Minimum-Energy Multiwavelet Frames with Arbitrary Integer Dilation Factor

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 [10–22], 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 [24–26]. 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 ℤ, ℝ, and ℂ denote the set of integers, real numbers, and complex numbers respectively; 𝑎∈ℤ with 𝑎⩾2, 𝜔_𝑗=cos(2𝑗𝜋/𝑎)+𝑖sin(2𝑗𝜋/𝑎), 𝑗=0,1,…,𝑎−1.

A multiscaling function vector (refinable function vector) is a vector-valued function: Φ=𝜙₁(𝑥),…,𝜙_𝑟(𝑥)^𝑇,𝜙_𝑙(𝑥)∈𝐿²(ℝ),𝑙=1,…,𝑟,(2.1) which satisfies a two-scale matrix refinement equation of the form: Φ(𝑥)=_𝑘∈ℤ𝑃_𝑘Φ(𝑎𝑥−𝑘),𝑥∈ℝ,(2.2)𝑟 is called the multiplicity of Φ, the integer 𝑎 is said to be dilation factor. The recursion coefficients {𝑃_𝑘}_𝑘∈ℤ are 𝑟×𝑟 matrices.

The Fourier transform of the formula (2.2) is Φ(𝜔)=𝑃(𝑧)Φ𝜔𝑎,𝑧=𝑒^{−𝑖𝜔/𝑎},(2.3) where 𝑃(𝑧)=1𝑎_𝑘∈ℤ𝑃_𝑘𝑧^𝑘.(2.4)𝑃(𝑧) is the symbol of the matrix sequence {𝑃_𝑘}_𝑘∈ℤ.

The multiresolution analysis (MRA) with multiplicity 𝑟 and dilation factor 𝑎 generated by Φ(𝑥) is defined as 𝑉_𝑗=span𝜙_{𝜏,𝑗,𝑘}∶1⩽𝜏⩽𝑟,𝑘∈ℤ,𝑗∈ℤ,(2.5) where 𝜙_{𝜏,𝑗,𝑘}=𝑎^𝑗/2𝜙_𝜏(𝑎^𝑗𝑥−𝑘), and the sequence of closed subspace of 𝐿²(ℝ) has the following properties:(1) 𝑉_𝑗⊂𝑉_𝑗+1, 𝑗∈ℤ.(2) ⋃_𝑗∈ℤ𝑉_𝑗=𝐿²(ℝ), ⋂_𝑗∈ℤ𝑉_𝑗={0};(3) 𝑓(𝑥)∈𝑉_𝑗⇔𝑓(𝑎𝑥)∈𝑉_𝑗+1, for all 𝑗∈ℤ;(4) 𝑓(𝑥)∈𝑉_j⇔𝑓(𝑥−𝑎^−𝑗𝑘)∈𝑉_𝑗, for all 𝑘,𝑗∈ℤ;(5) {𝜙_𝜏,0,𝑘∶1⩽𝜏⩽𝑟,𝑘∈𝐙} forms a Riesz basis of 𝑉₀;

Definition 2.1. A finite family vector-valued function Ψ^𝑖=(𝜓^𝑖1,…,𝜓^𝑖𝑟)^𝑇, 𝑖=1,…,𝑁 generates a multiwavelet frames for 𝐿²(ℝ), if there exist constants 0<𝐴⩽𝐵<∞ such that for any 𝑓(𝑥)∈𝐿²(ℝ)𝐴‖𝑓‖²⩽^𝑁_𝑟𝑖=1_𝜏=1_{𝑗,𝑘∈ℤ}|||𝑓,𝜓_{𝑖𝜏,𝑗,𝑘}|||²⩽𝐵‖𝑓‖²,(2.6) where 𝜓^{𝑖𝜏,𝑗,𝑘}=𝑎^𝑗/2𝜓^𝑖𝜏(𝑎^𝑗𝑥−𝑘).

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,…,Ψ^𝑁} generates a frame multiresolution analysis associated the vector-valued function Φ(𝑥), if the finite family Ψ^𝑖=(𝜓^𝑖1,…,𝜓^𝑖𝑟)^𝑇, 𝑖=1,…,𝑁 satisfies the formulation (2.6) with 𝜓^𝑖𝜏∈𝑉₁, 𝑖=1,…,𝑁; 𝜏=1,…,𝑟.

Definition 2.3. Let Φ(𝑥)=(𝜙₁(𝑥),…,𝜙_𝑟(𝑥))^𝑇, with 𝜙_𝜏∈𝐿^∞(ℝ)∩𝐿²(ℝ), 𝜏=1,…,𝑟, Φ continuous at 0 and Φ(0)≠0, be a multiscaling vector-valued function that generates the nested subspace {𝑉_𝑗}_𝑗∈𝐙 in the sense of (2.5). Then a finite family vector-valued function {Ψ¹,…,Ψ^𝑁}⊂𝑉₁ is called a minimum-energy multiwavelet frames associated with Φ(𝑥), if for for all 𝑓∈𝐿²(ℝ)^𝑟_𝜏=1_𝑘∈ℤ||𝑓,𝜙_𝜏,1,𝑘||²=^𝑟_𝜏=1_𝑘∈ℤ||𝑓,𝜙_𝜏,0,𝑘||²+^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈ℤ|||𝑓,𝜓_{𝑖𝜏,0,𝑘}|||².(2.7)

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

Remark 2.5. The formula (2.7) is equivalent to the following formulas: ^𝑟_𝜏=1_𝑘∈ℤ𝑓,𝜙_𝜏,1,𝑘𝜙_𝜏,1,𝑘=^𝑟_𝜏=1_𝑘∈ℤ𝑓,𝜙_𝜏,0,𝑘𝜙_𝜏,0,𝑘+^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈ℤ𝑓,𝜓^{𝑖𝜏,0,𝑘}𝜓^{𝑖𝜏,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),…,𝜙_𝑟(𝑥))^𝑇 with 𝜙_𝜏∈𝐿^∞(ℝ)∩𝐿²(ℝ), 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,…,Ψ^𝑁}⊂𝑉₁, then Ψ^𝑙(𝑥)=_𝑘∈ℤ𝑄^𝑙𝑘Φ(𝑎𝑥−𝑘),(3.1) where {𝑄^𝑙𝑘}_𝑘∈ℤ, 𝑙=1,…,𝑁 are 𝑟×𝑟 matrices. Using Fourier transform on (3.1), we can get their symbols as follows: 𝑄_𝑙(𝑧)=1𝑎_𝑘∈ℤ𝑄^𝑙𝑘𝑧^k,𝑙=1,…,𝑁.(3.2)

With 𝑃(𝑧), 𝑄_𝑙(𝑧), 𝑙=1,…,𝑁, we formulate the (𝑁+1)𝑟×𝑎𝑟 block matrix as follows: 𝑅(𝑧)=⎡⎢⎢⎢⎢⎢⎢⎣𝑃(𝑧)𝑃𝜔¹𝑄₁𝑧⋯𝑃𝜔_𝑎−1𝑧𝑄⋮(𝑧)𝑄₁⋮𝜔₁𝑧⋯𝑄₁𝜔⋮_𝑎−1𝑁𝑧(𝑧)𝑄_𝑁𝜔₁𝑧⋯𝑄_𝑁𝜔_𝑎−1𝑧⎤⎥⎥⎥⎥⎥⎥⎦,(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), 𝑄_𝑙(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 {𝑉_𝑗}_𝑗∈𝐙. Then the following statements are equivalent:(1){Ψ¹,…,Ψ^𝑁} is a minimum-energy multiwavelet frames associated with Φ(𝑥):(2) 𝑅^∗(𝑧)𝑅(𝑧)=𝐼_𝑎𝑟𝑓𝑜𝑟∀|𝑧|=1;(3.4)(3)𝛼_{𝑚𝑙,𝑖𝑗}=0,∀𝑚,𝑙∈ℤ;𝑖,𝑗=1,…,𝑟,(3.5)where 𝛼_{𝑚𝑙,𝑖𝑗}=_{𝑟𝑘∈𝐙}_𝜏=1𝑃^{𝜏𝑖∗𝑙−𝑎𝑘}𝑃^{𝜏𝑗𝑚−𝑎𝑘}+^𝑁_𝑡=1𝑄^{𝑡,𝜏𝑖∗𝑙−𝑎𝑘}𝑄^{𝑡,𝜏𝑗𝑚−𝑎𝑘}−𝑎𝛿^{𝑚𝑙,𝑖𝑗}𝛿_{𝑚𝑙,𝑖𝑗},=10,,𝑚𝑒𝑙𝑠=𝑒.𝑙,𝑖=𝑗,(3.6)

