valued function {Ψ1,,Ψ𝑁}𝑉1 is called a minimum-energy multiwavelet frames associated with Φ(𝑥), if for for all 𝑓𝐿2()𝑟𝜏=1𝑘||𝑓,𝜙𝜏,1,𝑘||2=𝑟𝜏=1𝑘||𝑓,𝜙𝜏,0,𝑘||2+𝑁𝑟𝑖=1𝜏=1𝑘|||𝑓,𝜓𝑖𝜏,0,𝑘|||2.(2.7)

Remark 2.4. By the Parseval identity, minimum-energy multiwavelet frames {Ψ1,,Ψ𝑁} must be tight frames for 𝐿2() with frames bound equal to 1.

Remark 2.5. The formula (2.7) is equivalent to the following formulas: 𝑟𝜏=1𝑘𝑓,𝜙𝜏,1,𝑘𝜙𝜏,1,𝑘=𝑟𝜏=1𝑘𝑓,𝜙𝜏,0,𝑘𝜙𝜏,0,𝑘+𝑁𝑟𝑖=1𝜏=1𝑘𝑓,𝜓𝑖𝜏,0,𝑘𝜓𝑖𝜏,0,𝑘.(2.8) The interpretation of minimum energy will be clarified later.

3. Main Result

In this section, we will give a complete characterization of minimum-energy multiwavelet frames associated with some given multiscaling vector-valued function in term of their two-scale symbols. Let Φ(𝑥)=(𝜙1(𝑥),,𝜙𝑟(𝑥))𝑇 with 𝜙𝜏𝐿()𝐿2(), 𝜏=1,,𝑟, Φ continuous at 0, and Φ(0)0 be a multiscaling vector-valued function which satisfies (2.2)–(2.5). Consider {Ψ1,,Ψ𝑁}𝑉1, then Ψ𝑙(𝑥)=𝑘𝑄𝑙𝑘Φ(𝑎𝑥𝑘),(3.1) where {𝑄𝑙𝑘}𝑘, 𝑙=1,,𝑁 are 𝑟×𝑟 matrices. Using Fourier transform on (3.1), we can get their symbols as follows: 𝑄𝑙1(𝑧)=𝑎𝑘𝑄𝑙𝑘𝑧k,𝑙=1,,𝑁.(3.2)

With 𝑃(𝑧), 𝑄𝑙(𝑧), 𝑙=1,,𝑁, we formulate the (𝑁+1)𝑟×𝑎𝑟 block matrix as follows: 𝜔𝑅(𝑧)=𝑃(𝑧)𝑃1𝑧𝜔𝑃𝑎1𝑧𝑄1(𝑧)𝑄1𝜔1𝑧𝑄1𝜔𝑎1𝑧𝑄𝑁(𝑧)𝑄𝑁𝜔1𝑧𝑄𝑁𝜔𝑎1𝑧,(3.3) and the 𝑅(𝑧) denotes the complex conjugate of the transpose of 𝑅(𝑧).

The following theorem presents the equivalent characterizations of the minimum-energy multiwavelet frames with arbitrary integer dilation factor.

Theorem 3.1. Suppose that every element of the symbols, 𝑃(𝑧), 𝑄𝑙(𝑧), 𝑙=1,,𝑁, in (2.4) and (3.2) is a Laurent polynomial, and the multiscaling vector-valued function Φ(𝑥) associated with 𝑃(𝑧) generates a nested subspace {𝑉𝑗}𝑗𝐙. Then the following statements are equivalent:(1){Ψ1,,Ψ𝑁} is a minimum-energy multiwavelet frames associated with Φ(𝑥):(2) 𝑅(𝑧)𝑅(𝑧)=𝐼𝑎𝑟𝑓𝑜𝑟|𝑧|=1;(3.4)(3)𝛼𝑚𝑙,𝑖𝑗=0,𝑚,𝑙;𝑖,𝑗=1,,𝑟,(3.5)where 𝛼𝑚𝑙,𝑖𝑗=𝑟𝑘𝐙𝜏=1𝑃𝜏𝑖𝑙𝑎𝑘𝑃𝜏𝑗𝑚𝑎𝑘+𝑁𝑡=1𝑄𝑡,𝜏𝑖𝑙𝑎𝑘𝑄𝑡,𝜏𝑗𝑚𝑎𝑘𝑎𝛿𝑚𝑙,𝑖𝑗,𝛿𝑚𝑙,𝑖𝑗=1,𝑚=𝑙,𝑖=𝑗,0,𝑒𝑙𝑠𝑒.(3.6)

