# Minimum-Energy Multiwavelet Frame on the Interval[0,1]

Authors: Fengjuan Zhu; Yongdong Huang; Xiao Feng; Qiufu Li
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

Drawing inspiration from the idea of combining multiwavelets on the interval with frame theory organically, we study minimum-energy multiwavelet frame on the interval[0,1] (MEMWFI). Firstly, left boundary multiscaling
functions, right boundary multiscaling functions, and the definition of MEMWFI are put forward, and the equivalent characterizations of
MEMWFI are given. Then, two algorithms of constructing MEMWFI are proposed. Finally, the decomposition formula, reconstruction formulas, and numerical examples are given.


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## Body

## 1. Introduction

The notion of frame was introduced by Duffin and Schaeffer [1]. It is a generalization of basis and can be viewed as some kind of “overcomplete basis.” A basis in a Hilbert space allows each element to be written as a linear combination of the elements in the basis, and the combination is unique. An overcomplete frame also allows one to represent each element via it, but the representation is not unique. This property plays an important role in mathematics, signal analysis, time-frequency analysis, and so on [2–20]. In nearly thirty years, frame theory has been growing rapidly. Wavelet frames are a class of important frames among all kinds of frames.LetH be a complex, separable Hilbert space. A sequence {fk:k∈Z} of elements in H is a frame for H if there exist constants A,B>0 such that(1)Af2≤∑k∈Zf,fk2≤Bf2,∀f∈H.The numbers A,B are called frame bounds. A frame {fk:k∈Z} is tight if we can choose A=B as frame bounds. A frame {fk:k∈Z} is exact if it ceases to be a frame when an arbitrary element is removed. When A=B=1, then (2)f=∑k∈Zf,fkfk,∀f∈H.Chui and He [6] proposed the concept of minimum-energy wavelet frames, which can reduce the computational complexity and maintain the numerical stability, and do not need to search dual frames in the decomposition and reconstruction of functions. Therefore, many people paid a lot of attention to the study of minimum-energy wavelet frames. Petukhov [8] studied minimum-energy wavelet frames with symmetry. Huang and Cheng [16] studied minimum-energy wavelet frames with arbitrary integer dilation factor. Gao and Cao [17] researched minimum-energy wavelets frames on the interval and its applications. Huang and Li [18] studied the algorithms of constructing minimum-energy wavelet frames on the interval. Recently, Huang et al. [19] and Liang and Zhao [20] have been concerned with the algorithms of constructing minimum-energy multiwavelet frames. In [19], the authors gave the definitions of multiwavelet frame multiresolution analysis for L2(R) and minimum-energy multiwavelet frame. But, in many practical applications, one is often interested in the problems which are confined to a finite interval, such as the differential equation solving, image processing, and signal analysis. Therefore, we should pay more attention to minimum-energy multiwavelet frames on the interval; under this case, we can combine the features (symmetry, compact support, and regularity) of multiwavelets on the interval with the superiority (redundancy, good numerical stability, and lower computational complexity) of minimum-energy wavelet frames.Throughout this paper, letZ,R denote the set of integers and real numbers, respectively. N denotes the set of positive integers. Let a,r≥2, and a,r∈N.The organization of this paper is as follows. In Section2, we give the main results; left boundary multiscaling functions, right boundary multiscaling functions, and the definition of MEMWFI are put forward; the equivalent characterizations of MEMWFI, two algorithms for constructing MEMWFI, and the decomposition and reconstruction formulas of MEMWFI are proposed. Finally, numerical examples are given in Section 3.


## 2. Main Results

In order to give the definitions of MEMWFI, left boundary multiscaling functions, and right boundary multiscaling functions, we letγ>0,j0=min{j:aj≥γ,j∈Z}.Definition 1.
If a closed subspaces sequenceVj([0,1]),j≥j0 in L2([0,1]) satisfies(1)
Vj([0,1])⊂Vj+1([0,1]),j≥j0;(2)
⋃j≥j0Vj([0,1])‍ is dense in L2([0,1]);(3)
⋂j≥j0Vj([0,1])‍=Vj0([0,1]);(4)
f(x)∈Vj([0,1])⇔f(ax)∈Vj+1([0,1]), j≥j0,j∈Z;(5)
f(x)∈Vj([0,1])⇔f(x+a-jk)∈Vj([0,1]), j≥j0,j,k∈Z;(6)
there exists a vector-valued functionϕ(x)=(ϕ1,ϕ2,…,ϕr)T,ϕ1,ϕ2,…,ϕr∈Vj0([0,1]) with suppϕ=[0,γ] such that (3)ϕτaj0x-k,1≤τ≤r,0≤k≤Kj0forms a frame ofVj0([0,1]), where Kj0 is the biggest integer number to make suppϕ(aj0x)∪suppϕ(aj0x-Kj0)⊂[0,1],then we sayVj([0,1])(j≥j0) is a multiwavelet frame multiresolution analysis on the interval with dilation factor a and multiplicity r or a multiwavelet frame multiresolution analysis on the interval [0,1] and denote it as MWFMRA (Vj[0,1]). The function ϕ(x) is vector-valued scaling function (multiscaling function).For a MWFMRA(Vj[0,1]), we let Vj+1([0,1])=Vj([0,1])+Wj([0,1]). If we can find vector-valued functions ψ1(x),ψ2(x),…,ψM(x) such that {ψτi(aj0x-k):1≤τ≤r,1≤i≤M,0≤k≤Kj0} is a frame for Wj0([0,1]), then we say {ψ1(x),ψ2(x),…,ψM(x)} is a multiwavelet frame on the interval, where ψi(x)=(ψ1i(x),ψ2i(x),…,ψri(x))T,i=1,2,…,M.