Proof. By using the two-scale relations (2.2) and (3.1) and notation 𝛼_{𝑚𝑙,𝑖𝑗} for for all 𝑓∈𝐿²(ℝ), (2.8) can be written as _𝑙∈ℤ_{𝑟𝑚∈ℤ}_𝑟𝑖=1_𝑗=1𝛼_{𝑚𝑙,𝑖𝑗}⟨𝑓,𝜙_𝑖(𝑎𝑥−𝑚)⟩𝜙_𝑗(𝑎𝑥−𝑙)=0.(3.7) On the other hand, (3.4) can be reformulated as 𝑃^∗(𝑧)𝑃(𝑧)+^𝑁_𝑡=1𝑃_∗𝑄^∗𝑡(𝑧)𝑄_𝑡(𝑧)=𝐼_𝑟,(𝑧)𝑃𝜔_𝑗𝑧+^𝑁_𝑡=1𝑗=1,2,…,𝑎−𝑄1^∗𝑡;(𝑧)∀𝑄|𝑧_𝑡|𝜔=_𝑗1𝑧,=0_𝑟,(3.8) and it is equivalent to ^𝑎−1_𝑘=0𝑃^∗𝜔_𝑘𝑧𝑃(𝑧)+^𝑁_{𝑡=1𝑎−1𝑘=0}𝑃_∗𝑄^∗𝑡𝜔_𝑘𝑧𝑄_𝑡(𝑧)=𝐼_𝑟,(𝑧)−^𝑎−1_𝑘=1𝑃^∗𝜔_𝑘𝑧𝑃(𝑧)+^𝑁_𝑡=1𝑄^∗𝑡(𝑧)−^𝑎−1_𝑘=1_𝑎−1𝑄^∗𝑡𝜔_𝑘𝑧𝑄_𝑡(𝑧)=𝐼_𝑟,_𝑘=0𝑃^∗𝜔_𝑘𝑧−2𝑃^∗𝜔_𝑙𝑧𝑃(𝑧)+^𝑁_𝑡=1^𝑎−1_𝑘=0𝑙=1,2,…,𝑎−1;𝑄∀^∗𝑡|𝑧|𝜔=_𝑘𝑧1.−2𝑄^∗𝑡𝜔_𝑙𝑧𝑄_𝑡(𝑧)=𝐼_𝑟,(3.9) With |𝑧|=1,𝑧^𝑘=𝑧^−𝑘,𝜔^𝑘𝑙=𝜔^𝑙𝑘=𝜔^𝑘𝑙, and ^𝑎−1_𝑙=0𝜔^𝑘𝑙=^𝑎−1_𝑙=0𝜔^𝑙𝑘=0𝜔^𝑘𝑎𝜔_𝑘≠1=1,(3.10) the formulation (3.9) is equivalent to _𝑘_∈ℤ𝑃^{∗−𝑎𝑘}𝑧^𝑎𝑘𝑃(𝑧)+^𝑁_𝑡=1_𝑘∈ℤ_𝑎−1𝑄^{𝑡∗−𝑎𝑘}𝑧^𝑎𝑘𝑄_𝑡(𝑧)=𝐼_𝑟,_𝑙=1_𝑘∈ℤ𝑃^{∗𝑙−𝑎𝑘}𝑧^{𝑎𝑘−𝑙}𝑃(𝑧)+^𝑁_𝑡=1^𝑎−1_𝑙=1_𝑘∈ℤ𝑄^{𝑡∗𝑙−𝑎𝑘}𝑧^{𝑎𝑘−𝑙}𝑄^𝑡_𝑎−1(𝑧)=(𝑎−1)𝐼_𝑟,_𝑙=1𝑒^{−(2𝑠𝑙𝜋/𝑎)𝑖}_𝑘_∈ℤ𝑃^{∗𝑙−𝑎𝑘}𝑧^{𝑎𝑘−𝑙}𝑃(𝑧)+^𝑁_𝑡=1^𝑎−1_𝑙=1𝑒^{−(2𝑠𝑙𝜋/𝑎)𝑖}_𝑘_∈ℤ𝑄^{𝑡∗𝑙−𝑎𝑘}𝑧^{𝑎𝑘−𝑙}𝑄^𝑡𝑠=1,2,…,𝑎−1;∀|𝑧|=1.(𝑧)=−𝐼_𝑟,(3.11)
Using the properties of roots of unity, the Vandermonde matrix and Cramer's rule, the above equation is equivalent to _𝑘_∈ℤ𝑃^{∗−𝑎𝑘}𝑧^𝑎𝑘𝑃(𝑧)+^𝑁_𝑡=1_{𝑘∈ℤ𝑘}_∈ℤ𝑄^{𝑡∗−𝑎𝑘}𝑧^𝑎𝑘𝑄_𝑡(𝑧)=𝐼_𝑟,𝑃^{∗1−𝑎𝑘}𝑧^𝑎𝑘−1𝑃(𝑧)+^𝑁_𝑡=1_{𝑘∈ℤ𝑘}_∈ℤ⋮𝑄^{𝑡∗1−𝑎𝑘}𝑧^𝑎𝑘−1𝑄_𝑡(𝑧)=𝐼_𝑟,𝑃^{∗𝑎−1−𝑎𝑘}𝑧^{𝑎𝑘−𝑎+1}𝑃(𝑧)+^𝑁_𝑡=1_𝑘∈ℤ𝑄^{𝑡∗𝑎−1−𝑎𝑘}𝑧^{𝑎𝑘−𝑎+1}𝑄_𝑡(𝑧)=𝐼_𝑟.(3.12)
We multiply the identities in (3.12) by Φ(𝜔/𝑎)𝑧^𝑙, 𝑙=0,1,…,𝑎−1, respectively, where 𝑧=𝑒^{−𝑖𝜔/𝑎}, to get _𝑘𝑃^{∗𝑙−𝑎𝑘}𝑧^𝑎𝑘𝑃(𝑧)Φ𝜔𝑎+^𝑁_𝑡=1𝑄^{𝑡∗𝑙−𝑎𝑘}𝑧^𝑎𝑘𝑄_𝑡(𝑧)Φ𝜔𝑎=Φ𝜔𝑎𝑧^𝑙,𝑙=0,…,𝑎−1.(3.13) Hence, (3.12) is equivalent to _𝑘𝑃^{∗𝑙−𝑎𝑘}𝑧^𝑎𝑘Φ(𝜔)+^𝑁_𝑡𝑖=1𝑄^{𝑡∗𝑙−𝑎𝑘}𝑧^𝑎𝑘Ψ^𝑡(𝜔)=Φ𝜔𝑎𝑒^{−𝑖𝑙𝜔/𝑎},𝑙=0,…,𝑎−1(3.14) or _𝑘𝑃^{∗𝑙−𝑎𝑘}𝑧^𝑎𝑘Φ(𝑥−𝑘)+^𝑁_𝑡=1𝑄^{𝑡∗𝑙−𝑎𝑘}𝑧^𝑎𝑘Ψ^𝑡(𝑥−𝑘)=𝑎Φ(𝑎𝑥−𝑙),𝑙=0,…,𝑎−1,(3.15) which can be reformulated as _𝑘𝑃^{∗𝑙−𝑎𝑘}𝑧^𝑎𝑘Φ(𝑥−𝑘)+^𝑁_𝑡=1𝑄^{𝑡∗𝑙−𝑎𝑘}𝑧^𝑎𝑘Ψ^𝑡(𝑥−𝑘)=𝑎Φ(𝑎𝑥−𝑙).(3.16) By using the two-scaling relations (2.2) and (3.1), we can rewrite (3.16) as _{𝑚𝑎𝑗=1}𝛼_{𝑚𝑙,𝑖𝑗}𝜙_𝑗(𝑎𝑥−𝑚)=0,𝑖=1,…,𝑟;∀𝑙∈ℤ.(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 𝑓∈𝐿²(ℝ) be any compactly supported function. By using the properties that for every fixed 𝑚, 𝛼_{𝑚𝑙,𝑖𝑗}=0 expect for finitely many 𝑙,𝑖,𝑗, then the functional 𝛽_𝑙𝑗(𝑓)=_𝑚𝑟_𝑖=1𝛼_{𝑚𝑙,𝑖𝑗}⟨𝑓,𝜙_𝑖(𝑎𝑥−𝑚)⟩(3.18) just has finite nonzero for 𝑙∈ℤ, 𝑗=1,…,𝑟.
Using the property of Fourier transform, we obtain _{𝑙𝑟𝑗=1}𝛽_𝑙𝑗(𝑓)𝜙_𝑗(𝜔)𝑒^{−𝑖𝑙𝜔/𝑎}=0.(3.19) Since 𝜙_𝑙(𝜔) is nontrivial function, then 𝛽_𝑙𝑗(𝑓)=0, 𝑙∈ℤ, 𝑗=1,…,𝑟, in other words, we have 𝑓,_{𝑚𝑟𝑖=1}𝛼_{𝑚𝑙,𝑖𝑗}𝜙_𝑖(𝑎𝑥−𝑚)=0,𝑙∈ℤ,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(ℝ). By choosing 𝑓 to be this function, it follows that _{𝑚𝑟𝑖=1}𝛼_{𝑚𝑙,𝑖𝑗}𝜙_𝑖(𝑎𝑥−𝑚)=0,(3.21) which implies that the trigonometric polynomial ∑_𝑚∑^𝑟𝑖=1𝛼_{𝑚𝑙,𝑖𝑗}𝜙_𝑖(𝜔)𝑒^{−𝑖𝑚𝜔} is identically equal to 0 so that 𝛼_{𝑚𝑙,𝑖𝑗}=0, for all 𝑚,𝑙∈ℤ;𝑖,𝑗=1,…,𝑟.
We complete the proof of Theorem 3.1 because the set of compactly supported functions is dense in 𝐿²(ℝ).