Proof. By using the two-scale relations (2.2) and (3.1) and notation 𝛼𝑚𝑙,𝑖𝑗 for for all 𝑓𝐿2(), (2.8) can be written as 𝑙𝑟𝑚𝑟𝑖=1𝑗=1𝛼𝑚𝑙,𝑖𝑗𝑓,𝜙𝑖(𝑎𝑥𝑚)𝜙𝑗(𝑎𝑥𝑙)=0.(3.7) On the other hand, (3.4) can be reformulated as 𝑃(𝑧)𝑃(𝑧)+𝑁𝑡=1𝑄𝑡(𝑧)𝑄𝑡(𝑧)=𝐼𝑟,𝑃𝜔(𝑧)𝑃𝑗𝑧+𝑁𝑡=1𝑄𝑡(𝑧)𝑄𝑡𝜔𝑗𝑧=0𝑟,𝑗=1,2,,𝑎1;|𝑧|=1,(3.8) and it is equivalent to 𝑎1𝑘=0𝑃𝜔𝑘𝑧𝑃(𝑧)+𝑁𝑡=1𝑎1𝑘=0𝑄𝑡𝜔𝑘𝑧𝑄𝑡(𝑧)=𝐼𝑟,𝑃(𝑧)𝑎1𝑘=1𝑃𝜔𝑘𝑧𝑃(𝑧)+𝑁𝑡=1𝑄𝑡(𝑧)𝑎1𝑘=1𝑄𝑡𝜔𝑘𝑧𝑄𝑡(𝑧)=𝐼𝑟,𝑎1𝑘=0𝑃𝜔𝑘𝑧2𝑃𝜔𝑙𝑧𝑃(𝑧)+𝑁𝑡=1𝑎1𝑘=0𝑄𝑡𝜔𝑘𝑧2𝑄𝑡𝜔𝑙𝑧𝑄𝑡(𝑧)=𝐼𝑟,𝑙=1,2,,𝑎1;|𝑧|=1.(3.9) With |𝑧|=1,𝑧𝑘=𝑧𝑘,𝜔𝑘𝑙=𝜔𝑙𝑘=𝜔𝑘𝑙, and 𝑎1𝑙=0𝜔𝑘𝑙=𝑎1𝑙=0𝜔𝑙𝑘=0𝜔𝑘1𝑎𝜔𝑘=1,(3.10) the formulation (3.9) is equivalent to 𝑘𝑃𝑎𝑘𝑧𝑎𝑘𝑃(𝑧)+𝑁𝑡=1𝑘𝑄𝑡𝑎𝑘𝑧𝑎𝑘𝑄𝑡(𝑧)=𝐼𝑟,𝑎1𝑙=1𝑘𝑃𝑙𝑎𝑘𝑧𝑎𝑘𝑙𝑃(𝑧)+𝑁𝑡=1𝑎1𝑙=1𝑘𝑄𝑡𝑙𝑎𝑘𝑧𝑎𝑘𝑙𝑄𝑡(𝑧)=(𝑎1)𝐼𝑟,𝑎1𝑙=1𝑒(2𝑠𝑙𝜋/𝑎)𝑖𝑘𝑃𝑙𝑎𝑘𝑧𝑎𝑘𝑙𝑃(𝑧)+𝑁𝑡=1𝑎1𝑙=1𝑒(2𝑠𝑙𝜋/𝑎)𝑖𝑘𝑄𝑡𝑙𝑎𝑘𝑧𝑎𝑘𝑙𝑄𝑡(𝑧)=𝐼𝑟,𝑠=1,2,,𝑎1;|𝑧|=1.(3.11)
Using the properties of roots of unity, the Vandermonde matrix and Cramer's rule, the above equation is equivalent to 𝑘𝑃𝑎𝑘𝑧𝑎𝑘𝑃(𝑧)+𝑁𝑡=1𝑘𝑄𝑡𝑎𝑘𝑧𝑎𝑘𝑄𝑡(𝑧)=𝐼𝑟,𝑘𝑃1𝑎𝑘𝑧𝑎𝑘1𝑃(𝑧)+𝑁𝑡=1𝑘𝑄𝑡1𝑎𝑘𝑧𝑎𝑘1𝑄𝑡(𝑧)=𝐼𝑟,𝑘𝑃𝑎1𝑎𝑘𝑧𝑎𝑘𝑎+1𝑃(𝑧)+𝑁𝑡=1𝑘𝑄𝑡𝑎1𝑎𝑘𝑧𝑎𝑘𝑎+1𝑄𝑡(𝑧)=𝐼𝑟.(3.12)
We multiply the identities in (3.12) by Φ(𝜔/𝑎)𝑧𝑙, 𝑙=0,1,,𝑎1, respectively, where 𝑧=𝑒𝑖𝜔/𝑎, to get 𝑘𝑃𝑙𝑎𝑘𝑧𝑎𝑘Φ𝜔𝑃(𝑧)𝑎+𝑁𝑡=1𝑄𝑡𝑙𝑎𝑘𝑧𝑎𝑘𝑄𝑡Φ𝜔(𝑧)𝑎=Φ𝜔𝑎𝑧𝑙,𝑙=0,,𝑎1.(3.13) Hence, (3.12) is equivalent to 𝑘𝑃𝑙𝑎𝑘𝑧𝑎𝑘Φ(𝜔)+𝑁𝑡𝑖=1𝑄𝑡𝑙𝑎𝑘𝑧𝑎𝑘Ψ𝑡=Φ𝜔(𝜔)𝑎𝑒𝑖𝑙𝜔/𝑎,𝑙=0,,𝑎1(3.14) or 𝑘𝑃𝑙𝑎𝑘𝑧𝑎𝑘Φ(𝑥𝑘)+𝑁𝑡=1𝑄𝑡𝑙𝑎𝑘𝑧𝑎𝑘Ψ𝑡(𝑥𝑘)=𝑎Φ(𝑎𝑥𝑙),𝑙=0,,𝑎1,(3.15) which can be reformulated as 𝑘𝑃𝑙𝑎𝑘𝑧𝑎𝑘Φ(𝑥𝑘)+𝑁𝑡=1𝑄𝑡𝑙𝑎𝑘𝑧𝑎𝑘Ψ𝑡(𝑥𝑘)=𝑎Φ(𝑎𝑥𝑙).(3.16) By using the two-scaling relations (2.2) and (3.1), we can rewrite (3.16) as 𝑚𝑎𝑗=1𝛼𝑚𝑙,𝑖𝑗𝜙𝑗(𝑎𝑥𝑚)=0,𝑖=1,,𝑟;𝑙.(3.17)
In conclusion, the proof of Theorem 3.1 reduces to the proof of the equivalence of (3.5), (3.7), and (3.17).
It is obvious that (3.5)(3.17)(3.7). To show (3.7)(3.5), let 𝑓𝐿2() be any compactly supported function. By using the properties that for every fixed 𝑚, 𝛼𝑚𝑙,𝑖𝑗=0 expect for finitely many 𝑙,𝑖,𝑗, then the functional 𝛽𝑙𝑗(𝑓)=𝑚𝑟𝑖=1𝛼𝑚𝑙,𝑖𝑗𝑓,𝜙𝑖(𝑎𝑥𝑚)(3.18) just has finite nonzero for 𝑙, 𝑗=1,,𝑟.
Using the property of Fourier transform, we obtain 𝑙𝑟𝑗=1𝛽𝑙𝑗𝜙(𝑓)𝑗(𝜔)𝑒𝑖𝑙𝜔/𝑎=0.(3.19) Since 𝜙𝑙(𝜔) is nontrivial function, then 𝛽𝑙𝑗(𝑓)=0, 𝑙, 𝑗=1,,𝑟, in other words, we have 𝑓,𝑚𝑟𝑖=1𝛼𝑚𝑙,𝑖𝑗𝜙𝑖(𝑎𝑥𝑚)=0,𝑙,𝑗=1,,𝑟.(3.20)
Then the series in the above equation is a finite sum and hence represents a compactly supported function in 𝐿2(). By choosing 𝑓 to be this function, it follows that 𝑚𝑟𝑖=1𝛼𝑚𝑙,𝑖𝑗𝜙𝑖(𝑎𝑥𝑚)=0,(3.21) which implies that the trigonometric polynomial 𝑚𝑟𝑖=1𝛼𝑚𝑙,𝑖𝑗𝜙𝑖(𝜔)𝑒𝑖𝑚𝜔 is identically equal to 0 so that 𝛼𝑚𝑙,𝑖𝑗=0, for all 𝑚,𝑙;𝑖,𝑗=1,,𝑟.
We complete the proof of Theorem 3.1 because the set of compactly supported functions is dense in 𝐿2().