### 2.1. Left and Right Boundary Multiscaling Functions

Letϕ(x) be a vector-valued scaling function and generate a multiwavelet frame multiresolution analysis for L2(R) with corresponding multiwavelet frame Ψ=ψ1,ψ2,…,ψM. Suppose ϕ(x) satisfies the refinable equation (4)ϕx=∑k=0γ1Pkϕax-kwith suppϕ=[0,γ].Letγ1=(a-1)γ,N=[γ/a].Theorem 2.
Define left boundary multiscaling functionsϕj,kL and right boundary multiscaling functions ϕj,kR,s as follows:(5)ϕj,kLx≔∑n=-γ+1k-NCk,nϕj,nx0,1,0≤k≤N-1,ϕj,kR,sx≔∑n=aj-γ+1+kaj-1Ck,nsϕj,nx0,1,0≤k≤N-1,1≤s≤a-1,where Ck,n,Ck,ns are r×r coefficient matrices. Denote (6)Φj≔ϕτ,j,kL,k=0,1,…,N-1;ϕτ,j,k,k=0,1,…,aj-γ;ϕj,kτ,R,s,k=0,1,…,N-1,1≤s≤a-1:1≤τ≤r.
IfCk,n and {Ck,ns} make {ϕj,kL,k=0,1,…,N-1} and {ϕj,kR,s,k=0,1,…,N-1,s=1,2,…,a-1} satisfy frame condition, respectively, then there exist coefficient matrices Pk,nL,Pk,nR,s,pk,nL,pk,nR,s, such that the matrix-valued refinable equations of Φj are expressed as follows: (7)aϕj,kL=∑n=0N-1Pk,nLϕj+1,nL+∑n=0ak+N1pk,nLϕj+1,n,0≤k≤N-1,aϕj,k=∑n=akak+γ1Pn-akϕj+1,n,0≤k≤aj-γ,aϕj,kR,s=∑n=0ak+N2pk,nR,sϕj+1,aj+1-γ-n+∑n=0N-1Pk,nR,sϕj+1,nR,s,0≤k≤N-1,1≤s≤a-1,where N1=γ1-aN,N2=(a-1)γ-aN, and Φj can form a frame of Vj([0,1]) when j≥j0.Proof.
The proof is similar to Theorem 2.2 in [17].Theorem 3.
Letϕj,kL and ϕj,kR,s be the left boundary multiscaling functions and right boundary multiscaling functions defined in Theorem 2. Let vector-valued functions {ψj,kL,i,1≤i≤M,k=0,1,…,N-1}, {ψj,ki,1≤i≤M,k=0,1,…,aj-γ}, {ψj,kR,s,i,1≤i≤M,k=0,1,…,N-1,s=1,2,…,a-1} be expressed as follows: (8)aψj,kL,i=∑n=0N-1Qk,nL,iϕj+1,nL+∑n=0ak+N1qk,nL,iϕj+1,n,0≤k≤N-1,aψj,ki=∑n=akak+γ1Qn-akiϕj+1,n,0≤k≤aj-γ,aψj,kR,s,i=∑n=0ak+N2qk,nR,s,iϕj+1,aj+1-γ-n+∑n=0N-1Qμ:k,nR,s,iϕj+1,nR,s,0≤k≤N-1,s=1,2,…,a-1,where N1=γ1-aN,N2=(a-1)γ-aN. Denote(9)Ψj≔ψτ,j,kL,i,k=0,1,…,N-1;ψτ,j,ki,k=0,1,…,aj-γ;ψτ,j,kR,s,i,k=0,1,…,N-1,1≤s≤a-1:1≤i≤M,1≤τ≤r. If coefficient matrices Qk,nL,i,qk,nL,i,Qk,nR,s,i,qk,nR,s,i such that {ψj,kL,i,k=0,1,…,N-1,i=1,2,…,M} and {ψj,kR,s,i,k=0,1,…,N-1,s=1,2,…,a-1,i=1,2,…,M} satisfy frame condition, respectively, then Ψj is a frame of Wj([0,1]) for any j≥j0.Proof.
The proof is similar to Theorem 3.2 in [17] and Theorem 3.2 in [18].Definition 4.
LetΦj,Ψj,j∈Z be defined in Theorems 2 and 3, respectively. If any f∈L2([0,1]), one has (10)∑θ∈Φj0+1f,θ2=∑ηϕ∈Φj0f,ηϕ2+∑ηψ∈Ψj0f,ηψ2,and then the family of functions Ψj0 are called a minimum-energy multiwavelet frame on the interval [0,1] (MEMWFI) associated Φj0with dilation factor a and multiplicity r.
By Parseval identity, (10) is equivalent to (11)∑θ∈Φj0+1f,θθ=∑ηϕ∈Φj0f,ηϕηϕ+∑ηψ∈Ψj0f,ηψηψ,f∈L20,1.


### 2.2. Construction of MEMWFI

For convenience, we denote the following:(1)
lj=aj-γ,λ=a(N-1).(2)
Sj={k:-N≤k≤aj-γ+(a-1)N}.(3)
Φj=(…,ϕj,kT,…)T is |Sj|r dimension column vector; and ϕj,k are left boundary multiscaling functions and right boundary multiscaling functions defined in Theorem 2: (12)ϕj,k=ϕj,N+kL,k=-N,…,-1,ϕj,k,k=0,…,aj-γ,ϕj,k-lj-1R,1,k=lj+1,…,lj+N,ϕj,k-lj-N-1R,2,k=lj+N+1,…,lj+2N,⋮ϕj,k-lj-a-2N-1R,a-1,k=lj+a-2N+1,…,lj+a-1N.(4)
Ψji=(…,ψj,kiT,…)T are |Sj|r dimension column vector; and ψj,ki are multiwavelet frames defined in Theorem 3: (13)ψj,ki=ψj,N+kL,i,k=-N,…,-1,ψj,ki,k=0,…,aj-γ,ψj,k-lj-1R,1,i,k=lj+1,…,lj+N,ψj,k-lj-N-1R,2,i,k=lj+N+1,…,lj+2N,⋮ψj,k-lj-a-2N-1R,a-1,i,k=lj+a-2N+1,…,lj+a-1N,wherei=1,2,…,M. If Ψi is seen as a set of functions, then Ψj=Ψj1,…,ΨjM.Define(14)P≔P11P12P22P32P33,where P11,P12,P22,P32,P33 are block matrices as follows: (15)P11=P0,0LP0,1L⋯P0,N-1LP1,0LP1,1L⋯P1,N-1L⋮⋮⋮PN-1,0LPN-1,1L⋯PN-1,N-1L,P12=p0,0L⋯p0,N1Lp1,0L⋯⋯p1,a+N1L⋮⋱pN-1,0L⋯⋯⋯pN-1,λ+N1L,P22=P0⋯Pa⋯Pγ1P0⋯Pa⋯Pγ1⋱⋱⋱⋱P0⋯Pa⋯Pγ1,P32=p1p2⋮pa-1,P33=diag⁡P1,P2,…,Pa-1,pi=p0,N2R,i⋯p0,0R,ip1,a+N2R,i⋯⋯h1,0R,i⋮⋮pN-1,λ+N2R,i⋯⋯⋯pN-1,0R,i,Pi=P0,0R,i⋯P0,N-1R,i⋮⋮PN-1,0R,i⋯PN-1,N-1R,i,i=1,…,a-1.Similarly, we can define(16)Qi≔Q11iQ12iQ22iQ32iQ33i,i=1,2,…,M,where Q11i,Q12i,Q22i,Q32i,Q33i have the same form as P11,P12,P22,P32,P33, respectively.Then, by Theorems2 and 3, we have (17)aΦj=PΦj+1,aΨji=QiΦj+1i.The Fourier transform of (17) is (18)aΦ^j=PzΦ^j+1,aΨ^ji=QizΦ^j+1i,where (19)Pz=1aDj-1PDj+1,Qiz=1aDj-1QiDj+1,Dj=diag⁡eiω-N/ajIr,…,eiωlj+λ/ajIr.With theseP(z),Qi(z),Φj,Ψj,i=1,2,…,M,j≥j0, we will give the characterizations of MEMWFI.Theorem 5.
The following statements are equivalent:(1)
Ψj0 is a MEMWFI.(2)
For anyz=1, (20)aPz∗Pz+∑i=1MQiz∗Qiz=Isj+1r.(3)
For anym,l∈Sj+1, (21)∑k∈SjPk,m∗Pk,l+∑i=1MQk,mi∗Qk,li-aδm,lIr=0r,where Pk,m,Qk,mi denote the r×r block matrices from ((k-1)r+1)th row to krth row and ((m-1)r+1)th row to mrth column in P,Qi, respectively, and P(z)∗ denotes the complex conjugate of the transpose of P(z).Proof.
By using Theorems2 and 3, (11) can be rewritten as (22)∑k∈Sj+1∑τ=1rf,ϕτ,j+1,kϕτ,j+1,k=∑k∈Sj∑τ=1rf,ϕτ,j,kϕτ,j,k+∑i=1M∑k∈Sj∑τ=1rf,ψτ,j,kiψτ,j,ki.For convenience, for every f∈L2([0,1]), we denote f,ϕj+1,k as r dimension column vector cj+1,k, where the τth component of cj+1,k is f,ϕτ,j+1,k. Similarly, we denote f,ψj+1,ki as r dimension column vectors dj+1,ki. Then, the former formula is equivalent to (23)∑k∈Sj+1f,ϕj+1,kTϕj+1,k=∑k∈Sjf,ϕj,kTϕj,k+∑i=1M∑k∈Sjf,ψj,kiTψj,ki=∑k∈Sjf,1a∑l∈Sj+1Pk,lϕj+1,lT1a∑l∈Sj+1Pk,lϕj+1,l+∑i=1M∑k∈Sjf,1a∑l∈Sj+1Qk,liϕj+1,lT1a∑l∈Sj+1Qk,liϕj+1,l=1a∑l∈Sj+1∑m∈Sj+1∑k∈Sjf,ϕj+1,mTPk,m∗Pk,l+∑i=1MQk,mi∗Qk,liϕj+1,l,which is equivalent to (24)∑l∈Sj+1∑m∈Sj+1f,ϕj+1,mT·∑k∈SjPk,m∗Pk,l+∑i=1MQk,mi∗Qk,li-aδm,lIrϕj+1,l=0or (25)∑l∈Sj+1∑m∈Sj+1f,ϕj+1,mTαm,lϕj+1,l=0,∀f∈L20,1,where αm,l=∑k∈Sj‍Pk,m∗Pk,l+∑i=1M‍Qk,mi∗Qk,li-aδm,lIr.
We multiply the same identities in (20) by Φ^j+1(ω/a), to get (26)aaPz∗Φ^jω+∑i=1MQiz∗Ψ^jiω=Φ^j+1ωa,and this is equivalent to (27)aDj-1PDj+1∗Φ^jω+∑i=1MDj-1QiDj+1∗Ψ^jiω=Φ^j+1ωa⟺P∗aΦjx+∑i=1MQi∗aΦjx=aΦj+1x⟺P∗PΦj+1x+∑i=1MQi∗QiΦj+1x=aΦj+1x⟺P∗P+∑i=1MQi∗QiΦj+1x=aΦj+1xor (28)∑m∈Sj+1αm,lϕj+1,l=0,∀l∈Sj+1.
In other words, the proof of theorem can be reduced to the proof of the equivalence of (21) and (25). It is obvious that (21) ⇒ (28) ⇒ (25). To show (25) ⇒ (21), take any f∈L2([0,1]) and define (29)βlf≔∑m∈Sj+1f,ϕj+1,mTαm,l,∀l∈Sj+1.The Fourier transform of (25) is (30)∑l∈Sj+1βlfe-ilω/aj+1=0.Then βlf=0,∀l∈Sj+1; that is, (31)f,∑m∈Sj+1αm,l∗ϕj+1,m=0,l∈Sj+1.
Becausef∈L2([0,1]) is a compact support function, (32)βlf=0,∀l∈Sj+1,∀f∈L20,1;that is,(33)f,∑m∈Sj+1αm,l∗ϕj+1,m=0,l∈Sj+1,∀f∈L20,1.Then, we can get (34)∑m∈Sj+1αm,lϕj+1,m=0,l∈Sj+1,and this is equivalent to αm,l=0r,∀m,l∈Sj+1.
Related to the proof,0 is zero vector with appropriate size, and 0r is r×r zero matrix.Theorem5 characterizes the necessary and sufficient condition for the existence of the MEMWFI associated with ϕ. But it is not a good choice to use this theorem to construct MEMWFI with arbitrary dilation a. For convenience, we need to present some algorithms of constructing MEMWFI.