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),…,𝜙_𝑟(𝑥))^𝑇, with Φ continuous at 0 and Φ(0)≠0. Let {Ψ¹,…,Ψ^𝑁} be the minimum-energy multiwavelet frames associated with it, then ^𝑟_𝑖=1||𝑝_𝑖𝑗(𝜔_𝑙𝑧)||²⩽1∀|𝑧|=1,1⩽𝑗⩽𝑟,0⩽𝑙⩽𝑎−1,(3.22)_𝑎−1_𝑟𝑙=0_𝑗=1||𝑝_𝑖𝑗(𝜔_𝑙𝑧)||²⩽1∀|𝑧|=1,1⩽𝑖⩽𝑟.(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 𝑓(𝑧)=𝑝₁₁(𝑧)⋯𝑝_1𝑟(𝑧)⋯𝑝₁₁𝜔_𝑎−1𝑧⋯𝑝_1𝑟𝜔_𝑎−1𝑧,(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)=𝐼_𝑎𝑟,(3.25) or equivalently, 𝐹(𝑧)^∗𝐹(𝑧)=𝐼_𝑎𝑟−𝑓(𝑧)^∗𝑓(𝑧), which is a nonnegative definite Hermitian matrix for |𝑧|=1 so that det𝐼_𝑎𝑟−𝑓(𝑧)^∗𝑓(𝑧)⩾0∀|𝑧|=1,(3.26) and this gives ^𝑎−1_𝑟𝑙=0_𝑗=1||𝑝_1𝑗(𝜔_𝑙𝑧)||²⩽1∀|𝑧|=1.(3.27)
In fact, we have ⎛⎜⎜⎝𝐼_𝑎𝑟𝑓(𝑧)^∗𝑓(𝑧)1⎞⎟⎟⎠⎛⎜⎜⎝𝐼^𝑎𝑟−𝑓(𝑧)^∗−𝑓(𝑧)1⎞⎟⎟⎠=⎛⎜⎜⎝𝐼^𝑎𝑟det⎛⎜⎜⎝⎛⎜⎜⎝𝐼_𝑎𝑟−𝑓(0𝑧)^∗𝑓(𝑧)1−𝑓(0𝑧)𝑓(𝑧)_∗⎞⎟⎟⎠,𝑓(𝑧)^∗𝑓(𝑧)1⎞⎟⎟⎠⎞⎟⎟⎠=det⎛⎜⎜⎝⎛⎜⎜⎝𝐼^𝑎𝑟𝑓(𝑧)^∗01−𝑓(𝑧)𝑓(𝑧)^∗det⎛⎜⎜⎝⎛⎜⎜⎝𝐼_𝑎𝑟⎞⎟⎟⎠⎞⎟⎟⎠,−𝑓(𝑧)^∗−𝑓(𝑧)1⎞⎟⎟⎠⎞⎟⎟⎠=det⎛⎜⎜⎝⎛⎜⎜⎝𝐼^𝑎𝑟−𝑓(𝑧)^∗01−𝑓(𝑧)𝑓(𝑧)^∗⎞⎟⎟⎠⎞⎟⎟⎠,(3.28) then det𝐼_𝑎𝑟−𝑓(𝑧)^∗𝑓(𝑧)1−𝑓(𝑧)𝑓(𝑧)^∗=1−𝑓(𝑧)𝑓(𝑧)^∗²,(3.29) and it gives 1−𝑓(𝑧)𝑓(𝑧)^∗⩾0, for all |𝑧|=1, that is, ^𝑎−1_𝑟𝑙=0_𝑗=1||𝑝_𝑖𝑗(𝜔_𝑙𝑧)||²⩽1∀|𝑧|=1,1⩽𝑖⩽𝑟.(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), 𝑄_𝑗(𝑧), 𝑗=1,…,𝑁 in their polyphase forms: 𝑃(𝑧)=√𝑎𝑎𝑃¹𝑄(𝑧^𝑎)+𝑧𝑃₂(𝑧^𝑎)+⋯+𝑧^𝑎−1𝑃_𝑎(𝑧^𝑎),(3.31)_𝑗(𝑧)=√𝑎𝑎𝑄^𝑗1(𝑧^𝑎)+𝑧𝑄_𝑗2(𝑧^𝑎)+⋯+𝑧^𝑎−1𝑄_𝑗𝑎(𝑧^𝑎),𝑗=1,…,𝑁,(3.32) where 𝑃_𝑖(𝑧), 𝑄_𝑖𝑗(𝑧), 𝑖=1,…,𝑎; 𝑗=1,…,𝑁 are 𝑟×𝑟 matrices and their every element is Laurent polynomial. Observe that 𝑅(𝑧)√𝑎𝑎⎡⎢⎢⎢⎢⎢⎢⎣𝐼^𝑟𝑧⁻¹𝐼_𝑟⋯𝑧^1−𝑎𝐼_𝑟𝐼_𝑟𝜔₁𝑧⁻¹𝐼_𝑟⋯𝜔₁𝑧^1−𝑎𝐼_𝑟𝐼⋮⋮⋮_𝑟𝜔_𝑎−1𝑧⁻¹𝐼_𝑟⋯𝜔_𝑎−1𝑧^1−𝑎𝐼_𝑟⎤⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎢⎣𝑄𝑃₁₁₁(𝑧^𝑎)𝑃₂(𝑧^𝑎)⋯𝑃_𝑎(𝑧^𝑎)𝑄⋮(𝑧^𝑎)𝑄₁₂⋮(𝑧^𝑎)⋯𝑄_1𝑎⋮_𝑁1(𝑧^𝑎)(𝑧^𝑎)𝑄_𝑁2(𝑧^𝑎)⋯𝑄_𝑁𝑎(𝑧^𝑎)⎤⎥⎥⎥⎥⎥⎥⎦.(3.33)
Therefore, we have 𝑎⎡⎢⎢⎢⎢⎢⎢⎣𝑃¹𝑄₁₁(𝑧^𝑎)𝑃₂(𝑧^𝑎)⋯𝑃_𝑎(𝑧^𝑎)𝑄⋮(𝑧^𝑎)𝑄₁₂⋮(𝑧^𝑎)⋯𝑄_1𝑎⋮_𝑁1(𝑧^𝑎)(𝑧^𝑎)𝑄_𝑁2(𝑧^𝑎)⋯𝑄_𝑁𝑎(𝑧^𝑎)⎤⎥⎥⎥⎥⎥⎥⎦^∗⎡⎢⎢⎢⎢⎢⎢⎣𝑄𝑃₁₁₁(𝑧^𝑎)𝑃₂(𝑧^𝑎)⋯𝑃_𝑎(𝑧^𝑎)𝑄⋮(𝑧^𝑎)𝑄₁₂⋮(𝑧^𝑎)⋯𝑄_1𝑎⋮_𝑁1(𝑧^𝑎)=⎡⎢⎢⎢⎢⎢⎢⎣𝐼_𝑟(𝑧^𝑎)𝑄_𝑁2(𝑧^𝑎)⋯𝑄_𝑁𝑎(𝑧^𝑎)⎤⎥⎥⎥⎥⎥⎥⎦𝑧⁻¹𝐼_𝑟⋯𝑧^1−𝑎𝐼_𝑟𝐼_𝑟𝜔₁𝑧⁻¹𝐼_𝑟⋯𝜔₁𝑧^1−𝑎𝐼_𝑟𝐼⋮⋮⋮_𝑟𝜔_𝑎−1𝑧⁻¹𝐼_𝑟⋯𝜔_𝑎−1𝑧^1−𝑎𝐼_𝑟⎤⎥⎥⎥⎥⎥⎥⎦^∗𝑅(𝑧)^∗𝑅(𝑧)⎡⎢⎢⎢⎢⎢⎢⎣𝐼^𝑟𝑧⁻¹𝐼_𝑟⋯𝑧^1−𝑎𝐼_𝑟𝐼_𝑟𝜔₁𝑧⁻¹𝐼_𝑟⋯𝜔₁𝑧^1−𝑎𝐼_𝑟𝐼⋮⋮⋮_𝑟𝜔_𝑎−1𝑧⁻¹𝐼_𝑟⋯𝜔_𝑎−1𝑧^1−𝑎𝐼_𝑟⎤⎥⎥⎥⎥⎥⎥⎦,(3.34) and it follows from (3.4), that ⎡⎢⎢⎢⎢⎢⎢⎣𝑄𝑃₁₁₁(𝑧^𝑎)𝑃₂(𝑧^𝑎)⋯𝑃_𝑎(𝑧^𝑎)Q⋮(𝑧^𝑎)𝑄₁₂⋮(𝑧^𝑎)⋯𝑄_1𝑎⋮_𝑁1(𝑧^𝑎)(𝑧^𝑎)𝑄_𝑁2(𝑧^𝑎)⋯𝑄_𝑁𝑎(𝑧^𝑎)⎤⎥⎥⎥⎥⎥⎥⎦^∗⎡⎢⎢⎢⎢⎢⎢⎣𝑄𝑃₁₁₁(𝑧^𝑎)𝑃₂(𝑧^𝑎)⋯𝑃_𝑎(𝑧^𝑎)𝑄⋮(𝑧^𝑎)𝑄₁₂⋮(𝑧^𝑎)⋯𝑄_1𝑎⋮_𝑁1(𝑧^𝑎)(𝑧^𝑎)𝑄_𝑁2(𝑧^𝑎)⋯𝑄_𝑁𝑎(𝑧^𝑎)⎤⎥⎥⎥⎥⎥⎥⎦=𝐼^𝑎𝑟,∀|𝑧|=1.(3.35) And it is easy to obtain (3.35) from (3.4).
For convenience, we denote 𝑧^𝑎=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),…,𝜙_𝑟(𝑥))^𝑇 with Φ continuous at 0 and Φ(0)≠0, its symbol 𝑃(𝑧) satisfies ^𝑟_{𝑖=1𝑎−1𝑟𝑙=0}_𝑗=1||𝑝_𝑖𝑗𝜔_𝑙𝑧||<1,∀|𝑧|=1.(3.36) Then there exist minimum-energy multiwavelet frames associated with Φ.

Proof. Let 𝑃_𝑗(𝑧),𝑗=1,…,𝑎 be the polynomial components of 𝑃(𝑧), that is, 𝑃(𝑧)=√𝑎𝑎𝑃¹(𝑧^𝑎)+𝑧𝑃₂(𝑧^𝑎)+⋯+𝑧^𝑎−1𝑃_𝑎(𝑧^𝑎).(3.37)
Using (3.34) and (3.35), we can get ^𝑟_𝑎𝑖=1_𝑟𝑙=1_𝑗=1||𝑝^𝑙𝑖𝑗(𝑢)||²<1.(3.38)
Then we can find 𝑟 real numbers 𝑥₁,𝑥₂,…,𝑥_𝑟, with ^𝑟_𝑖=1𝑥_𝑖=1,^𝑎_𝑟𝑙=1_𝑗=1||𝑝^𝑙𝑖𝑗(𝑢)||²<𝑥_𝑖,1⩽𝑖⩽𝑟.(3.39) By the Riesz lemma [27, Lemma 6.13], we can find Laurent polynomials 𝑃^𝑖𝑎+1(𝑧),𝑖=1,…,𝑟 satisfying ^𝑎_𝑟𝑙=1_𝑗=1||𝑝^𝑙𝑖𝑗(𝑢)||²+||𝑝^𝑖𝑎+1(𝑢)||²=𝑥_𝑖,1⩽𝑖⩽𝑟.(3.40)
For every 𝑖∈{1,…,𝑟}, using the method in the reference [15,Theorem 3] on the unit vector 1√𝑥_𝑖𝑃^1𝑖1(𝑧)⋯𝑃^1𝑖𝑟(𝑧)⋯𝑃^𝑎𝑖1(𝑧)⋯𝑃^𝑎𝑖𝑟(𝑧)𝑃^𝑖𝑎+1(𝑧),(3.41) we can get a matrix 𝑅^𝑖(𝑧)=1√𝑥_𝑖⎛⎜⎜⎜⎜⎜⎜⎝𝑄𝑃^1𝑖1(𝑧)⋯𝑃^1𝑖𝑟(𝑧)⋯P^𝑎𝑖1(𝑧)⋯𝑃^𝑎𝑖𝑟(𝑧)𝑃^𝑖𝑎+1(𝑧)_𝑖111𝑄⋯(𝑧)⋯𝑄⋯^𝑖𝑟11(𝑧)⋯𝑄⋯^𝑖11𝑎(𝑧)⋯𝑄⋯^{𝑖𝑟1𝑎}_𝑖1𝑎1(𝑧)𝑄^𝑖1,𝑎+1(𝑧)(𝑧)⋯𝑄^𝑎𝑟11(𝑧)⋯𝑄^{𝑖1𝑎𝑎}(𝑧)⋯𝑄^{𝑖𝑟𝑎𝑎}(𝑧)𝑄^{𝑖𝑎,𝑎+1}(𝑧)⎞⎟⎟⎟⎟⎟⎟⎠,(3.42) which satisfies 𝑅^𝑖(𝑧)^∗𝑅^𝑖(𝑧)=𝐼_𝑎𝑟+1.
Therefor, the block matrix 𝑅(𝑧)=⎛⎜⎜⎜⎜⎜⎜⎝√√𝑥₁𝑅¹(𝑧)√𝑥₂⋮𝑅²(𝑧)𝑥_𝑎𝑅^𝑎(𝑧)⎞⎟⎟⎟⎟⎟⎟⎠(3.43) satisfies 𝑅(𝑧)^∗𝑅(𝑧)=𝐼_𝑎𝑟+1.
We can get matrix 𝑅(𝑧) which satisfies 𝑅(𝑧)^∗𝑅(𝑧)=𝐼_𝑎𝑟, 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),…,𝜙_𝑟(𝑥))^𝑇 with Φ continuous at 0 and Φ(0)≠0 a compactly supported multiscaling vector-valued function. If the block matrix 𝑃(𝑧)𝑃𝜔₁𝑧⋯𝑃𝜔_𝑎−1𝑧(3.44) satisfies standard orthogonal by row, then there exist a minimum-energy multiwavelet frames associated with the function Φ.