Theorem 3.1 characterizes the necessary and sufficient condition for the existence of the minimum-energy multiwavelet frames associated with Φ. However it is not a good choice to use this theorem to construct the minimum-energy multiwavelet frames. For convenience, we need to present some sufficient conditions in terms of the symbols.

In this paper, we just discuss the minimum-energy frames with compact support, that is, every element of symbols is Laurent polynomial.

Theorem 3.2. A compactly supported refinable vector-valued function Φ(𝑥)=(𝜙1(𝑥),,𝜙𝑟(𝑥))𝑇, with Φ continuous at 0 and Φ(0)0. Let {Ψ1,,Ψ𝑁} be the minimum-energy multiwavelet frames associated with it, then 𝑟𝑖=1||𝑝𝑖𝑗(𝜔𝑙||𝑧)21|𝑧|=1,1𝑗𝑟,0𝑙𝑎1,(3.22)𝑎1𝑟𝑙=0𝑗=1||𝑝𝑖𝑗(𝜔𝑙||𝑧)21|𝑧|=1,1𝑖𝑟.(3.23)

Proof. Using Theorem 3.1, it is clear to show that the 𝑙2-norm of every row vector of the symbol for Φ is less than 1, in other words, (3.22) is valid. In order to prove (3.23), let 𝑖=1. First, we set 𝑝𝑓(𝑧)=11(𝑧)𝑝1𝑟(𝑧)𝑝11𝜔𝑎1𝑧𝑝1𝑟𝜔𝑎1𝑧,(3.24) and the rest of 𝑅(𝑧) removed 𝑓(𝑧) as 𝐹(𝑧). Then we can reformulate (3.4) as 𝑓(𝑧)𝑓(𝑧)+𝐹(𝑧)𝐹(𝑧)=𝐼𝑎𝑟,(3.25) or equivalently, 𝐹(𝑧)𝐹(𝑧)=𝐼𝑎𝑟𝑓(𝑧)𝑓(𝑧), which is a nonnegative definite Hermitian matrix for |𝑧|=1 so that 𝐼det𝑎𝑟𝑓(𝑧)𝑓(𝑧)0|𝑧|=1,(3.26) and this gives 𝑎1𝑟𝑙=0𝑗=1||𝑝1𝑗(𝜔𝑙||𝑧)21|𝑧|=1.(3.27)
In fact, we have 𝐼𝑎𝑟𝑓(𝑧)𝐼𝑓(𝑧)1𝑎𝑟𝑓(𝑧)=𝐼𝑓(𝑧)1𝑎𝑟𝑓(𝑧)𝑓(𝑧)001𝑓(𝑧)𝑓(𝑧),𝐼det𝑎𝑟𝑓(𝑧)𝐼𝑓(𝑧)1=det𝑎𝑟𝑓(𝑧)01𝑓(𝑧)𝑓(𝑧),𝐼det𝑎𝑟𝑓(𝑧)𝐼𝑓(𝑧)1=det𝑎𝑟𝑓(𝑧)01𝑓(𝑧)𝑓(𝑧),(3.28) then 𝐼det𝑎𝑟𝑓(𝑧)𝑓(𝑧)1𝑓(𝑧)𝑓(𝑧)=1𝑓(𝑧)𝑓(𝑧)2,(3.29) and it gives 1𝑓(𝑧)𝑓(𝑧)0, for all |𝑧|=1, that is, 𝑎1𝑟𝑙=0𝑗=1||𝑝𝑖𝑗(𝜔𝑙||𝑧)21|𝑧|=1,1𝑖𝑟.(3.30) The proof of Theorem 3.2 is completed.