### 2.3. Algorithms of Constructing MEMWFI

With matricesP,Qi,i=1,2,…,M, we formulate the following block matrix: (35)M=P∗,Q1∗,Q2∗,…,QM∗,and then, (21), the characterization of MEMWFI, can be rewritten as (36)MM∗=aISj0r.Now we present two algorithms of constructing MEMWFI starting from a multiscaling function and a scalar scaling function, respectively.Algorithm 1.
(1) Take a proper multiscaling functionϕ(x).
(2) Calculate the parametersγ,γ1,N=γ/a,j0=logaγ,N1,N2.
(3) Choose proper coefficient matricesCk,n,0≤k≤N-1,-γ+1≤n≤k-N and Ck,ns,0≤k≤N-1,aj0-γ+1+k≤n≤aj0-1,1≤s≤a-1, and construct boundary multiscaling functions using (5); in addition, we can get Φj0.
(4) Use Theorem2, ascertain coefficient matrices Pk,nL,0≤k,n≤N-1;pk,nL,0≤k≤N-1,0≤n≤ak+N1 and pk,nR,s,0≤k≤N-1,0≤n≤ak+N2;Pk,nR,s,0≤k,n≤N-1;1≤s≤a-1 to get the block matrix P.
(5) Take coefficient matricesQk,nL,i,0≤k,n≤N-1;qk,nL,i,0≤k≤N-1,0≤n≤ak+N1; Qni,0≤n≤γ1 and qk,nR,s,i,0≤k≤N-1,0≤n≤ak+N2;Qk,nR,s,i,0≤k,n≤N-1;1≤s≤a-1,1≤i≤M to construct the block matrixes Qi,i=1,…,M, and ensure that they satisfy (21) with P.
(6) Use the coefficient matricesQi which we have got in step (5) to construct MEMWFI Ψj0=Ψj01,…,Ψj0M, by Theorem 3.
(7) Draw the graphs ofΦj0 and Ψj0 by Matlab.Algorithm 2.
(1) Take a proper scalar scaling functionφ(x) (the quadratic sum of recursion coefficients can not be larger than 1).
(2) Useφ(x) to get refinable vector-valued function ϕ(x); for example, (37)ϕx=φx,φx,…,φxrT,ϕx=φx,φx-1,…,φx-r+1T,ϕx=φax,φax-1,…,φax-r+1T,and so on; we can find the sequence of matrices Pk which satisfies refinable equation with ϕ. We should know that the necessary and sufficient conditions for MEMWFI in Theorem 5 reduce to sufficient conditions when the set of components in the vector-valued function ϕ(x) is linearly dependent.
(3) Turn to steps (3)–(7) of Algorithm1.We should note that the refinable vector-valued functionϕ in Algorithm 2 can not be a multiscaling function.