Proof. Let 𝑃_𝑗(𝑧), 𝑗=1,…,𝑎 are the polynomial components of 𝑃(𝑧), that is, 𝑃(𝑧)=√𝑎𝑎𝑃¹(𝑧^𝑎)+𝑧𝑃₂(𝑧^𝑎)+⋯+𝑧^𝑎−1𝑃_𝑎(𝑧^𝑎),(3.45) with (3.34) and (3.35), we can know that the block matrix 𝑁(𝑢)=𝑃₁(𝑢)𝑃₂(𝑢)⋯𝑃_𝑎(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)𝐻₁(𝑢)=⎛⎜⎜⎝1𝑁(𝑢)⎞⎟⎟⎠,(3.47) with 𝑁(𝑢) also a matrix standard orthogonal by row.
By mathematical induction, there are 𝑟 paraunitary matrices 𝐻₁(𝑢),…,𝐻_𝑟(𝑢) satisfying 𝑁(𝑢)𝐻₁(𝑢)⋯𝐻_𝑎(𝑢)=⎛⎜⎜⎜⎜⎝⋮10⋱⋯⋯⋮01⋮00⋱⋯⋯⋮00⎞⎟⎟⎟⎟⎠_{𝑟×𝑎𝑟},(3.48) then the matrix 𝑁(𝑢) is equivalent to the front 𝑟 rows in the paraunitary matrix 𝐻₁(𝑢)^∗𝐻₂(𝑢)^∗⋯𝐻_𝑎(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 𝑃_𝑎+1(𝑢) to make the block matrix 𝑃₁(𝑢)𝑃₂(𝑢)⋯𝑃_𝑎(𝑢)𝑃_𝑎+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),…,𝜙_𝑟(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, ^𝑎−1_𝑟𝑙=0_𝑗=1||𝑝_𝑖𝑗𝜔_𝑙𝑧||²<1,∀|𝑧|=1,1⩽𝑖⩽𝑟,(3.50) and there exists a matrix 𝑃_𝑎+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),…,𝛼_𝑛(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),…,𝛽_𝑚(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 𝑃_𝑎+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,…,Ψ^𝑁}. Now, we consider the projection operators 𝐏_𝑗 of 𝐿²(ℝ) onto the nested subspace 𝑉_𝑗 defined by 𝐏_𝑗𝑓∶=^𝑟_𝜏=1_𝑘∈ℤ𝑓,𝜙_{𝜏,𝑗,𝑘}𝜙_{𝜏,𝑗,𝑘}.(4.1) Then the formula (2.8) can be rewritten as 𝐏_𝑗+1𝑓−𝐏_𝑗𝑓∶=^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈ℤ𝑓,𝜓^{𝑖𝜏,𝑗,𝑘}𝜓^{𝑖𝜏,𝑗,𝑘}.(4.2) In other words, the error term 𝑔_𝑗=𝐏_𝑗+1𝑓−𝐏_𝑗𝑓 between consecutive projections is given by the frame expansion: 𝑔_𝑗=^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈ℤ𝑓,𝜓^{𝑖𝜏,𝑗,𝑘}𝜓^{𝑖𝜏,𝑗,𝑘}.(4.3)

Suppose that the error term 𝑔_𝑗 has other expansion in terms of the frames {Ψ¹,…,Ψ^𝑁}, that is, 𝑔_𝑗=^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈ℤ𝑐_{𝜏,𝑗,𝑘}𝜓^{𝑖𝜏,𝑗,𝑘}.(4.4) Then by using both (4.3) and (4.4), we have 𝑔_𝑗,𝑓=^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈ℤ|||𝑓,𝜓_{𝑖𝜏,𝑗,𝑘}|||²=^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈𝐙𝑐_{𝜏,𝑗,𝑘}𝑓,𝜓^{𝑖𝜏,𝑗,𝑘},(4.5) and this derives 0⩽^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈𝐙|||𝑐_{𝜏,𝑗,𝑘}−𝑓,𝜓^{𝑖𝜏,𝑗,𝑘}|||²=^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈𝐙||𝑐_{𝜏,𝑗,𝑘}||²−2^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈𝐙𝑐_{𝜏,𝑗,𝑘}𝑓,𝜓^{𝑖𝜏,𝑗,𝑘}+^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈ℤ|||𝑓,𝜓_{𝑖𝜏,𝑗,𝑘}|||²=^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈𝐙||𝑐_{𝜏,𝑗,𝑘}||²−^𝑁_𝑟𝑖=1_𝜏=1_𝑘∈ℤ|||𝑓,𝜓_{𝑖𝜏,𝑗,𝑘}|||².(4.6)

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

We next discuss minimum-energy multiwavelet frames decomposition and reconstruction. For any 𝑓∈𝐿²(ℝ), define the vector coefficients as follows: 𝐜_𝑗,𝑘∶=𝑓,Φ_𝑗,𝑘,𝐝_𝑗,𝑘∶=𝑓,Ψ^{𝑖𝑗,𝑘}𝑖=1,…,𝑁.(4.7) The inner product of 𝑓 with vector-valued Φ_𝑗,𝑘, Ψ^{𝑖𝑗,𝑘}, 𝑖=1,…,𝑁 is a vector, its every component is the inner product of 𝑓 with the corresponding component of Φ_𝑗,𝑘, Ψ^{𝑖𝑗,𝑘}, 𝑖=1,…,𝑁.

(1) Decomposition Algorithm
suppose the vector coefficients {𝐜_𝑗+1,𝑙∶𝑙∈𝐙} are known. By the two-scale relations (2.2) and (3.1), we have Φ_𝑗,𝑙(𝑥)=1√𝑎_𝑘∈ℤ𝑃_{𝑘−𝑎𝑙}Φ_𝑗+1,𝑘(𝑥),Ψ^{𝑖𝑗,𝑙}(𝑥)=1√𝑎_𝑘∈ℤ𝑄^{𝑖𝑘−𝑎𝑙}Ψ^{𝑖𝑗+1,𝑘}(𝑥),𝑖=1,…,𝑁.(4.8) Then, the decomposition algorithm is given as 𝐜_𝑗,𝑙=1√𝑎_𝑘∈ℤ𝑃_{𝑘−𝑎𝑙}𝐜_𝑗+1,𝑘,𝐝^{𝑖𝑗,𝑙}=1√𝑎_𝑘∈ℤ𝑄^{𝑖𝑘−𝑎𝑙}𝐝^{𝑖𝑗+1,𝑘},𝑖=1,…,𝑁.(4.9)

(2) Reconstruction Algorithm
from (3.16), it follow that Φ_𝑗+1,𝑙(𝑥)=1√𝑎_𝑘𝑃^{∗𝑙−𝑎𝑘}Φ_𝑗,𝑘(𝑥)+^𝑁_𝑖=1𝑄^{𝑖∗𝑙−𝑎𝑘}Ψ^{𝑖𝑗,𝑘}(𝑥).(4.10) Taking the inner products on both sides of this equality, we get 𝐜_𝑗+1,𝑙=1√𝑎_𝑘𝑃^{∗𝑙−𝑎𝑘}𝐜_𝑗,𝑘+^𝑁_𝑖=1𝑄^{𝑖∗𝑙−𝑎𝑘}𝐝^{𝑖𝑗,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 𝑁^𝑎𝑚(𝑥) with dilation factor 𝑎 has the two-scale relation as follows: 𝑁^𝑎𝑚(𝜔)=𝑃^𝑎𝑚(𝑧)𝑁^𝑎𝑚𝜔𝑎,𝑃^𝑎𝑚(𝑧)=1+𝑧+⋯𝑎+𝑧^𝑎−1^𝑚,𝑧=𝑒^{−𝑖𝜔/𝑎}.(5.1)

In addition, if a scale wavelet 𝜙(𝑥) satisfies the refinable function 𝜙(𝑥)=^𝑘1_𝑘_=𝑘0𝑝_𝑘𝜙(𝑎𝑥−𝑘),(5.2) and let Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥−1),…,𝜙(𝑥−𝑟+1))^𝑇, then the vector-valued function Φ satisfies (2.2) with some matrixes {𝑃_𝑘}.

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 𝑃²²(𝑧)=14+12𝑧+14𝑧².(5.3) Take 𝜙(𝑥)=𝑁²²(𝑥), and the support of this function is [0,2]. The function satisfies 𝜙(𝑥)=14𝜙(2𝑥)+21𝜙(2𝑥−1)+14𝜙(2𝑥−2).(5.4)
Let Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥−1))^𝑇, and Φ(𝑥)=⎛⎜⎜⎝120012⎞⎟⎟⎠Φ(2𝑥)+⎛⎜⎜⎝120012⎞⎟⎟⎠Φ(2𝑥−1)+⎛⎜⎜⎝100212⎞⎟⎟⎠Φ(2𝑥−2)+⎛⎜⎜⎝210012⎞⎟⎟⎠Φ(2𝑥−3).(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)=√22⎛⎜⎜⎝⎛⎜⎜⎝120012⎞⎟⎟⎠+𝑢⎛⎜⎜⎝100212⎞⎟⎟⎠⎞⎟⎟⎠.(5.6) Take 𝑃₃(𝑢)=√22⎛⎜⎜⎜⎝⎛⎜⎜⎜⎝√22√2002⎞⎟⎟⎟⎠+𝑢⎛⎜⎜⎜⎝−√0022−√22⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠,(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)=√22⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝12𝑢121212−−0211−02𝑢1102𝑢012𝑢−2𝑢12−121222𝑢−2𝑢2𝑢⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠,(5.9) which satisfy the formula (3.35). Then we take symbols as 𝑄₁(𝑧)=1𝑄2⎛⎜⎜⎝⎛⎜⎜⎝−01−01⎞⎟⎟⎠+𝑧⎛⎜⎜⎝1001₂⎞⎟⎟⎠⎞⎟⎟⎠,(𝑧)=12⎛⎜⎜⎝⎛⎜⎜⎝−102012⎞⎟⎟⎠+𝑧⎛⎜⎜⎝−012012⎞⎟⎟⎠+𝑧²⎛⎜⎜⎝−010212⎞⎟⎟⎠+𝑧³⎛⎜⎜⎝−010212⎞⎟⎟⎠⎞⎟⎟⎠.(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.