Remark 3.3. By the proof of Theorem 3.2, we know that the restriction in Theorem 3.2 on the two-scale symbol 𝑃(𝑧) of a refinable vector-valued function Φ(𝑥) is a necessary condition for the existence of a minimum-energy frames associated with Φ(𝑥) via the rectangular unitary matrix extension approach, even if Φ(𝑥) is not compactly supported.

Remark 3.4. For a certain compactly supported refinable vector-valued function, it cannot exist in minimum-energy frames.
We write 𝑃(𝑧), 𝑄𝑗(𝑧), 𝑗=1,,𝑁 in their polyphase forms: 𝑃(𝑧)=𝑎𝑎𝑃1(𝑧𝑎)+𝑧𝑃2(𝑧𝑎)++𝑧𝑎1𝑃𝑎(𝑧𝑎),𝑄(3.31)𝑗(𝑧)=𝑎𝑎𝑄𝑗1(𝑧𝑎)+𝑧𝑄𝑗2(𝑧𝑎)++𝑧𝑎1𝑄𝑗𝑎(𝑧𝑎),𝑗=1,,𝑁,(3.32) where 𝑃𝑖(𝑧), 𝑄𝑖𝑗(𝑧), 𝑖=1,,𝑎; 𝑗=1,,𝑁 are 𝑟×𝑟 matrices and their every element is Laurent polynomial. Observe that 𝑅(𝑧)𝑎𝑎𝐼𝑟𝑧1𝐼𝑟𝑧1𝑎𝐼𝑟𝐼𝑟𝜔1𝑧1𝐼𝑟𝜔1𝑧1𝑎𝐼𝑟𝐼𝑟𝜔𝑎1𝑧1𝐼𝑟𝜔𝑎1𝑧1𝑎𝐼𝑟=𝑃1(𝑧𝑎)𝑃2(𝑧𝑎)𝑃𝑎(𝑧𝑎)𝑄11(𝑧𝑎)𝑄12(𝑧𝑎)𝑄1𝑎(𝑧𝑎)𝑄𝑁1(𝑧𝑎)𝑄𝑁2(𝑧𝑎)𝑄𝑁𝑎(𝑧𝑎).(3.33)
Therefore, we have 𝑎𝑃1(𝑧𝑎)𝑃2(𝑧𝑎)𝑃𝑎(𝑧𝑎)𝑄11(𝑧𝑎)𝑄12(𝑧𝑎)𝑄1𝑎(𝑧𝑎)𝑄𝑁1(𝑧𝑎)𝑄𝑁2(𝑧𝑎)𝑄𝑁𝑎(𝑧𝑎)𝑃1(𝑧𝑎)𝑃2(𝑧𝑎)𝑃𝑎(𝑧𝑎)𝑄11(𝑧𝑎)𝑄12(𝑧𝑎)𝑄1𝑎(𝑧𝑎)𝑄𝑁1(𝑧𝑎)𝑄𝑁2(𝑧𝑎)𝑄𝑁𝑎(𝑧𝑎)=𝐼𝑟𝑧1𝐼𝑟𝑧1𝑎𝐼𝑟𝐼𝑟𝜔1𝑧1𝐼𝑟𝜔1𝑧1𝑎𝐼𝑟𝐼𝑟𝜔𝑎1𝑧1𝐼𝑟𝜔𝑎1𝑧1𝑎𝐼𝑟𝑅(𝑧)𝐼𝑅(𝑧)𝑟𝑧1𝐼𝑟𝑧1𝑎𝐼𝑟𝐼𝑟𝜔1𝑧1𝐼𝑟𝜔1𝑧1𝑎𝐼𝑟𝐼𝑟𝜔𝑎1𝑧1𝐼𝑟𝜔𝑎1𝑧1𝑎𝐼𝑟,(3.34) and it follows from (3.4), that 𝑃1(𝑧𝑎)𝑃2(𝑧𝑎)𝑃𝑎(𝑧𝑎)𝑄11(𝑧𝑎)𝑄12(𝑧𝑎)𝑄1𝑎(𝑧𝑎)Q𝑁1(𝑧𝑎)𝑄𝑁2(𝑧𝑎)𝑄𝑁𝑎(𝑧𝑎)𝑃1(𝑧𝑎)𝑃2(𝑧𝑎)𝑃𝑎(𝑧𝑎)𝑄11(𝑧𝑎)𝑄12(𝑧𝑎)𝑄1𝑎(𝑧𝑎)𝑄𝑁1(𝑧𝑎)𝑄𝑁2(𝑧𝑎)𝑄𝑁𝑎(𝑧𝑎)=𝐼𝑎𝑟,|𝑧|=1.(3.35) And it is easy to obtain (3.35) from (3.4).
For convenience, we denote 𝑧𝑎=𝑢. Next, we present some theorems to give several sufficient conditions for existence of minimum-energy multiwavelet frames.

Theorem 3.5. A compactly supported vector-valued function Φ(𝑥)=(𝜙1(𝑥),,𝜙𝑟(𝑥))𝑇 with Φ continuous at 0 and Φ(0)0, its symbol 𝑃(𝑧) satisfies 𝑟𝑖=1𝑎1𝑟𝑙=0𝑗=1||𝑝𝑖𝑗𝜔𝑙𝑧||<1,|𝑧|=1.(3.36) Then there exist minimum-energy multiwavelet frames associated with Φ.