### 2.4. Decomposition Formula and Reconstruction Formula of MEMWFI

The decomposition formula and reconstruction formula of MEMWFI are obtained by Theorems2, 3, and 5, which are similar to those of orthogonal wavelets or multiwavelets on the interval. Firstly, for ∀f∈L2([0,1]), denote (38)cj,k=f,ϕj,k,cj,kL=f,ϕj,kL,cj,kR,s=f,ϕj,kR,s,dj,ki=f,ψj,ki,dj,kL,i=f,ψj,kL,i,dj,kR,s,i=f,ψj,kR,s,ii=1,…,M;s=1,…,a-1.Theorem 6.
The decomposition formula of MEMWFI is(39)acj,kL=∑n=0N-1Pk,nLcj+1,nL+∑n=0ak+N1pk,nLcj+1,nL,0≤k≤N-1acj,k=∑n=akak+γ1Pn-akcj+1,n,0≤k≤aj-γacj,kR,s=∑n=0ak+N2pk,nR,scj+1,aj+1-γ-n+∑n=0ak+N1Pk,nR,scj+1,nR,s,0≤k≤N-1,1≤s≤a-1adj,kL,i=∑n=0N-1Qk,nL,icj+1,nL+∑n=0ak+N1qk,nL,icj+1,n,0≤k≤N-1adj,ki=∑n=akak+γ1Qn-akicj+1,n,0≤k≤aj-γadj,kR,s,i=∑n=0ak+N2qk,nR,s,icj+1,aj+1-γ-n+∑n=0N-1Qk,nR,s,icj+1,nR,s,0≤k≤N-1,1≤s≤a-1,i=1,…,M,and the reconstruction formula of MEMWFI is (40)acj+1,nL=∑k=0N-1Pk,nLcj,kL+∑i=1M∑k=0N-1Qk,nL,idj,kL,i,0≤n≤N-1acj+1,nR,s=∑k=0N-1Pk,nR,scj,kR,s+∑i=1M∑k=0N-1Qk,nL,s,idj,kL,s,i,0≤n≤N-1,1≤s≤a-1acj+1,n=∑k=0N-1pk,nLcj,kL+∑i=0M∑k=0N-1qk,nLdj,kL,i+∑kPn-akcj,k+∑i=1M∑kQn-akidj,ki,0≤n≤N1acj+1,n=acj+1,N1+al+n′=∑k=l+1N-1pk,N1+al+n′Lcj,kL+∑i=1M∑k=l+1N-1qk,N1+al+n′Ldj,kL,i+∑kPN1+n′+al-kcj,k+∑i=1M∑kQN1+n′+al-kidj,ki,0≤l≤N-2,1≤n′≤a,N1+1≤n≤aN-1+N1acj+1,n=∑kPn-akcj,k+∑i=1M∑kQn-akidj,ki,a-1γ-a+1≤n≤aj+1-aγ+a-1(41)acj+1,n=acj+1,N-1-la+N2-n′+aj+1-aγ+a=∑s=1a-1∑k=l+1N-1pk,N-1-la+N2-n′R,scj,kR,s+∑i=1M∑s=1a-1∑k=l+1N-1qk,N-1-la+N2-n′R,s,idj,kR,s,i+∑kPN-1-l-ka+N2-n′+aj+1-aγ+acj,k+∑i=1M∑kQN-1-l-ka+N2-n′+aj+1-aγ+aidj,ki,0≤l≤N-2,0≤n′≤a-1,aj+1-aγ+a≤n≤aj+1-aγ+aN-1(42)acj+1,n=∑s=1a-1∑k=0N-1pk,aj+1-γ-nR,scj,kR,s+∑i=1M∑s=1a-1∑k=0N-1qk,aj+1-γ-nR,s,idj,kR,s,i+∑kPn-akcj,k+∑i=1M∑kQn-akidj,ki,aj+1-aγ+aN≤n≤aj+1-γ.Proof.
The decomposition formula can be obtained by Theorems2 and 3, and the reconstruction formula is gotten from Theorems 2, 3, and 5.


## 2.1. Left and Right Boundary Multiscaling Functions

Letϕ(x) be a vector-valued scaling function and generate a multiwavelet frame multiresolution analysis for L2(R) with corresponding multiwavelet frame Ψ=ψ1,ψ2,…,ψM. Suppose ϕ(x) satisfies the refinable equation (4)ϕx=∑k=0γ1Pkϕax-kwith suppϕ=[0,γ].Letγ1=(a-1)γ,N=[γ/a].Theorem 2.
Define left boundary multiscaling functionsϕj,kL and right boundary multiscaling functions ϕj,kR,s as follows:(5)ϕj,kLx≔∑n=-γ+1k-NCk,nϕj,nx0,1,0≤k≤N-1,ϕj,kR,sx≔∑n=aj-γ+1+kaj-1Ck,nsϕj,nx0,1,0≤k≤N-1,1≤s≤a-1,where Ck,n,Ck,ns are r×r coefficient matrices. Denote (6)Φj≔ϕτ,j,kL,k=0,1,…,N-1;ϕτ,j,k,k=0,1,…,aj-γ;ϕj,kτ,R,s,k=0,1,…,N-1,1≤s≤a-1:1≤τ≤r.
IfCk,n and {Ck,ns} make {ϕj,kL,k=0,1,…,N-1} and {ϕj,kR,s,k=0,1,…,N-1,s=1,2,…,a-1} satisfy frame condition, respectively, then there exist coefficient matrices Pk,nL,Pk,nR,s,pk,nL,pk,nR,s, such that the matrix-valued refinable equations of Φj are expressed as follows: (7)aϕj,kL=∑n=0N-1Pk,nLϕj+1,nL+∑n=0ak+N1pk,nLϕj+1,n,0≤k≤N-1,aϕj,k=∑n=akak+γ1Pn-akϕj+1,n,0≤k≤aj-γ,aϕj,kR,s=∑n=0ak+N2pk,nR,sϕj+1,aj+1-γ-n+∑n=0N-1Pk,nR,sϕj+1,nR,s,0≤k≤N-1,1≤s≤a-1,where N1=γ1-aN,N2=(a-1)γ-aN, and Φj can form a frame of Vj([0,1]) when j≥j0.Proof.
The proof is similar to Theorem 2.2 in [17].Theorem 3.
Letϕj,kL and ϕj,kR,s be the left boundary multiscaling functions and right boundary multiscaling functions defined in Theorem 2. Let vector-valued functions {ψj,kL,i,1≤i≤M,k=0,1,…,N-1}, {ψj,ki,1≤i≤M,k=0,1,…,aj-γ}, {ψj,kR,s,i,1≤i≤M,k=0,1,…,N-1,s=1,2,…,a-1} be expressed as follows: (8)aψj,kL,i=∑n=0N-1Qk,nL,iϕj+1,nL+∑n=0ak+N1qk,nL,iϕj+1,n,0≤k≤N-1,aψj,ki=∑n=akak+γ1Qn-akiϕj+1,n,0≤k≤aj-γ,aψj,kR,s,i=∑n=0ak+N2qk,nR,s,iϕj+1,aj+1-γ-n+∑n=0N-1Qμ:k,nR,s,iϕj+1,nR,s,0≤k≤N-1,s=1,2,…,a-1,where N1=γ1-aN,N2=(a-1)γ-aN. Denote(9)Ψj≔ψτ,j,kL,i,k=0,1,…,N-1;ψτ,j,ki,k=0,1,…,aj-γ;ψτ,j,kR,s,i,k=0,1,…,N-1,1≤s≤a-1:1≤i≤M,1≤τ≤r. If coefficient matrices Qk,nL,i,qk,nL,i,Qk,nR,s,i,qk,nR,s,i such that {ψj,kL,i,k=0,1,…,N-1,i=1,2,…,M} and {ψj,kR,s,i,k=0,1,…,N-1,s=1,2,…,a-1,i=1,2,…,M} satisfy frame condition, respectively, then Ψj is a frame of Wj([0,1]) for any j≥j0.Proof.
The proof is similar to Theorem 3.2 in [17] and Theorem 3.2 in [18].Definition 4.
LetΦj,Ψj,j∈Z be defined in Theorems 2 and 3, respectively. If any f∈L2([0,1]), one has (10)∑θ∈Φj0+1f,θ2=∑ηϕ∈Φj0f,ηϕ2+∑ηψ∈Ψj0f,ηψ2,and then the family of functions Ψj0 are called a minimum-energy multiwavelet frame on the interval [0,1] (MEMWFI) associated Φj0with dilation factor a and multiplicity r.
By Parseval identity, (10) is equivalent to (11)∑θ∈Φj0+1f,θθ=∑ηϕ∈Φj0f,ηϕηϕ+∑ηψ∈Ψj0f,ηψηψ,f∈L20,1.