(a)

(b)

(c)

Figure 1 

Example 5.2. With 𝑎=2, the symbol of the B-spline 𝑁²³(𝑥) is 𝑃²³(𝑧)=18+38𝑧+38𝑧²+18𝑧³.(5.11) Take 𝜙(𝑥)=𝑁²³(𝑥), and the support of this function is [0,3].(1) Let Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥−1))^𝑇, and Φ(𝑥)=⎛⎜⎜⎝140014⎞⎟⎟⎠Φ(2𝑥)+⎛⎜⎜⎝201012⎞⎟⎟⎠Φ(2𝑥−1)+⎛⎜⎜⎝11441+⎛⎜⎜⎝100441⎞⎟⎟⎠Φ(2𝑥−2)212⎞⎟⎟⎠Φ(2𝑥−3)+⎛⎜⎜⎝410014⎞⎟⎟⎠Φ(2𝑥−4).(5.12)The symbol of Φ has polyphase components as follows: 𝑃₁(𝑢)=√22⎛⎜⎜⎝⎛⎜⎜⎝140014⎞⎟⎟⎠+𝑢⎛⎜⎜⎝41114414⎞⎟⎟⎠+𝑢²⎛⎜⎜⎝100𝑃₂414⎞⎟⎟⎠⎞⎟⎟⎠,(𝑢)=√22⎛⎜⎜⎝⎛⎜⎜⎝120012⎞⎟⎟⎠+𝑢⎛⎜⎜⎝100212⎞⎟⎟⎠⎞⎟⎟⎠.(5.13) Take 𝑃₃(𝑢)=⎛⎜⎜⎜⎝−√24−√24𝑢−√2√2−12+2𝑢024𝑢+√24𝑢²√22𝑢0−2𝑢+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)=√2𝑄2⎛⎜⎜⎝⎛⎜⎜⎝10−01⎞⎟⎟⎠+𝑧⎛⎜⎜⎝−011₂0⎞⎟⎟⎠⎞⎟⎟⎠,(𝑧)=√22⎛⎜⎜⎜⎝⎛⎜⎜⎜⎝−√24−√2004⎞⎟⎟⎟⎠+𝑧²⎛⎜⎜⎜⎝√24√2√424√24⎞⎟⎟⎟⎠+𝑧⁴⎛⎜⎜⎜⎝−√0024−√2𝑄₃4⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠,(𝑧)=√22⎛⎜⎜⎝⎛⎜⎜⎝−104−014⎞⎟⎟⎠+𝑧⎛⎜⎜⎝120012⎞⎟⎟⎠+𝑧²⎛⎜⎜⎝−114−14414⎞⎟⎟⎠+𝑧³⎛⎜⎜⎝−0102−21⎞⎟⎟⎠+𝑧⁴⎛⎜⎜⎝100414⎞⎟⎟⎠⎞⎟⎟⎠.(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(𝑥−1),𝜙(𝑥−2))^𝑇, which satisfies Φ(𝑥)=14⎛⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝0010000003⎞⎟⎟⎟⎟⎠Φ(2𝑥+2)+⎛⎜⎜⎜⎜⎝0000031001⎞⎟⎟⎟⎟⎠Φ(2𝑥+1)+⎛⎜⎜⎜⎜⎜⎝1300111+⎛⎜⎜⎜⎜⎜⎝1110003⎞⎟⎟⎟⎟⎟⎠Φ(2𝑥)01300301⎞⎟⎟⎟⎟⎟⎠Φ(2𝑥−1)+⎛⎜⎜⎜⎜⎜⎝11130300111+⎛⎜⎜⎜⎜⎜⎝13⎞⎟⎟⎟⎟⎟⎠Φ(2𝑥−2)31100101331⎞⎟⎟⎟⎟⎟⎠Φ(2𝑥−3)+⎛⎜⎜⎜⎜⎜⎝10010130+⎛⎜⎜⎜⎜⎜⎝1000311⎞⎟⎟⎟⎟⎟⎠Φ(2𝑥−4)3110013⎞⎟⎟⎟⎟⎟⎠Φ(2𝑥−5)+⎛⎜⎜⎜⎜⎝1000013000⎞⎟⎟⎟⎟⎠Φ(2𝑥−6)+⎛⎜⎜⎜⎜⎝3100000000⎞⎟⎟⎟⎟⎠Φ(2𝑥−7)⎞⎟⎟⎟⎟⎟⎠(5.16)and the symbol of this multiscaling vector-valued function has the following polyphase components: 𝑃₁(𝑢)=√28⎛⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝0010000003⎞⎟⎟⎟⎟⎠+⎛⎜⎜⎜⎜⎜⎝13001110003⎞⎟⎟⎟⎟⎟⎠𝑢+⎛⎜⎜⎜⎜⎜⎝111303001113⎞⎟⎟⎟⎟⎟⎠𝑢²+⎛⎜⎜⎜⎜⎜⎝10010130311⎞⎟⎟⎟⎟⎟⎠𝑢³+⎛⎜⎜⎜⎜⎝1000010030⎞⎟⎟⎟⎟⎠𝑢⁴𝑃₂⎞⎟⎟⎟⎟⎟⎠,(𝑢)=√28⎛⎜⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝0100003001⎞⎟⎟⎟⎟⎠+⎛⎜⎜⎜⎜⎜⎝10111300301⎞⎟⎟⎟⎟⎟⎠𝑢+⎛⎜⎜⎜⎜⎜⎝131100101331⎞⎟⎟⎟⎟⎟⎠𝑢²+⎛⎜⎜⎜⎜⎜⎝10003110013⎞⎟⎟⎟⎟⎟⎠𝑢³+⎛⎜⎜⎜⎜⎝1000000300⎞⎟⎟⎟⎟⎠𝑢⁴⎞⎟⎟⎟⎟⎟⎠.(5.17)
Let 𝑃₃(𝑢)=√2×8⎛⎜⎜⎜⎜⎜⎝√6+√63𝑢0−√6+√63𝑢√−(𝒜)𝒜06𝑢0−√6√000−(ℬ)ℬ06𝑢²0−√6𝑢√06𝑢³−√6𝑢²⎞⎟⎟⎟⎟⎟⎠,(5.18) where 𝒜 denotes √6𝑢+(√6/3)𝑢², and ℬ 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 𝑄=₁√(𝑧)22++⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝𝑧𝑧−0₂⎛⎜⎜⎜⎜⎝0.0−.00.−2020870.3.2093.510207426035770239888755052680608291856079171412673604449478742041351327121633030835569283−4−−00.0−−.01.006108..19.005986475059715512014664429847193356037196788535801078991900332770592263048829489099298551147−−−000−−...010100.974.0.90198143029608681540084098280692681845645817885797889768166311230367699625176117049804193430⎞⎟⎟⎟⎟⎠947⎞⎟⎟⎟⎟⎠+𝑧₃⎛⎜⎜⎜⎜⎝−−−000.2..200474800902163139585610964699527520745454366418−−−000...10025921727925626095098548948941916167893376871626−−000..00.11118612828622368291239870841607905996187483682575⎞⎟⎟⎟⎟⎠+𝑧₄⎛⎜⎜⎜⎜⎝−−−000..01.052912772952662059089584984914961176839367817666−−000..00.11118612828622368291239870841607905996187483682575−−00.00.0.000635076902471250767346052088509809537521795765421⎞⎟⎟⎟⎟⎠+𝑧₅⎛⎜⎜⎜⎜⎝−−00.00.0.111816218826223862192938078146709509691784386174478−0−.000.0.0060305796042721570637406502858089059357271975514959000⎞⎟⎟⎟⎟⎠𝑄⎛⎜⎜⎜⎜⎝−0−.000.00.0603057960427215706374065028580890593572719755149590000₂00⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠,=√(𝑧)22++⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝𝑧𝑧−0₂⎛⎜⎜⎜⎜⎝0.0−000...01000.084.900154941145856710653527380045239798500628166020026627308536196240002937360904815475115743181543−−57000.3..−20001065..500.058000719155908090538904701581839037279071805992775114951676387306510789373896203512393819150−637.6000.0.23−59020095−5..80103081408.99330933111208926501312667156401640633721113777622430536015097309266954286⎞⎟⎟⎟⎟⎠126704543⎞⎟⎟⎟⎟⎠+𝑧₃⎛⎜⎜⎜⎜⎝−−000..20.23202398610361911911749441479717973674218891914284−−000..00.07511190552105591568449710630181285878089083947484−−000...00020107343089807340338582620107342272541237655160497⎞⎟⎟⎟⎟⎠+𝑧₄⎛⎜⎜⎜⎜⎝−−000...00075111905521055915684497106301812858780890839464347−−000...00020107343089807340338582620107342272541237655160497−0−0.0.000.0062084426365686119124170832744190017842454239031464⎞⎟⎟⎟⎟⎠+𝑧₅⎛⎜⎜⎜⎜⎝−−000..00.02010734308980734033858262010734227254123764316752544−−000.0..00000642834266615816119284770312404980147422404946471000⎞⎟⎟⎟⎟⎠𝑄⎛⎜⎜⎜⎜⎝−−000.0..0000064283426661581611928477031240498014742240494647100000₃0⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠,=(𝑧√)22++⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝𝑧𝑧−₂−⎛⎜⎜⎜⎜⎝00−0−..−0.020000.540.3.02704923272083574472677698923285069492010284680224428029384318205989142606822031302596671495450−0.0.0−−−25.1000749...21000488763917363697367897251853678475623750753316577334179811971660517249705081680395519081−43200.01.001.570..60093.303013937887135047304582098172750012374005916225271245799961500838565666412379920044260⎞⎟⎟⎟⎟⎠38148⎞⎟⎟⎟⎟⎠+𝑧₃⎛⎜⎜⎜⎜⎝−−0.000..660492638307650887140601636549255674096483107846−000..01.0618067909871351745975466048066249527794184753236000...110067609393497661138099499786711780628088902212884⎞⎟⎟⎟⎟⎠+𝑧₄⎛⎜⎜⎜⎜⎝−000..01.0618067909871351745975466048066249527794184753286000...101670096933974616381990994867117807286880029122848000..00.052336564465457842769069168507960202599334630487266⎞⎟⎟⎟⎟⎠+𝑧₅⎛⎜⎜⎜⎜⎝000...101670096933974616381990994867117807286880029111277000...00023565344654678527490691685679002625093946303359764000⎞⎟⎟⎟⎟⎠⎛⎜⎜⎜⎜⎝000...00023565344654678527490691685679002625093946303359764000000⎞⎟⎟⎟⎟⎠⎞⎟⎟⎟⎟⎠.(5.19)
They are the symbols of the minimum-energy multiwavelet frames associated with Φ. The graphs of them are shown in Figure 3.
We can discover from Figure 3 that every component of the minimum-energy frames is smooth. When r = 3, it is very difficult to construct the minimum-energy multiwavelet frames with symmetry.