Proof. Let 𝑃𝑗(𝑧),𝑗=1,,𝑎 be the polynomial components of 𝑃(𝑧), that is, 𝑃(𝑧)=𝑎𝑎𝑃1(𝑧𝑎)+𝑧𝑃2(𝑧𝑎)++𝑧𝑎1𝑃𝑎(𝑧𝑎).(3.37)
Using (3.34) and (3.35), we can get 𝑟𝑎𝑖=1𝑟𝑙=1𝑗=1||𝑝𝑙𝑖𝑗||(𝑢)2<1.(3.38)
Then we can find 𝑟 real numbers 𝑥1,𝑥2,,𝑥𝑟, with 𝑟𝑖=1𝑥𝑖=1,𝑎𝑟𝑙=1𝑗=1||𝑝𝑙𝑖𝑗||(𝑢)2<𝑥𝑖,1𝑖𝑟.(3.39) By the Riesz lemma [27, Lemma 6.13], we can find Laurent polynomials 𝑃𝑖𝑎+1(𝑧),𝑖=1,,𝑟 satisfying 𝑎𝑟𝑙=1𝑗=1||𝑝𝑙𝑖𝑗||(𝑢)2+||𝑝𝑖𝑎+1||(𝑢)2=𝑥𝑖,1𝑖𝑟.(3.40)
For every 𝑖{1,,𝑟}, using the method in the reference [15,Theorem 3] on the unit vector 1𝑥𝑖𝑃1𝑖1(𝑧)𝑃1𝑖𝑟(𝑧)𝑃𝑎𝑖1(𝑧)𝑃𝑎𝑖𝑟(𝑧)𝑃𝑖𝑎+1,(𝑧)(3.41) we can get a matrix 𝑅𝑖1(𝑧)=𝑥𝑖𝑃1𝑖1(𝑧)𝑃1𝑖𝑟(𝑧)P𝑎𝑖1(𝑧)𝑃𝑎𝑖𝑟(𝑧)𝑃𝑖𝑎+1𝑄(𝑧)𝑖111(𝑧)𝑄𝑖𝑟11(𝑧)𝑄𝑖11𝑎(𝑧)𝑄𝑖𝑟1𝑎(𝑧)𝑄𝑖1,𝑎+1(𝑄𝑧)𝑖1𝑎1(𝑧)𝑄𝑎𝑟11(𝑧)𝑄𝑖1𝑎𝑎(𝑧)𝑄𝑖𝑟𝑎𝑎(𝑧)𝑄𝑖𝑎,𝑎+1(𝑧),(3.42) which satisfies 𝑅𝑖(𝑧)𝑅𝑖(𝑧)=𝐼𝑎𝑟+1.
Therefor, the block matrix 𝑅(𝑧)=𝑥1𝑅1(𝑧)𝑥2𝑅2(𝑧)𝑥𝑎𝑅𝑎(𝑧)(3.43) satisfies 𝑅(𝑧)𝑅(𝑧)=𝐼𝑎𝑟+1.
We can get matrix 𝑅(𝑧) which satisfies 𝑅(𝑧)𝑅(𝑧)=𝐼𝑎𝑟, after adjusting the rows of 𝑅(𝑧) and removing the last column of it, and the 𝑟 rows in the front of matrix 𝑅(𝑧) are the polynomial components of the symbol 𝑃(𝑧).
Then we complete proof of Theorem 3.5 using the formulas (3.34), (3.32), and Theorem 3.1.

Theorem 3.5 requests the sum of 𝑙2-norm for every row in the matrix symbol 𝑃(𝑧) associated with the vector-valued function Φ. Then we can find a minimum-energy multiwavelet frames associated with the function using the theorem. The condition in Theorem 3.5 is too stringent compared with the sufficient conditions in Theorem 3.2. We can get the following theorem by strengthening the structure of the matrix symbol 𝑃(𝑧).

Theorem 3.6. Let Φ(𝑥)=(𝜙1(𝑥),,𝜙𝑟(𝑥))𝑇 with Φ continuous at 0 and Φ(0)0 a compactly supported multiscaling vector-valued function. If the block matrix 𝜔𝑃(𝑧)𝑃1𝑧𝜔𝑃𝑎1𝑧(3.44) satisfies standard orthogonal by row, then there exist a minimum-energy multiwavelet frames associated with the function Φ.

Proof. Let 𝑃𝑗(𝑧), 𝑗=1,,𝑎 are the polynomial components of 𝑃(𝑧), that is, 𝑃(𝑧)=𝑎𝑎𝑃1(𝑧𝑎)+𝑧𝑃2(𝑧𝑎)++𝑧𝑎1𝑃𝑎(𝑧𝑎),(3.45) with (3.34) and (3.35), we can know that the block matrix 𝑃𝑁(𝑢)=1(𝑢)𝑃2(𝑢)𝑃𝑎(𝑢)𝑟×𝑎𝑟(3.46) satisfies standard orthogonal by row.
Now, we use the method in the reference [15, Theorem 3] to deal with the first unit row vector 𝑁1(𝑢) in the matrix 𝑁(𝑢). And, we can find a paraunitary matrix 𝐻1(𝑢) which satisfies 𝑁1(𝑢)𝐻1(𝑢)=𝑒1=(1,0,,0)𝑎𝑟 and 𝑁(𝑢)𝐻11(𝑢)=𝑁(𝑢),(3.47) with 𝑁(𝑢) also a matrix standard orthogonal by row.
By mathematical induction, there are 𝑟 paraunitary matrices 𝐻1(𝑢),,𝐻𝑟(𝑢) satisfying 𝑁(𝑢)𝐻1(𝑢)𝐻𝑎(𝑢)=10000100𝑟×𝑎𝑟,(3.48) then the matrix 𝑁(𝑢) is equivalent to the front 𝑟 rows in the paraunitary matrix 𝐻1(𝑢)𝐻2(𝑢)𝐻𝑎(𝑢).
Using the formulation (3.34), (3.35), and Theorem 3.1, we completed the proof of this theorem.