## 2.2. Construction of MEMWFI

For convenience, we denote the following:(1)
lj=aj-γ,λ=a(N-1).(2)
Sj={k:-N≤k≤aj-γ+(a-1)N}.(3)
Φj=(…,ϕj,kT,…)T is |Sj|r dimension column vector; and ϕj,k are left boundary multiscaling functions and right boundary multiscaling functions defined in Theorem 2: (12)ϕj,k=ϕj,N+kL,k=-N,…,-1,ϕj,k,k=0,…,aj-γ,ϕj,k-lj-1R,1,k=lj+1,…,lj+N,ϕj,k-lj-N-1R,2,k=lj+N+1,…,lj+2N,⋮ϕj,k-lj-a-2N-1R,a-1,k=lj+a-2N+1,…,lj+a-1N.(4)
Ψji=(…,ψj,kiT,…)T are |Sj|r dimension column vector; and ψj,ki are multiwavelet frames defined in Theorem 3: (13)ψj,ki=ψj,N+kL,i,k=-N,…,-1,ψj,ki,k=0,…,aj-γ,ψj,k-lj-1R,1,i,k=lj+1,…,lj+N,ψj,k-lj-N-1R,2,i,k=lj+N+1,…,lj+2N,⋮ψj,k-lj-a-2N-1R,a-1,i,k=lj+a-2N+1,…,lj+a-1N,wherei=1,2,…,M. If Ψi is seen as a set of functions, then Ψj=Ψj1,…,ΨjM.Define(14)P≔P11P12P22P32P33,where P11,P12,P22,P32,P33 are block matrices as follows: (15)P11=P0,0LP0,1L⋯P0,N-1LP1,0LP1,1L⋯P1,N-1L⋮⋮⋮PN-1,0LPN-1,1L⋯PN-1,N-1L,P12=p0,0L⋯p0,N1Lp1,0L⋯⋯p1,a+N1L⋮⋱pN-1,0L⋯⋯⋯pN-1,λ+N1L,P22=P0⋯Pa⋯Pγ1P0⋯Pa⋯Pγ1⋱⋱⋱⋱P0⋯Pa⋯Pγ1,P32=p1p2⋮pa-1,P33=diag⁡P1,P2,…,Pa-1,pi=p0,N2R,i⋯p0,0R,ip1,a+N2R,i⋯⋯h1,0R,i⋮⋮pN-1,λ+N2R,i⋯⋯⋯pN-1,0R,i,Pi=P0,0R,i⋯P0,N-1R,i⋮⋮PN-1,0R,i⋯PN-1,N-1R,i,i=1,…,a-1.Similarly, we can define(16)Qi≔Q11iQ12iQ22iQ32iQ33i,i=1,2,…,M,where Q11i,Q12i,Q22i,Q32i,Q33i have the same form as P11,P12,P22,P32,P33, respectively.Then, by Theorems2 and 3, we have (17)aΦj=PΦj+1,aΨji=QiΦj+1i.The Fourier transform of (17) is (18)aΦ^j=PzΦ^j+1,aΨ^ji=QizΦ^j+1i,where (19)Pz=1aDj-1PDj+1,Qiz=1aDj-1QiDj+1,Dj=diag⁡eiω-N/ajIr,…,eiωlj+λ/ajIr.With theseP(z),Qi(z),Φj,Ψj,i=1,2,…,M,j≥j0, we will give the characterizations of MEMWFI.Theorem 5.
The following statements are equivalent:(1)
Ψj0 is a MEMWFI.(2)
For anyz=1, (20)aPz∗Pz+∑i=1MQiz∗Qiz=Isj+1r.(3)
For anym,l∈Sj+1, (21)∑k∈SjPk,m∗Pk,l+∑i=1MQk,mi∗Qk,li-aδm,lIr=0r,where Pk,m,Qk,mi denote the r×r block matrices from ((k-1)r+1)th row to krth row and ((m-1)r+1)th row to mrth column in P,Qi, respectively, and P(z)∗ denotes the complex conjugate of the transpose of P(z).Proof.
By using Theorems2 and 3, (11) can be rewritten as (22)∑k∈Sj+1∑τ=1rf,ϕτ,j+1,kϕτ,j+1,k=∑k∈Sj∑τ=1rf,ϕτ,j,kϕτ,j,k+∑i=1M∑k∈Sj∑τ=1rf,ψτ,j,kiψτ,j,ki.For convenience, for every f∈L2([0,1]), we denote f,ϕj+1,k as r dimension column vector cj+1,k, where the τth component of cj+1,k is f,ϕτ,j+1,k. Similarly, we denote f,ψj+1,ki as r dimension column vectors dj+1,ki. Then, the former formula is equivalent to (23)∑k∈Sj+1f,ϕj+1,kTϕj+1,k=∑k∈Sjf,ϕj,kTϕj,k+∑i=1M∑k∈Sjf,ψj,kiTψj,ki=∑k∈Sjf,1a∑l∈Sj+1Pk,lϕj+1,lT1a∑l∈Sj+1Pk,lϕj+1,l+∑i=1M∑k∈Sjf,1a∑l∈Sj+1Qk,liϕj+1,lT1a∑l∈Sj+1Qk,liϕj+1,l=1a∑l∈Sj+1∑m∈Sj+1∑k∈Sjf,ϕj+1,mTPk,m∗Pk,l+∑i=1MQk,mi∗Qk,liϕj+1,l,which is equivalent to (24)∑l∈Sj+1∑m∈Sj+1f,ϕj+1,mT·∑k∈SjPk,m∗Pk,l+∑i=1MQk,mi∗Qk,li-aδm,lIrϕj+1,l=0or (25)∑l∈Sj+1∑m∈Sj+1f,ϕj+1,mTαm,lϕj+1,l=0,∀f∈L20,1,where αm,l=∑k∈Sj‍Pk,m∗Pk,l+∑i=1M‍Qk,mi∗Qk,li-aδm,lIr.
We multiply the same identities in (20) by Φ^j+1(ω/a), to get (26)aaPz∗Φ^jω+∑i=1MQiz∗Ψ^jiω=Φ^j+1ωa,and this is equivalent to (27)aDj-1PDj+1∗Φ^jω+∑i=1MDj-1QiDj+1∗Ψ^jiω=Φ^j+1ωa⟺P∗aΦjx+∑i=1MQi∗aΦjx=aΦj+1x⟺P∗PΦj+1x+∑i=1MQi∗QiΦj+1x=aΦj+1x⟺P∗P+∑i=1MQi∗QiΦj+1x=aΦj+1xor (28)∑m∈Sj+1αm,lϕj+1,l=0,∀l∈Sj+1.
In other words, the proof of theorem can be reduced to the proof of the equivalence of (21) and (25). It is obvious that (21) ⇒ (28) ⇒ (25). To show (25) ⇒ (21), take any f∈L2([0,1]) and define (29)βlf≔∑m∈Sj+1f,ϕj+1,mTαm,l,∀l∈Sj+1.The Fourier transform of (25) is (30)∑l∈Sj+1βlfe-ilω/aj+1=0.Then βlf=0,∀l∈Sj+1; that is, (31)f,∑m∈Sj+1αm,l∗ϕj+1,m=0,l∈Sj+1.
Becausef∈L2([0,1]) is a compact support function, (32)βlf=0,∀l∈Sj+1,∀f∈L20,1;that is,(33)f,∑m∈Sj+1αm,l∗ϕj+1,m=0,l∈Sj+1,∀f∈L20,1.Then, we can get (34)∑m∈Sj+1αm,lϕj+1,m=0,l∈Sj+1,and this is equivalent to αm,l=0r,∀m,l∈Sj+1.
Related to the proof,0 is zero vector with appropriate size, and 0r is r×r zero matrix.Theorem5 characterizes the necessary and sufficient condition for the existence of the MEMWFI associated with ϕ. But it is not a good choice to use this theorem to construct MEMWFI with arbitrary dilation a. For convenience, we need to present some algorithms of constructing MEMWFI.