(a)

(b)

(c)

(d)

Figure 2 

(a)

(b)

(c)

(d)

Figure 3 

5.2. 𝑎=3

Example 5.3. With 𝑎=3, the symbol of the B-spline 𝜙(𝑥)=𝑁³³(𝑥) is 𝑃³³(𝑧)=1+3𝑧+6𝑧²+7𝑧³+6𝑧⁴+3𝑧⁵+𝑧⁶27.(5.20) Take 𝜙(𝑥)=𝑁³³(𝑥), and the support of this function is [0,4].(1) Let Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥−1))^𝑇, and this vector-valued function satisfies Φ(𝑥)=19⎛⎜⎜⎝⎛⎜⎜⎝13+0⎛⎜⎜⎝027⎞⎟⎟⎠Φ(3𝑥)+⎛⎜⎜⎝32003⎞⎟⎟⎠Φ(3𝑥−1)+⎛⎜⎜⎝03027⎞⎟⎟⎠Φ(3𝑥−2)21332⎞⎟⎟⎠Φ(3𝑥−3)+⎛⎜⎜⎝333223⎞⎟⎟⎠Φ(3𝑥−4)+⎛⎜⎜⎝32317+⎛⎜⎜⎝7002⎞⎟⎟⎠Φ(3𝑥−5)23⎞⎟⎟⎠Φ(3𝑥−6)+⎛⎜⎜⎝30032⎞⎟⎟⎠Φ(3𝑥−7)+⎛⎜⎜⎝30201⎞⎟⎟⎠Φ(3𝑥−8)⎞⎟⎟⎠.(5.21)The symbol of Φ(𝑥) is 𝑃(𝑧)=127⎛⎜⎜⎝⎛⎜⎜⎝13002⎞⎟⎟⎠+𝑧⎛⎜⎜⎝32003⎞⎟⎟⎠+𝑧²⎛⎜⎜⎝37002⎞⎟⎟⎠+𝑧³⎛⎜⎜⎝721332⎞⎟⎟⎠+𝑧⁴⎛⎜⎜⎝3332+𝑧₅23⎞⎟⎟⎠⎛⎜⎜⎝323172⎞⎟⎟⎠+𝑧⁶⎛⎜⎜⎝70023⎞⎟⎟⎠+𝑧⁷⎛⎜⎜⎝30032⎞⎟⎟⎠+𝑧⁸⎛⎜⎜⎝30021⎞⎟⎟⎠⎞⎟⎟⎠,(5.22) and its polynomial components are 𝑃₁(𝑢)=√327⎛⎜⎜⎝⎛⎜⎜⎝13002⎞⎟⎟⎠+⎛⎜⎜⎝721332⎞⎟⎟⎠𝑢+⎛⎜⎜⎝70203⎞⎟⎟⎠𝑢²𝑃₂⎞⎟⎟⎠,(𝑢)=√327⎛⎜⎜⎝⎛⎜⎜⎝32003⎞⎟⎟⎠+⎛⎜⎜⎝332323⎞⎟⎟⎠𝑢+⎛⎜⎜⎝30032⎞⎟⎟⎠𝑢²𝑃₃⎞⎟⎟⎠,(𝑢)=√327⎛⎜⎜⎝⎛⎜⎜⎝37002⎞⎟⎟⎠+⎛⎜⎜⎝323172⎞⎟⎟⎠𝑢+⎛⎜⎜⎝30201⎞⎟⎟⎠𝑢²⎞⎟⎟⎠,(5.23) then 𝑃₁(𝑢)𝑃₁(𝑢)^∗+𝑃₂(𝑢)𝑃₂(𝑢)^∗+𝑃₃(𝑢)𝑃₃(𝑢)^∗=1486⎛⎜⎜⎝50𝑢+143+50𝑢5𝑢0²+14350+143𝑢+50𝑢₂𝑢+5050𝑢+143+50𝑢⎞⎟⎟⎠.(5.24)(i) This example satisfies the conditions in Theorem 3.5. Let 𝑃₄(𝑢)=√327⎛⎜⎜⎝55𝑢−−55𝑢𝑢₂⎞⎟⎟⎠.(5.25)The sum of 𝑙²-norm for every row in the matrix in (3.34) formed by 𝑃₁(𝑢), 𝑃₂(𝑢), 𝑃₃(𝑢), and 𝑃₄(𝑢) is equivalent to 1. Using Theorem 3.5, we can get the symbols of the minimum-energy frames associated with Φ the following: 𝑄₁(𝑧)=127⎛⎜⎜⎝⎛⎜⎜⎝−1−3002⎞⎟⎟⎠+𝑧⎛⎜⎜⎝−032−03⎞⎟⎟⎠+𝑧²⎛⎜⎜⎝−3−7002⎞⎟⎟⎠+𝑧³⎛⎜⎜⎝7−21−33+𝑧₄2⎞⎟⎟⎠⎛⎜⎜⎝33−322−3⎞⎟⎟⎠+𝑧⁵⎛⎜⎜⎝3−23−17+𝑧₆2⎞⎟⎟⎠⎛⎜⎜⎝70023⎞⎟⎟⎠+𝑧⁷⎛⎜⎜⎝30032⎞⎟⎟⎠+𝑧⁸⎛⎜⎜⎝300𝑄₂21⎞⎟⎟⎠⎞⎟⎟⎠,(𝑧)=127⎛⎜⎜⎜⎝⎛⎜⎜⎜⎝0−9√3002⎞⎟⎟⎟⎠+𝑧⎛⎜⎜⎜⎝9√32000⎞⎟⎟⎟⎠+𝑧³⎛⎜⎜⎜⎝00−90√32⎞⎟⎟⎟⎠+𝑧⁴⎛⎜⎜⎜⎝9√003𝑄₃20⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠,(𝑧)=127⎛⎜⎜⎜⎝⎛⎜⎜⎜⎝9√32−3√003⎞⎟⎟⎟⎠+𝑧⎛⎜⎜⎜⎝−3√33√3002⎞⎟⎟⎟⎠+𝑧³⎛⎜⎜⎜⎝9√0032−3√+𝑧₄3⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝−30√033√3𝑄₄2⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠,(𝑧)=127⎛⎜⎜⎜⎝⎛⎜⎜⎜⎝92335−330035⎞⎟⎟⎟⎠+𝑧⎛⎜⎜⎜⎝−3335−52130035⎞⎟⎟⎟⎠+𝑧²⎛⎜⎜⎜⎝273+𝑧₃03500⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝9002335−3335⎞⎟⎟⎟⎠+𝑧⁴⎛⎜⎜⎜⎝−300335−521335⎞⎟⎟⎟⎠+𝑧⁵⎛⎜⎜⎜⎝27003𝑄35₅0⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠,(𝑧)=127⎛⎜⎜⎜⎝⎛⎜⎜⎜⎝4520133√420566⎞⎟⎟⎟⎠+𝑧⎛⎜⎜⎜⎝√42566−45200133⎞⎟⎟⎟⎠+𝑧²⎛⎜⎜⎜⎝−452+𝑧₃013300⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝45002133√42566⎞⎟⎟⎟⎠+𝑧⁴⎛⎜⎜⎜⎝4050√266−452133⎞⎟⎟⎟⎠+𝑧⁵⎛⎜⎜⎜⎝−45002𝑄133₆0⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠,(𝑧)=127⎛⎜⎜⎜⎝⎛⎜⎜⎜⎝14√0952√30395⎞⎟⎟⎟⎠+𝑧⎛⎜⎜⎜⎝2√30395√20945⎞⎟⎟⎟⎠+𝑧²⎛⎜⎜⎜⎝24√95−√+𝑧₃0095⎞⎟⎟⎟⎠⎛⎜⎜⎜⎝1040√952√3395⎞⎟⎟⎟⎠+𝑧⁴⎛⎜⎜⎜⎝30302√95√2945⎞⎟⎟⎟⎠+𝑧⁵⎛⎜⎜⎜⎝2040√95−√95⎞⎟⎟⎟⎠⎞⎟⎟⎟⎠.(5.26) The graphs of Φ and the minimum-energy frames associated with it are shown in Figure 4.
From this figure, we can discover that every component of the minimum-energy frames is smooth. The first vector-valued function of frames is antisymmetry and the second function vanishes.(ii) In fact, this example also satisfies the conditions of Corollary 3.7. Take 𝑃₄(𝑢)=√327⎛⎜⎜⎜⎝5+5𝑢4325√−5𝑢−5𝑢₂2−5√2𝑢0−432𝑢05√2𝑢−5√2𝑢²⎞⎟⎟⎟⎠(5.27)and with 𝑃₁(𝑢), 𝑃₂(𝑢), and 𝑃₃(𝑢) to form matrix (3.49), which satisfies standard orthogonal by row. By Theorem 3.6, we can get minimum-energy multiwavelet frames associated with Φ.(2) Let Φ(𝑥)=(𝜙(𝑥),𝜙(𝑥−1),𝜙(𝑥−2))^𝑇. This vector-valued function satisfies Φ(𝑥)=19⎛⎜⎜⎜⎜⎝⎛⎜⎜⎜⎜⎝001+⎛⎜⎜⎜⎜⎜⎝0010020037⎞⎟⎟⎟⎟⎠Φ(3𝑥+2)+⎛⎜⎜⎜⎜⎝0000013001⎞⎟⎟⎟⎟⎠Φ(3𝑥+1)+⎛⎜⎜⎜⎜⎝1300001002⎞⎟⎟⎟⎟⎠Φ(3𝑥)00310003⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−1)+⎛⎜⎜⎜⎜⎜⎝27031200301⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−2)+⎛⎜⎜⎜⎜⎜⎝73121+⎛⎜⎜⎜⎜⎜⎝211300102⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−3)123700313⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−4)+⎛⎜⎜⎜⎜⎜⎝112370031231⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−5)+⎛⎜⎜⎜⎜⎜⎝131003121+⎛⎜⎜⎜⎜⎜⎝020101312⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−6)12373⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−7)+⎛⎜⎜⎜⎜⎜⎝10010237032⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−8)+⎛⎜⎜⎜⎜⎜⎝10003700+⎛⎜⎜⎜⎜⎝002001001321⎞⎟⎟⎟⎟⎟⎠Φ(3𝑥−9)𝑃3⎞⎟⎟⎟⎟⎠Φ(3𝑥−10)+⎛⎜⎜⎜⎜⎝1000013000⎞⎟⎟⎟⎟⎠Φ(3𝑥−11)+⎛⎜⎜⎜⎜⎝1003000000⎞⎟⎟⎟⎟⎠Φ(3𝑥−12)₁⎞⎟⎟⎟⎟⎠,(𝑢)𝑃₁(𝑢)^∗+𝑃₂(𝑢)𝑃₂(𝑢)^∗+𝑃₃(𝑢)𝑃₃(𝑢)^∗=1×⎛⎜⎜⎜⎜⎜⎝1729𝑢²+50𝑢+141+50𝑢+𝑢²1𝑢³+50𝑢²+141𝑢+50+𝑢𝑢1⁴+50𝑢³+141𝑢²+501𝑢+1𝑢+50+141𝑢+50𝑢²+𝑢³1𝑢²+50𝑢+141+50𝑢+𝑢²1𝑢³+50𝑢²+1411+50𝑢+141𝑢₂𝑢+50+𝑢+50𝑢³+𝑢⁴1𝑢+50+141𝑢+50𝑢²+𝑢³1𝑢²+50𝑢+141+50𝑢+𝑢²⎞⎟⎟⎟⎟⎟⎠.(5.28)
Let 𝑃₄(𝑢)=⎛⎜⎜⎜⎜⎝−𝑥𝑥0((𝑢𝑢))𝑢−𝑥𝑥0((𝑢𝑢))𝑢𝑥−(𝑥𝑢0)(𝑢𝑢)₂𝑦0(𝑢)𝑦(0𝑢)𝑢0000𝑦(𝑢)𝑢²⎞⎟⎟⎟⎟⎠,(5.29) where 𝑥𝑦((𝑢𝑢))==00..446026570366866265210768875753974+−00.1.465732640190281718642864571935724𝑢𝑢+−00.0.00038387927565564156842229508353965𝑢²𝑢,₂,(5.30) then 𝑃₁(𝑢)𝑃₁(𝑢)^∗+𝑃₂(𝑢)𝑃₂(𝑢)^∗+𝑃₃(𝑢)𝑃₃(𝑢)^∗+𝑃₄(𝑢)𝑃₄(𝑢)^∗=𝐼₃.(5.31) By Corollary 3.7 and Theorem 3.6, we know that the existence of the minimum-energy multiwavelet frames is associated with Φ.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4 