Theorem 3.6 requests that the multiscaling vector-valued function's symbol 𝑃(𝑧) satisfies standard orthogonal by row. This means the 𝑙2-norm of every row in 𝑃(𝑧) is 1. If the 𝑙2-norm of every row in 𝑃(𝑧) is less than 1 strictly, and we can find a matrix 𝑃𝑎+1(𝑢) to make the block matrix 𝑃1(𝑢)𝑃2(𝑢)𝑃𝑎(𝑢)𝑃𝑎+1(𝑢)(3.49) satisfy standard orthogonal by row, then there exist minimum-energy multiwavelet frames associated with the function Φ.

Corollary 3.7. Let Φ(𝑥)=(𝜙1(𝑥),,𝜙𝑟(𝑥))𝑇 with Φ continuous at 0 and Φ(0)0 a compactly supported multiscaling vector-valued function. If the 𝑙2-norm of every row in 𝑃(𝑧) is less than 1 strictly, that is, 𝑎1𝑟𝑙=0𝑗=1||𝑝𝑖𝑗𝜔𝑙𝑧||2<1,|𝑧|=1,1𝑖𝑟,(3.50) and there exists a matrix 𝑃𝑎+1(𝑢) to make (3.49) satisfy standard orthogonal by row, then there exist minimum-energy multiwavelet frames associated with the function Φ.

By Theorem 3.1, if we can find some row vectors 𝛼1(𝑧),,𝛼𝑛(𝑧) with multiplicity 𝑎𝑟 and the matrix in (3.3) formed by the vectors and the symbol of Φ satisfies standard orthogonal by column, there exist a minimum-energy multiwavelet frames associated with Φ, and vice versa. However, the number of columns in the symbol of Φ is so larger, that it is not easy to find the frames using Theorem 3.1. Corollary 3.7 requests some column vectors 𝛽1(𝑢),,𝛽𝑚(𝑢) with multiplicity 𝑟 and the matrix in (3.49) formed by the vectors and the polynomial components of 𝑃(𝑧) satisfies standard orthogonal by row, then we can find a minimum-energy frames associated with Φ. Obviously, the problem is vastly simplified.

For some multiscaling vector-valued function with small multiplicity which satisfies the conditions in Theorem 3.2, the matrix 𝑃𝑎+1(𝑢) that makes the block matrix in (3.49) satisfied standard orthogonal by column can be found using the method of undetermined coefficients. We will give some examples later.

4. Decomposition and Reconstruction Formulas of Minimum-Energy Multiwavelet Frames

Suppose the multiscaling vector-valued function Φ has an associated minimum-energy multiwavelet frames {Ψ1,,Ψ𝑁}. Now, we consider the projection operators 𝐏𝑗 of 𝐿2() onto the nested subspace 𝑉𝑗 defined by 𝐏𝑗𝑓=𝑟𝜏=1𝑘𝑓,𝜙𝜏,𝑗,𝑘𝜙𝜏,𝑗,𝑘.(4.1) Then the formula (2.8) can be rewritten as 𝐏𝑗+1𝑓𝐏𝑗𝑓=𝑁𝑟𝑖=1𝜏=1𝑘𝑓,𝜓𝑖𝜏,𝑗,𝑘𝜓𝑖𝜏,𝑗,𝑘.(4.2) In other words, the error term 𝑔𝑗=𝐏𝑗+1𝑓𝐏𝑗𝑓 between consecutive projections is given by the frame expansion: 𝑔𝑗=𝑁𝑟𝑖=1𝜏=1𝑘𝑓,𝜓𝑖𝜏,𝑗,𝑘𝜓𝑖𝜏,𝑗,𝑘.(4.3)

Suppose that the error term 𝑔𝑗 has other expansion in terms of the frames {Ψ1,,Ψ𝑁}, that is, 𝑔𝑗=𝑁𝑟𝑖=1𝜏=1𝑘𝑐𝜏,𝑗,𝑘𝜓𝑖𝜏,𝑗,𝑘.(4.4) Then by using both (4.3) and (4.4), we have 𝑔𝑗=,𝑓𝑁𝑟𝑖=1𝜏=1𝑘|||𝑓,𝜓𝑖𝜏,𝑗,𝑘|||2=𝑁𝑟𝑖=1𝜏=1𝑘𝐙𝑐𝜏,𝑗,𝑘𝑓,𝜓𝑖𝜏,𝑗,𝑘,(4.5) and this derives 0𝑁𝑟𝑖=1𝜏=1𝑘𝐙|||𝑐𝜏,𝑗,𝑘𝑓,𝜓𝑖𝜏,𝑗,𝑘|||2=𝑁𝑟𝑖=1𝜏=1𝑘𝐙||𝑐𝜏,𝑗,𝑘||22𝑁𝑟𝑖=1𝜏=1𝑘𝐙𝑐𝜏,𝑗,𝑘𝑓,𝜓𝑖𝜏,𝑗,𝑘+𝑁𝑟𝑖=1𝜏=1𝑘|||𝑓,𝜓𝑖𝜏,𝑗,𝑘|||2=𝑁𝑟𝑖=1𝜏=1𝑘𝐙||𝑐𝜏,𝑗,𝑘||2𝑁𝑟𝑖=1𝜏=1𝑘|||𝑓,𝜓𝑖𝜏,𝑗,𝑘|||2.(4.6)

This inequality means that the coefficients of the error term 𝑔𝑗 in (4.3) have minimal 𝑙2-norm among all sequences {𝑐𝜏,𝑗,𝑘} which satisfy (4.4).