## 2.3. Algorithms of Constructing MEMWFI

With matricesP,Qi,i=1,2,…,M, we formulate the following block matrix: (35)M=P∗,Q1∗,Q2∗,…,QM∗,and then, (21), the characterization of MEMWFI, can be rewritten as (36)MM∗=aISj0r.Now we present two algorithms of constructing MEMWFI starting from a multiscaling function and a scalar scaling function, respectively.Algorithm 1.
(1) Take a proper multiscaling functionϕ(x).
(2) Calculate the parametersγ,γ1,N=γ/a,j0=logaγ,N1,N2.
(3) Choose proper coefficient matricesCk,n,0≤k≤N-1,-γ+1≤n≤k-N and Ck,ns,0≤k≤N-1,aj0-γ+1+k≤n≤aj0-1,1≤s≤a-1, and construct boundary multiscaling functions using (5); in addition, we can get Φj0.
(4) Use Theorem2, ascertain coefficient matrices Pk,nL,0≤k,n≤N-1;pk,nL,0≤k≤N-1,0≤n≤ak+N1 and pk,nR,s,0≤k≤N-1,0≤n≤ak+N2;Pk,nR,s,0≤k,n≤N-1;1≤s≤a-1 to get the block matrix P.
(5) Take coefficient matricesQk,nL,i,0≤k,n≤N-1;qk,nL,i,0≤k≤N-1,0≤n≤ak+N1; Qni,0≤n≤γ1 and qk,nR,s,i,0≤k≤N-1,0≤n≤ak+N2;Qk,nR,s,i,0≤k,n≤N-1;1≤s≤a-1,1≤i≤M to construct the block matrixes Qi,i=1,…,M, and ensure that they satisfy (21) with P.
(6) Use the coefficient matricesQi which we have got in step (5) to construct MEMWFI Ψj0=Ψj01,…,Ψj0M, by Theorem 3.
(7) Draw the graphs ofΦj0 and Ψj0 by Matlab.Algorithm 2.
(1) Take a proper scalar scaling functionφ(x) (the quadratic sum of recursion coefficients can not be larger than 1).
(2) Useφ(x) to get refinable vector-valued function ϕ(x); for example, (37)ϕx=φx,φx,…,φxrT,ϕx=φx,φx-1,…,φx-r+1T,ϕx=φax,φax-1,…,φax-r+1T,and so on; we can find the sequence of matrices Pk which satisfies refinable equation with ϕ. We should know that the necessary and sufficient conditions for MEMWFI in Theorem 5 reduce to sufficient conditions when the set of components in the vector-valued function ϕ(x) is linearly dependent.
(3) Turn to steps (3)–(7) of Algorithm1.We should note that the refinable vector-valued functionϕ in Algorithm 2 can not be a multiscaling function.


## 2.4. Decomposition Formula and Reconstruction Formula of MEMWFI

The decomposition formula and reconstruction formula of MEMWFI are obtained by Theorems2, 3, and 5, which are similar to those of orthogonal wavelets or multiwavelets on the interval. Firstly, for ∀f∈L2([0,1]), denote (38)cj,k=f,ϕj,k,cj,kL=f,ϕj,kL,cj,kR,s=f,ϕj,kR,s,dj,ki=f,ψj,ki,dj,kL,i=f,ψj,kL,i,dj,kR,s,i=f,ψj,kR,s,ii=1,…,M;s=1,…,a-1.Theorem 6.
The decomposition formula of MEMWFI is(39)acj,kL=∑n=0N-1Pk,nLcj+1,nL+∑n=0ak+N1pk,nLcj+1,nL,0≤k≤N-1acj,k=∑n=akak+γ1Pn-akcj+1,n,0≤k≤aj-γacj,kR,s=∑n=0ak+N2pk,nR,scj+1,aj+1-γ-n+∑n=0ak+N1Pk,nR,scj+1,nR,s,0≤k≤N-1,1≤s≤a-1adj,kL,i=∑n=0N-1Qk,nL,icj+1,nL+∑n=0ak+N1qk,nL,icj+1,n,0≤k≤N-1adj,ki=∑n=akak+γ1Qn-akicj+1,n,0≤k≤aj-γadj,kR,s,i=∑n=0ak+N2qk,nR,s,icj+1,aj+1-γ-n+∑n=0N-1Qk,nR,s,icj+1,nR,s,0≤k≤N-1,1≤s≤a-1,i=1,…,M,and the reconstruction formula of MEMWFI is (40)acj+1,nL=∑k=0N-1Pk,nLcj,kL+∑i=1M∑k=0N-1Qk,nL,idj,kL,i,0≤n≤N-1acj+1,nR,s=∑k=0N-1Pk,nR,scj,kR,s+∑i=1M∑k=0N-1Qk,nL,s,idj,kL,s,i,0≤n≤N-1,1≤s≤a-1acj+1,n=∑k=0N-1pk,nLcj,kL+∑i=0M∑k=0N-1qk,nLdj,kL,i+∑kPn-akcj,k+∑i=1M∑kQn-akidj,ki,0≤n≤N1acj+1,n=acj+1,N1+al+n′=∑k=l+1N-1pk,N1+al+n′Lcj,kL+∑i=1M∑k=l+1N-1qk,N1+al+n′Ldj,kL,i+∑kPN1+n′+al-kcj,k+∑i=1M∑kQN1+n′+al-kidj,ki,0≤l≤N-2,1≤n′≤a,N1+1≤n≤aN-1+N1acj+1,n=∑kPn-akcj,k+∑i=1M∑kQn-akidj,ki,a-1γ-a+1≤n≤aj+1-aγ+a-1(41)acj+1,n=acj+1,N-1-la+N2-n′+aj+1-aγ+a=∑s=1a-1∑k=l+1N-1pk,N-1-la+N2-n′R,scj,kR,s+∑i=1M∑s=1a-1∑k=l+1N-1qk,N-1-la+N2-n′R,s,idj,kR,s,i+∑kPN-1-l-ka+N2-n′+aj+1-aγ+acj,k+∑i=1M∑kQN-1-l-ka+N2-n′+aj+1-aγ+aidj,ki,0≤l≤N-2,0≤n′≤a-1,aj+1-aγ+a≤n≤aj+1-aγ+aN-1(42)acj+1,n=∑s=1a-1∑k=0N-1pk,aj+1-γ-nR,scj,kR,s+∑i=1M∑s=1a-1∑k=0N-1qk,aj+1-γ-nR,s,idj,kR,s,i+∑kPn-akcj,k+∑i=1M∑kQn-akidj,ki,aj+1-aγ+aN≤n≤aj+1-γ.Proof.
The decomposition formula can be obtained by Theorems2 and 3, and the reconstruction formula is gotten from Theorems 2, 3, and 5.


## 3. Numerical Examples

In this section, we present some numerical examples to show the effectiveness of the proposed algorithm with dilator factorsa=2,a=3.N
m
a
(
x
) is mth order cardinal B-spline with dilation factor a; the refinement equation of Nma(x) is the following: (43)N^maω=hmazN^maωa,hmaz=1+z+⋯+za-1am,where z=e-iω/a.