5.3. 𝑎=4

With 𝑎=4, the symbol of the B-spline 𝜙(𝑥)=𝑁⁴⁴(𝑥) is 𝑃⁴⁴(𝑧)=1+4𝑧+10𝑧²+20𝑧³+31𝑧⁴+40𝑧⁵+44𝑧⁶+40𝑧⁷+31𝑧⁸+20𝑧⁹+10𝑧¹⁰+4𝑧¹¹+𝑧¹²256.(5.32) Take 𝜙(𝑥)=𝑁⁴⁴(𝑥), and the support of this function is [0,5], the symbol is 𝑃(𝑧)=1256⎛⎜⎜⎝⎛⎜⎜⎝1020⎞⎟⎟⎠+𝑧⎛⎜⎜⎝2050⎞⎟⎟⎠+𝑧²⎛⎜⎜⎝50100⎞⎟⎟⎠+𝑧₃⎛⎜⎜⎝1031+𝑧₄002⎞⎟⎟⎠⎛⎜⎜⎝3121220⎞⎟⎟⎠+𝑧⁵⎛⎜⎜⎝220252⎞⎟⎟⎠+𝑧₆⎛⎜⎜⎝2522100⎞⎟⎟⎠+𝑧₇⎛⎜⎜⎝203110321+𝑧₈2⎞⎟⎟⎠⎛⎜⎜⎝3132110220⎞⎟⎟⎠𝑧⁹⎛⎜⎜⎝1200252⎞⎟⎟⎠+𝑧₁₀⎛⎜⎜⎝252220⎞⎟⎟⎠+𝑧₁₁⎛⎜⎜⎝220311+𝑧₁₂2⎞⎟⎟⎠⎛⎜⎜⎝3010210⎞⎟⎟⎠+𝑧¹³⎛⎜⎜⎝10005⎞⎟⎟⎠+𝑧₁₄⎛⎜⎜⎝0502⎞⎟⎟⎠+𝑧₁₅⎛⎜⎜⎝0201⎞⎟⎟⎠⎞⎟⎟⎠.(5.33) The polynomial components of 𝑃(𝑧)𝐴=𝜋𝑟²𝑃₁(𝑢)=1128⎛⎜⎜⎝⎛⎜⎜⎝1020⎞⎟⎟⎠+𝑢⎛⎜⎜⎝3121220⎞⎟⎟⎠+𝑢²⎛⎜⎜⎝3132110220⎞⎟⎟⎠+𝑢³⎛⎜⎜⎝3010𝑃210₂⎞⎟⎟⎠⎞⎟⎟⎠,(𝑢)=1128⎛⎜⎜⎝⎛⎜⎜⎝2050⎞⎟⎟⎠+𝑢⎛⎜⎜⎝220252⎞⎟⎟⎠+𝑢²𝑃⎛⎜⎜⎝1200252⎞⎟⎟⎠+𝑢₃⎛⎜⎜⎝1005₃0⎞⎟⎟⎠⎞⎟⎟⎠,(𝑢)=1128⎛⎜⎜⎝⎛⎜⎜⎝50100⎞⎟⎟⎠+𝑢⎛⎜⎜⎝2521200⎞⎟⎟⎠+𝑢²𝑃⎛⎜⎜⎝252220⎞⎟⎟⎠+𝑢₃⎛⎜⎜⎝0502₄⎞⎟⎟⎠⎞⎟⎟⎠,(𝑢)=1128⎛⎜⎜⎝⎛⎜⎜⎝1031002⎞⎟⎟⎠+𝑢⎛⎜⎜⎝2100332112⎞⎟⎟⎠+𝑢²⎛⎜⎜⎝2203112⎞⎟⎟⎠+𝑢³⎛⎜⎜⎝0201⎞⎟⎟⎠⎞⎟⎟⎠,(5.34) and they satisfy the conditions in Theorem 3.5. If we take 𝑃₅(𝑢)=1128⎛⎜⎜⎝𝑎𝑎𝑢++𝑏𝑏𝑢𝑢²++𝑐𝑢𝑐𝑢²³⎞⎟⎟⎠,(5.35) where 𝑎=−4467+√195393534467+√19539353𝑏=−3823+√1288,195393534467+√19539353𝑐=1288,4467+√195393532,(5.36) then the sum of 𝑙²-norm for every row of matrix in (3.49) formed by 𝑃₁(𝑢), 𝑃₂(𝑢), 𝑃₃(𝑢), 𝑃₄(𝑢), and 𝑃₅(𝑢) is equivalent to 1. Using the method in Theorem 3.5, we can get 𝑄₁(𝑢)=⎛⎜⎜⎝++0.𝑧𝑧0₂⎛⎜⎜⎝201.404462585091351400332068463542806.80402.819037120033207805675721866382⎞⎟⎟⎠17⎞⎟⎟⎠+𝑧₃⎛⎜⎜⎝0.107232705771632170.21446550154326434⎞⎟⎟⎠−𝑧₄⎛⎜⎜⎝0.214465501543264340.33242150489205973⎞⎟⎟⎠−𝑧₅⎛⎜⎜⎝−00.1.092710424960545816554737286147394−00.1.054525879437100838068667512584688⎞⎟⎟⎠−𝑧₆⎛⎜⎜⎝−00.1.054525879437100838068667512584688−0.01.11109732833287657757717613423147⎞⎟⎟⎠−𝑧₇⎛⎜⎜⎝−0.01.11109732833287657757717613423147−0.00.72616444966555115534931256148364⎞⎟⎟⎠−𝑧₈⎛⎜⎜⎝−0.00.726164449665551155349312561483640−.005.33392747231852428591250752967335⎞⎟⎟⎠−𝑧₉⎛⎜⎜⎝00..0149471012197054080685576728712779400..105258547549710983866063761350458⎞⎟⎟⎠−𝑧₁₀⎛⎜⎜⎝00..10525854754971098386606376135045800..1011149232893589667850717811453245⎞⎟⎟⎠−𝑧₁₁⎛⎜⎜⎝00..011141292398539866870517871154235400..007065664991685318573290175216816⎞⎟⎟⎠−𝑧₁₂⎛⎜⎜⎝00.0.0706566499168531857329017521681600.0.0503298747539812923561053762360355⎞⎟⎟⎠−𝑧₁₃⎛⎜⎜⎝0.04411175000856272740.02845910936036305⎞⎟⎟⎠−𝑧₁₄⎛⎜⎜⎝0.028459109360363050.014229590680181525⎞⎟⎟⎠−𝑧₁₅⎛⎜⎜⎝0.0142295906801815250.00569180387207261⎞⎟⎟⎠𝑄⎛⎜⎜⎝0.005691803872072610.002845910936036305₂⎞⎟⎟⎠,(𝑢)=−−−⎛⎜⎜⎝𝑧𝑧0₂⎛⎜⎜⎝.−1407.67599577109431044837227635047103.1000.0954175192060183029365045534372⎞⎟⎟⎠49⎞⎟⎟⎠−𝑧₃⎛⎜⎜⎝0.0515961309605332490.065859630320155815⎞⎟⎟⎠−𝑧₄⎛⎜⎜⎝0.0658596303201558150.084679340092975693⎞⎟⎟⎠−𝑧₅⎛⎜⎜⎝00.0.12487074975473913843817428695048700..0110800994571112703812036534851427⎞⎟⎟⎠−𝑧₆⎛⎜⎜⎝−0.00.1685099751141174331207633087112300..0005910549765153896650553313920469⎞⎟⎟⎠−𝑧₇⎛⎜⎜⎝00..000591054976515389665055331392046900.0.0063568159906233342602115257861254⎞⎟⎟⎠−𝑧₈⎛⎜⎜⎝00..0006356815990623334260211525786125400.0.0081486079953141079321907653689132⎞⎟⎟⎠−𝑧₉⎛⎜⎜⎝0.02804743018314890870.018095110731063812⎞⎟⎟⎠−𝑧₁₀⎛⎜⎜⎝0.0180951107310638120.009047550865531906⎞⎟⎟⎠−𝑧₁₁⎛⎜⎜⎝0.0090475508655319060.0036190203462127624⎞⎟⎟⎠𝑄⎛⎜⎜⎝0.00361902034621276240.0018095101731063812₃⎞⎟⎟⎠,(𝑢)=−−−⎛⎜⎜⎝𝑧𝑧0₂⎛⎜⎜⎝.00.5073611537056088610580206632319088704.1−7400.60586316093727078556205282830⎞⎟⎟⎠53⎞⎟⎟⎠−𝑧₃⎛⎜⎜⎝0.0509089905927742250.08491200583693081⎞⎟⎟⎠−𝑧₄⎛⎜⎜⎝0.084912005836930810.123083550988310513⎞⎟⎟⎠−𝑧₅⎛⎜⎜⎝00..0015570691630026288108728633839988600.0.1079470309833669352573546404252037⎞⎟⎟⎠−𝑧₆⎛⎜⎜⎝00..0003917537953866955050364241405807740−.000.64586619967874875726578282025533⎞⎟⎟⎠−𝑧₇⎛⎜⎜⎝00..000540896098698945797267772422225530.000.018944971827035981306698380980112⎞⎟⎟⎠−𝑧₈⎛⎜⎜⎝0.000.0189449718270359813066983809801120.00.01029370389335659958583341404551036⎞⎟⎟⎠−𝑧₉⎛⎜⎜⎝0.01509602028078389860.0097393609553444507⎞⎟⎟⎠−𝑧₁₀⎛⎜⎜⎝0.00973936095534445070.0048696804776722253⎞⎟⎟⎠−𝑧₁₁⎛⎜⎜⎝0.00486968047767222530.00194787309106889012⎞⎟⎟⎠𝑄⎛⎜⎜⎝0.001947873091068890120.00097393609553444506₄⎞⎟⎟⎠,(𝑢)=−−−⎛⎜⎜⎝𝑧𝑧0₂⎛⎜⎜⎝.00.5073611537056088610580206632319088704.107.40050089306083929075596207272432⎞⎟⎟⎠25⎞⎟⎟⎠−𝑧₃⎛⎜⎜⎝−0.656197708525880530.08491200583693081⎞⎟⎟⎠−𝑧₄⎛⎜⎜⎝0.084912005836930810.123083550988310513⎞⎟⎟⎠−𝑧₅⎛⎜⎜⎝00..0015570691630026288108728633839988600.0.1079470309833669352573546404252037⎞⎟⎟⎠−𝑧₆⎛⎜⎜⎝00..00039175379538669550503642414058077400.0.0054089609869894579726777242222553⎞⎟⎟⎠−𝑧₇⎛⎜⎜⎝0−.000.645866199678748757265782820255330.000.018944971827035981306698380980112⎞⎟⎟⎠−𝑧₈⎛⎜⎜⎝0.000.0189449718270359813066983809801120.00.01029370389335659958583341404551036⎞⎟⎟⎠−𝑧₉⎛⎜⎜⎝0.01509602028078389860.0097393609553444507⎞⎟⎟⎠−𝑧₁₀⎛⎜⎜⎝0.00973936095534445070.0048696804776722253⎞⎟⎟⎠−𝑧₁₁⎛⎜⎜⎝0.00486968047767222530.00194787309106889012⎞⎟⎟⎠𝑄⎛⎜⎜⎝0.001947873091068890120.00097393609553444506₅⎞⎟⎟⎠,(𝑢)=⎛⎜⎜⎝−−0.