We next discuss minimum-energy multiwavelet frames decomposition and reconstruction. For any 𝑓𝐿2(), define the vector coefficients as follows: 𝐜𝑗,𝑘=𝑓,Φ𝑗,𝑘,𝐝𝑗,𝑘=𝑓,Ψ𝑖𝑗,𝑘𝑖=1,,𝑁.(4.7) The inner product of 𝑓 with vector-valued Φ𝑗,𝑘, Ψ𝑖𝑗,𝑘, 𝑖=1,,𝑁 is a vector, its every component is the inner product of 𝑓 with the corresponding component of Φ𝑗,𝑘, Ψ𝑖𝑗,𝑘, 𝑖=1,,𝑁.

(1) Decomposition Algorithm
suppose the vector coefficients {𝐜𝑗+1,𝑙𝑙𝐙} are known. By the two-scale relations (2.2) and (3.1), we have Φ𝑗,𝑙1(𝑥)=𝑎𝑘𝑃𝑘𝑎𝑙Φ𝑗+1,𝑘(𝑥),Ψ𝑖𝑗,𝑙1(𝑥)=𝑎𝑘𝑄𝑖𝑘𝑎𝑙Ψ𝑖𝑗+1,𝑘(𝑥),𝑖=1,,𝑁.(4.8) Then, the decomposition algorithm is given as 𝐜𝑗,𝑙=1𝑎𝑘𝑃𝑘𝑎𝑙𝐜𝑗+1,𝑘,𝐝𝑖𝑗,𝑙=1𝑎𝑘𝑄𝑖𝑘𝑎𝑙𝐝𝑖𝑗+1,𝑘,𝑖=1,,𝑁.(4.9)

(2) Reconstruction Algorithm
from (3.16), it follow that Φ𝑗+1,𝑙1(𝑥)=𝑎𝑘𝑃𝑙𝑎𝑘Φ𝑗,𝑘(𝑥)+𝑁𝑖=1𝑄𝑖𝑙𝑎𝑘Ψ𝑖𝑗,𝑘.(𝑥)(4.10) Taking the inner products on both sides of this equality, we get 𝐜𝑗+1,𝑙=1𝑎𝑘𝑃𝑙𝑎𝑘𝐜𝑗,𝑘+𝑁𝑖=1𝑄𝑖𝑙𝑎𝑘𝐝𝑖𝑗,𝑘.(4.11)

5. Numerical Examples

By Theorem 3.6, the orthogonal multiwavelet always have minimum-energy multiwavelet frames associated with them, for example, DGHM multiwavelet and Chui-Lian multiwavelet. These examples are trivial. In this section, we will construct some minimum-energy multiwavelet frames in general sense.

It is well known that the 𝑚th-order cardinal B-spline 𝑁𝑎𝑚(𝑥) with dilation factor 𝑎 has the two-scale relation as follows: 𝑁𝑎𝑚(𝜔)=𝑃𝑎𝑚𝑁(𝑧)𝑎𝑚𝜔𝑎,𝑃𝑎𝑚(𝑧)=1+𝑧++𝑧𝑎1𝑎𝑚,𝑧=𝑒𝑖𝜔/𝑎.(5.1)

In addition, if a scale wavelet 𝜙(𝑥) satisfies the refinable function 𝜙(𝑥)=𝑘1𝑘=𝑘0𝑝𝑘𝜙(𝑎𝑥𝑘),(5.2) and let Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥1),,𝜙(𝑥𝑟+1))𝑇, then the vector-valued function Φ satisfies (2.2) with some matrixes {𝑃𝑘}.

Below, upon these conclusions, using Theorem 3.5 and Corollary 3.7 in Section 3, the minimum-energy multiwavelet frames be presented with the dilation factors 𝑎=2, 𝑎=3, 𝑎=4, respectively.

5.1. 𝑎=2

Example 5.1. With 𝑎=2, the symbol of the B-spline 𝑁22(𝑥) is 𝑃221(𝑧)=4+121𝑧+4𝑧2.(5.3) Take 𝜙(𝑥)=𝑁22(𝑥), and the support of this function is [0,2]. The function satisfies 1𝜙(𝑥)=41𝜙(2𝑥)+21𝜙(2𝑥1)+4𝜙(2𝑥2).(5.4)
Let Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥1))𝑇, and 1Φ(𝑥)=212100Φ(2𝑥)+212100Φ(2𝑥1)+002121Φ(2𝑥2)+00212Φ(2𝑥3).(5.5) The coefficient matrixes in (5.5) are not unique.
And the symbol of Φ has polyphase components as follows: 𝑃1(𝑢)=𝑃2(𝑢)=221212100+𝑢00212.(5.6) Take 𝑃3(𝑢)=22222200+𝑢002222,(5.7) which satisfies 𝑃1(𝑢)𝑃1(𝑢)+𝑃2(𝑢)𝑃2(𝑢)+𝑃3(𝑢)𝑃3(𝑢)=𝐼2.(5.8)
Using Theorem 3.5, we can get matrix the following: 𝐶(𝑢)=2212121212𝑢2𝑢2𝑢2𝑢21101001012121212𝑢2𝑢2𝑢2𝑢2,(5.9) which satisfy the formula (3.35). Then we take symbols as 𝑄11(𝑧)=2,𝑄1001+𝑧100121(𝑧)=21212100+𝑧21200+𝑧2100212+𝑧3100212.(5.10)
The graphs of Φ and its minimum-energy frames are shown in Figure 1.
We may discover from Figure 1 that every component of minimum-energy frames is (anti)symmetrical.