### 3.1.a=2

Example 1.
Witha=2, the symbol of the B-spline N22(x) is (44)h22z=14+12z+14z2.Take ϕ1(x)=N2(x),ϕ2(x)=N2(x-1),ϕ(x)=(ϕ1(x),ϕ2(x))T, and ϕ(x) satisfies (45)ϕx=121100ϕ2x+1100ϕ2x-1+0011ϕ2x-2+0011ϕ2x-3.
In fact, forϕ(x), we can choose different set of coefficient matrices such that it satisfies the above equation.
Letγ=3=γ1,N=γ/a=1,j0=logaγ=2. Taking C0,-2=C0,31=02,C0,-1=C0,21=I2, M=3. Then, we can obtain a set of multiscaling functions on the interval by Theorem 2 and coefficient matrices by Theorem 5, (46)P=1211000000111111110000000011111111000000001111111100000011,Q1=121-10000001-11-11-11-1000000001-11-11-11-1000000001-11-1-11-11000000-11,Q2=123I2-20200-202-20200-202-2I22I20I2,Q3=123I200000-20200000-2020I20I26I2.
The graphs ofϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 1.
The graphs of the first row in Figure1 are the multiscaling function ϕ and its boundary multiscaling functions, and the rest of rows belong to MEMWFI associated with ϕ.
We can discover from Figure1 that each component of multiscaling function and MEMWFI is (anti)symmetrical. The boundary multiscaling functions have advantages of simple structure and some of them are (anti)symmetric.Figure 1
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 2 and multiplicity 2.Example 2.
Witha=2, the symbol of 3rd order B-spline N32(x) is (47)h32z=18+38z+38z2+18z3.Take ϕ1(x)=N3(2x),ϕ2(x)=N3(2x-1),ϕ(x)=(ϕ1(x),ϕ2(x))T, and ϕ satisfies (48)ϕx=141300ϕ2x+3113ϕ2x-1+0031ϕ2x-2.
Certainly, forϕ(x), we can choose different set of coefficient matrices such that it satisfies the above equation.
Letγ=2=γ1,N=γ/a=1,j0=logaγ=1. Taking C0,-1=C0,11=1/3I2,M=4. By Theorems 2 and 5, we can obtain that (49)P=14310013113133100001331131310013,Q1=143-2-13-190113-1-131-33-10000-13-31-13-11130-19-13-23,Q2=1460286270001927-192730-12030030-1203-192719270002862706,Q3=1460-2862700329003000-300-300030032900-2862706,Q4=1410010286810028681002868100-2868101001.
The graphs ofϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 2.
The graphs of the first row in Figure2 are multiscaling function ϕ and left (right) boundary multiscaling functions, and the rest of graphs belong to MEMWFI.
We can discover from Figure2 that each component of multiscaling function and MEMWFI is (anti)symmetric and smooth in this example. Left (right) boundary multiscaling functions are smooth and some of them are (anti)symmetric. Every pair of left boundary multiscaling functions and right boundary multiscaling functions are mutually symmetric.Figure 2
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 2 and multiplicity 2.Example 3.
The previous two examples are constructed whenr=2; we will give an example based on B-spline with r=3 here.
Witha=2, the symbol of 4th order B-spline N42(x) is (50)h42z=116+416z+616z2+416z3+116z4.Take ϕ1(x)=N4(3x),ϕ2(x)=N4(3x-1),ϕ3(x)=N4(3x-2),ϕ(x)=(ϕ1(x),ϕ2(x),ϕ3(x))T, and ϕ satisfies (51)ϕx=18146001000ϕ2x+410464014ϕ2x-1+000100641ϕ2x-2.
Letγ=2=γ1,N=γ/a=1,j0=logaγ=1. Taking C0,-1=C0,11=1/4I2,M=6. Then, by Theorems 2 and 5, we can get that (52)P=18410000464140001464441414641000000146410000001464114446441000144640000149×15,Q111,Q121=181900000232-12000122-62-42-12,Q221=18000-14140000000220-22000000014-14000,Q321,Q331=18-12-42-62212000-122320000019,Q112,Q122=1834-340000200-200000-3535000,Q222=18471674-700000000022000000000-7416747,Q322,Q332=1800035-350000-200200000-3434,Q113,Q123=18000152-15200001580-158000054-54,Q223=18134340-3000000002200000000-3044313,Q323,Q333=18-54540000-15801580000-152152000,Q114,Q124=181900000047400000033000,Q224=18000300-4300001000-10000-43003000,Q324,Q334=180003300000047400000019,Q225=18000-77000000020-20000000-77000,Q115,Q125=180002536441-25364410000202210-202210000-8082180821,Q325,Q335=1880821-808210000-202210202210000-25364412536441000,Q116,Q126=180006368417056000000829451764000000127112,Q226=18000-77000000020-20000000-77000,Q326,Q336=181271120000008294517640000006368417056000,Qi=Q11iQ12iQ22iQ32iQ33i9×15,i=1,2,…,6.
The graphs ofϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 3.
The graphs of the first row in Figure3 are multiscaling function ϕ and left (right) boundary multiscaling functions, and the rest rows belong to MEMWFI.
We can discover from Figure3 that each component of multiscaling function and MEMWFI is (anti)symmetric and smooth in this example. Left (right) boundary multiscaling functions are smooth and some of them are (anti)symmetric. Every pair of left boundary multiscaling functions and right boundary multiscaling functions are mutually symmetric.Figure 3
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 2 and multiplicity 3.


### 3.2.a=3

Example 4.
We will construct this numerical example based on 2nd B-splineN2(x). With a=3, the symbol of N23(x) is (53)h23z=1+2z+3z2+2z3+z49.Take ϕ1(x)=N2(x),ϕ2(x)=N2(x-1),ϕ(x)=(ϕ1(x),ϕ2(x))T, and ϕ satisfies (54)ϕx=131000ϕ3x+2000ϕ3x-1+3001ϕ3x-2+2002ϕ3x-3+1003ϕ3x-4+0002ϕ3x-5+0001ϕ3x-6.
In fact, we have other choices to get the refinable equations associated withϕ.
Letγ=3,γ1=(a-1)γ=6,N=γ/a=1,j0=logaγ=1. Taking (55)C0,-2=C0,21=C0,22=02,C0,-1=2C0,11=2C0,12=12I2,where 02 denotes 2×2 zero matrix, I2 denotes 2×2 unit matrix. Then, we can construct MEMWFI by Theorem 2 and Theorem 5 (see Figure 4). It is so lengthy that we do not write the expression of P,Q1,Q2,Q3,Q4,Q5,Q6 here. The graphs of ϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 4.
The graphs of the first row in Figure4 are multiscaling function ϕ and its left (right) boundary multiscaling functions, and the rest of rows belong to MEMWFI.
We can discover from Figure4 that each component of multiscaling function and MEMWFI is (anti)symmetric in this example. Left (right) boundary multiscaling functions have advantages of simple structure and some of them are (anti)symmetric. The second right boundary multiscaling function has the same graph with the first.Figure 4
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 3 and multiplicity 2.Example 5.
We will construct this numerical example based on 3rd B-splineN3(x). With a=3, the symbol of N33(x) is (56)h33z=1+3z+6z2+7z3+6z4+3z5+z627.Take ϕ1(x)=N3(x),ϕ2(x)=N3(x),ϕ(x)=(ϕ1(x),ϕ2(x))T, and ϕ satisfies (57)ϕx=191001ϕ3x+3003ϕ3x-1+6006ϕ3x-2+7007ϕ3x-3+6006ϕ3x-4+3003ϕ3x-5+1001ϕ3x-6.
Certainly, we have other choices to get the refinable equations associated withϕ. The components of the multiscaling function are linear dependence. It does not satisfy the assumption in Section 2. The sufficient and necessary conditions for minimum-energy frames in Theorem 5 reduce to sufficient conditions. We can still use Theorem 5 to get MEMWFI.
Nowγ=3,γ1=(a-1)γ=6,N=γ/a=1,j0=logaγ=1. Take (58)C0,-2=C0,21=C0,22=02,C0,-1=2C0,11=2C0,12=13I2.Then, we can construct MEMWFI by Theorem 2 and Theorem 5 (see Figure 5). It is so lengthy that we do not write the expression of P,Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8 here. The graphs of ϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 5.
The graphs of the first row in Figure5 are ϕ and its left (right) boundary multiscaling functions, and the rest of rows belong to MEMWFI associated with ϕ.Figure 5
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 3 and multiplicity 2.