𝑧𝑧0₂⎛⎜⎜⎝206.0090189510351474901778254141423−840.209.0648515503001708700847056963696⎞⎟⎟⎠19⎞⎟⎟⎠−𝑧₃⎛⎜⎜⎝0.045500780040666619−0.60082030844819463⎞⎟⎟⎠+𝑧₄⎛⎜⎜⎝0.1062863906704601240.172792910544901831⎞⎟⎟⎠+𝑧₅⎛⎜⎜⎝00..00124680990199511584043137825474647−0.00.02996608710563516077986705397329⎞⎟⎟⎠+𝑧₆⎛⎜⎜⎝−00.0.000986503710769576875916102337824−00.0.004458500305738208430968606168169⎞⎟⎟⎠+𝑧₇⎛⎜⎜⎝−00.0.0044585003057382084309686061681690.00.0610902812401338143458912904764341⎞⎟⎟⎠+𝑧₈⎛⎜⎜⎝0−.000.11096221846133916375094260071424410−.000.1079267097209615564749960013873211⎞⎟⎟⎠+𝑧₉⎛⎜⎜⎝0.01489095018033857670.009607060567960372⎞⎟⎟⎠+𝑧₁₀⎛⎜⎜⎝0.0096070605679603720.004803530283980186⎞⎟⎟⎠+𝑧₁₁⎛⎜⎜⎝0.0048035302839801860.00192141301359207441⎞⎟⎟⎠𝑄⎛⎜⎜⎝0.001921413013592074410.00096070605679603721₆⎞⎟⎟⎠,(𝑢)=⎛⎜⎜⎝−−0.𝑧𝑧0₂⎛⎜⎜⎝206.0090189510351474901778254141423−840.209.0648515503001708700847056963696⎞⎟⎟⎠19⎞⎟⎟⎠−𝑧₃⎛⎜⎜⎝0.0455007800406666190.106286390670460124⎞⎟⎟⎠+𝑧₄⎛⎜⎜⎝−0.600820308448194630.172792910544901831⎞⎟⎟⎠+𝑧₅⎛⎜⎜⎝00..00124680990199511584043137825474647−0.00.02996608710563516077986705397329⎞⎟⎟⎠+𝑧₆⎛⎜⎜⎝−00.0.000986503710769576875916102337824−00.0.004458500305738208430968606168169⎞⎟⎟⎠+𝑧₇⎛⎜⎜⎝−00.0.0044585003057382084309686061681690−.000.1109622184613391637509426007142441⎞⎟⎟⎠+𝑧₈⎛⎜⎜⎝0.000.6109028124013381434589129047643410−.000.1079267097209615564749960013873211⎞⎟⎟⎠+𝑧₉⎛⎜⎜⎝0.01489095018033857670.009607060567960372⎞⎟⎟⎠+𝑧₁₀⎛⎜⎜⎝0.0096070605679603720.004803530283980186⎞⎟⎟⎠+𝑧₁₁⎛⎜⎜⎝0.0048035302839801860.00192141301359207441⎞⎟⎟⎠𝑄⎛⎜⎜⎝0.001921413013592074410.00096070605679603721₇⎞⎟⎟⎠,(𝑢)=⎛⎜⎜⎝+−0.𝑧𝑧0₂⎛⎜⎜⎝809.4041653330876810203484159340606−290.4−302.305307670811313053249910581248⎞⎟⎟⎠642⎞⎟⎟⎠+𝑧₃⎛⎜⎜⎝0.0377133304908286420.125321920169119855⎞⎟⎟⎠+𝑧₄⎛⎜⎜⎝−0.1253219021691198550.48579940473999439⎞⎟⎟⎠+𝑧₅⎛⎜⎜⎝00..008591444969363871840431801795426−0.00.3433222355600608912195722951953164⎞⎟⎟⎠+𝑧₆⎛⎜⎜⎝00..00313523285661020494259370205692396−0.00.1063671721830303446940886228966482⎞⎟⎟⎠+𝑧₇⎛⎜⎜⎝−0.00.1063671721830303446940886228966482−0.00.0162654352112902118659914159188585⎞⎟⎟⎠+𝑧₈⎛⎜⎜⎝−0.00.016265435211290211865991415918858500.0.408353722959640407932999792453994⎞⎟⎟⎠+𝑧₉⎛⎜⎜⎝0.051499608144107520.0332256000929725936⎞⎟⎟⎠+𝑧₁₀⎛⎜⎜⎝0.03322560009297259360.0166128000464862968⎞⎟⎟⎠+𝑧₁₁⎛⎜⎜⎝0.01661280004648629680.0066451200185945188⎞⎟⎟⎠𝑄⎛⎜⎜⎝0.00664512001859451880.0033225600092972594₈⎞⎟⎟⎠,(𝑢)=⎛⎜⎜⎝++−𝑧𝑧0₂⎛⎜⎜⎝.6−200.2004035076885113104241370845803.002745101.506305293144308370574035⎞⎟⎟⎠790018⎞⎟⎟⎠+𝑧₃⎛⎜⎜⎝0.0323438707057900180.10234930218758696⎞⎟⎟⎠+𝑧₄⎛⎜⎜⎝0.102349302187586960.154827440823966293⎞⎟⎟⎠+𝑧₅⎛⎜⎜⎝−0.00.560250328000353698310394841390840.00.302765015115863519811420738564936⎞⎟⎟⎠+𝑧₆⎛⎜⎜⎝−00.0.0342367085511813211871825788360926400.0.0136233042358971750950769309031488⎞⎟⎟⎠+𝑧₇⎛⎜⎜⎝00..0013626314238807074065478960209168800.0.100625324190332616837652856597638⎞⎟⎟⎠+𝑧₈⎛⎜⎜⎝00.0.10062532419033261683765285659763800..01053428620754148832138916267289639⎞⎟⎟⎠+𝑧₉⎛⎜⎜⎝0.0505380303930981980.0326051803181278696⎞⎟⎟⎠+𝑧₁₀⎛⎜⎜⎝0.03260518031812786960.0163025901590639348⎞⎟⎟⎠+𝑧₁₁⎛⎜⎜⎝0.01661280004648629680.0065210306636255738⎞⎟⎟⎠⎛⎜⎜⎝0.00652103066362557380.0032605108318127869⎞⎟⎟⎠(5.37) they are the symbols of the minimum-energy multiwavelet frames associated with Φ. The graphs of the vector-valued functions are shown in Figure 5.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 5 

From Figure 5, we can find that every component of the minimum-energy frames is smooth, but not (anti)symmetrical.

6. Conclusions

In this paper, 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, then 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.

Acknowledgments

The authors acknowledge the support from the National Natural Science Foundation of China under the Project Grant no. 10961001 and 61261043, the Research Project of Beifang University of Nationalities under the Project no. 2012Y036, and Graduate Students Innovative Project under the Project no. 2011XYC035.  M. Li acknowledges the support from the 973 plan under the Project no. 2011CB302800 and the National Natural Science Foundation of China under the Project Grant no.s 61070214 and 60873264.