fig1
Figure 1

Example 5.2. With 𝑎=2, the symbol of the B-spline 𝑁23(𝑥) is 𝑃231(𝑧)=8+383𝑧+8𝑧2+18𝑧3.(5.11) Take 𝜙(𝑥)=𝑁23(𝑥), and the support of this function is [0,3].(1) Let Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥1))𝑇, and 1Φ(𝑥)=414100Φ(2𝑥)+212100Φ(2𝑥1)+4141414+1Φ(2𝑥2)002121Φ(2𝑥3)+00414Φ(2𝑥4).(5.12)The symbol of Φ has polyphase components as follows: 𝑃1(𝑢)=221414100+𝑢4141414+𝑢2100414,𝑃2(𝑢)=221212100+𝑢00212.(5.13) Take 𝑃3(𝑢)=2424𝑢2212+𝑢2024𝑢+24𝑢222𝑢𝑢02+𝑢22,(5.14) which satisfies 𝑃1(𝑢)𝑃1(𝑢)+𝑃2(𝑢)𝑃2(𝑢)+𝑃3(𝑢)𝑃3(𝑢)=𝐼2. Using Theorem 3.5, we can get the following symbols: 𝑄1(𝑧)=22,𝑄1100+𝑧00112(𝑧)=22242400+𝑧224242424+𝑧4002424,𝑄3(𝑧)=221414100+𝑧21200+𝑧214141414+𝑧3100212+𝑧4100414.(5.15) Then, we get the minimum-wavelet frames associated with Φ. The graphs of them are shown in Figure 2.
We can discover from Figure 2 that every component of the minimum-energy frames is (anti)symmetrical and smooth.(2) Take Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥1),𝜙(𝑥2))𝑇, which satisfies 1Φ(𝑥)=4100301000000Φ(2𝑥+2)+311000000Φ(2𝑥+1)+3111003+1000Φ(2𝑥)113013111000Φ(2𝑥1)+3013111003+1Φ(2𝑥2)3100113013111Φ(2𝑥3)+0003013+111Φ(2𝑥4)000310011311Φ(2𝑥5)+000000301Φ(2𝑥6)+000000300Φ(2𝑥7)(5.16)and the symbol of this multiscaling vector-valued function has the following polyphase components: 𝑃1(𝑢)=281003+1000000311100311000𝑢+3013111003𝑢2+110003013𝑢113+1100000030𝑢4,𝑃2(𝑢)=280131+100000011301311000𝑢+31001130131𝑢2+10003100113𝑢3+10000003𝑢004.(5.17)
Let 𝑃3(𝑢)=28×6+63𝑢06+63𝑢6𝑢600(𝒜)𝒜006𝑢26𝑢00()006𝑢36𝑢2,(5.18) where 𝒜 denotes 6𝑢+(6/3)𝑢2, and denotes 6𝑢2+(6/3)𝑢3, which satisfies 𝑃1(𝑢)𝑃1(𝑢)+𝑃2(𝑢)𝑃2(𝑢)+𝑃3(𝑢)𝑃3(𝑢)=𝐼3. Using Theorem 3.5, we can get 𝑄1=(𝑧)220.235478068164731050.19692476658003990.148914015596096930.082127850577445230.185554937811228980.191060266881261010.0273759501924817350.0618516459370762950.07928888488639749+𝑧0.90335897107423320.071129380939489170.0165068877336184970.06082861392110680.9056831509728050.038082577817574040.0202762046403689340.047041078935042140.9404994196162103+𝑧20.204023950497054680.051756588816736160.018282819017567320.24891156965574340.09796005999183860.0168632288699146550.07006381692245610.122222994441697720.11122693740098887+𝑧30.051756588816736160.018282819017567320.0060942730058557740.09796005999183860.0168632288699146550.0056210762899715510.122222994441697760.111226937400988870.03707564580032962+𝑧40.0182828190175673140.006094273005855771500.0168632288699146480.00562107628997154900.111226937400988770.037075645800329590+𝑧5,𝑄0.0060942730058557715000.005621076289971549000.03707564580032959002=(𝑧)220.099863251031038130.315898819955170330.00525389256461650240.08541249222343070.260795782879353230.29988930060312390.0144507588076074540.055103037075817090.30514319316774036+𝑧0.0163090668620558470.0059083791708918130.0193817133267621750.00455830620906111550.0109500916378539660.00312154273109820430.0117507606529947130.0050417124669621570.016260170595664+𝑧20.209161144717719180.0515155476128009440.020480438204743560.0236399744796289240.019509641388788480.0130833561325261470.232801119197348140.071025189001589370.007397082072217509+𝑧30.0515155476128009440.020480438204743560.00682681273491453360.0195096413887884370.0130833561325261470.004361118710842040.071025189001589360.0073970820722175090.0024656940240724914+𝑧40.0204804382047434620.00682681273491449400.0130833561325261540.00436111871084204700.00739708207221737540.0024656940240724610+𝑧5,𝑄0.006826812734914494000.004361118710842047000.002465694024072461003=(𝑧)220.047256900643218360.219496884733862030.16391307392755490.25233285244989610.72819957513164890.159184871024636740.0039046242203960510.0541877735064109150.7337722255518694+