## 3.1.a=2

Example 1.
Witha=2, the symbol of the B-spline N22(x) is (44)h22z=14+12z+14z2.Take ϕ1(x)=N2(x),ϕ2(x)=N2(x-1),ϕ(x)=(ϕ1(x),ϕ2(x))T, and ϕ(x) satisfies (45)ϕx=121100ϕ2x+1100ϕ2x-1+0011ϕ2x-2+0011ϕ2x-3.
In fact, forϕ(x), we can choose different set of coefficient matrices such that it satisfies the above equation.
Letγ=3=γ1,N=γ/a=1,j0=logaγ=2. Taking C0,-2=C0,31=02,C0,-1=C0,21=I2, M=3. Then, we can obtain a set of multiscaling functions on the interval by Theorem 2 and coefficient matrices by Theorem 5, (46)P=1211000000111111110000000011111111000000001111111100000011,Q1=121-10000001-11-11-11-1000000001-11-11-11-1000000001-11-1-11-11000000-11,Q2=123I2-20200-202-20200-202-2I22I20I2,Q3=123I200000-20200000-2020I20I26I2.
The graphs ofϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 1.
The graphs of the first row in Figure1 are the multiscaling function ϕ and its boundary multiscaling functions, and the rest of rows belong to MEMWFI associated with ϕ.
We can discover from Figure1 that each component of multiscaling function and MEMWFI is (anti)symmetrical. The boundary multiscaling functions have advantages of simple structure and some of them are (anti)symmetric.Figure 1
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 2 and multiplicity 2.Example 2.
Witha=2, the symbol of 3rd order B-spline N32(x) is (47)h32z=18+38z+38z2+18z3.Take ϕ1(x)=N3(2x),ϕ2(x)=N3(2x-1),ϕ(x)=(ϕ1(x),ϕ2(x))T, and ϕ satisfies (48)ϕx=141300ϕ2x+3113ϕ2x-1+0031ϕ2x-2.
Certainly, forϕ(x), we can choose different set of coefficient matrices such that it satisfies the above equation.
Letγ=2=γ1,N=γ/a=1,j0=logaγ=1. Taking C0,-1=C0,11=1/3I2,M=4. By Theorems 2 and 5, we can obtain that (49)P=14310013113133100001331131310013,Q1=143-2-13-190113-1-131-33-10000-13-31-13-11130-19-13-23,Q2=1460286270001927-192730-12030030-1203-192719270002862706,Q3=1460-2862700329003000-300-300030032900-2862706,Q4=1410010286810028681002868100-2868101001.
The graphs ofϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 2.
The graphs of the first row in Figure2 are multiscaling function ϕ and left (right) boundary multiscaling functions, and the rest of graphs belong to MEMWFI.
We can discover from Figure2 that each component of multiscaling function and MEMWFI is (anti)symmetric and smooth in this example. Left (right) boundary multiscaling functions are smooth and some of them are (anti)symmetric. Every pair of left boundary multiscaling functions and right boundary multiscaling functions are mutually symmetric.Figure 2
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 2 and multiplicity 2.Example 3.
The previous two examples are constructed whenr=2; we will give an example based on B-spline with r=3 here.
Witha=2, the symbol of 4th order B-spline N42(x) is (50)h42z=116+416z+616z2+416z3+116z4.Take ϕ1(x)=N4(3x),ϕ2(x)=N4(3x-1),ϕ3(x)=N4(3x-2),ϕ(x)=(ϕ1(x),ϕ2(x),ϕ3(x))T, and ϕ satisfies (51)ϕx=18146001000ϕ2x+410464014ϕ2x-1+000100641ϕ2x-2.
Letγ=2=γ1,N=γ/a=1,j0=logaγ=1. Taking C0,-1=C0,11=1/4I2,M=6. Then, by Theorems 2 and 5, we can get that (52)P=18410000464140001464441414641000000146410000001464114446441000144640000149×15,Q111,Q121=181900000232-12000122-62-42-12,Q221=18000-14140000000220-22000000014-14000,Q321,Q331=18-12-42-62212000-122320000019,Q112,Q122=1834-340000200-200000-3535000,Q222=18471674-700000000022000000000-7416747,Q322,Q332=1800035-350000-200200000-3434,Q113,Q123=18000152-15200001580-158000054-54,Q223=18134340-3000000002200000000-3044313,Q323,Q333=18-54540000-15801580000-152152000,Q114,Q124=181900000047400000033000,Q224=18000300-4300001000-10000-43003000,Q324,Q334=180003300000047400000019,Q225=18000-77000000020-20000000-77000,Q115,Q125=180002536441-25364410000202210-202210000-8082180821,Q325,Q335=1880821-808210000-202210202210000-25364412536441000,Q116,Q126=180006368417056000000829451764000000127112,Q226=18000-77000000020-20000000-77000,Q326,Q336=181271120000008294517640000006368417056000,Qi=Q11iQ12iQ22iQ32iQ33i9×15,i=1,2,…,6.
The graphs ofϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 3.
The graphs of the first row in Figure3 are multiscaling function ϕ and left (right) boundary multiscaling functions, and the rest rows belong to MEMWFI.
We can discover from Figure3 that each component of multiscaling function and MEMWFI is (anti)symmetric and smooth in this example. Left (right) boundary multiscaling functions are smooth and some of them are (anti)symmetric. Every pair of left boundary multiscaling functions and right boundary multiscaling functions are mutually symmetric.Figure 3
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 2 and multiplicity 3.


## 3.2.a=3

Example 4.
We will construct this numerical example based on 2nd B-splineN2(x). With a=3, the symbol of N23(x) is (53)h23z=1+2z+3z2+2z3+z49.Take ϕ1(x)=N2(x),ϕ2(x)=N2(x-1),ϕ(x)=(ϕ1(x),ϕ2(x))T, and ϕ satisfies (54)ϕx=131000ϕ3x+2000ϕ3x-1+3001ϕ3x-2+2002ϕ3x-3+1003ϕ3x-4+0002ϕ3x-5+0001ϕ3x-6.
In fact, we have other choices to get the refinable equations associated withϕ.
Letγ=3,γ1=(a-1)γ=6,N=γ/a=1,j0=logaγ=1. Taking (55)C0,-2=C0,21=C0,22=02,C0,-1=2C0,11=2C0,12=12I2,where 02 denotes 2×2 zero matrix, I2 denotes 2×2 unit matrix. Then, we can construct MEMWFI by Theorem 2 and Theorem 5 (see Figure 4). It is so lengthy that we do not write the expression of P,Q1,Q2,Q3,Q4,Q5,Q6 here. The graphs of ϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 4.
The graphs of the first row in Figure4 are multiscaling function ϕ and its left (right) boundary multiscaling functions, and the rest of rows belong to MEMWFI.
We can discover from Figure4 that each component of multiscaling function and MEMWFI is (anti)symmetric in this example. Left (right) boundary multiscaling functions have advantages of simple structure and some of them are (anti)symmetric. The second right boundary multiscaling function has the same graph with the first.Figure 4
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 3 and multiplicity 2.Example 5.
We will construct this numerical example based on 3rd B-splineN3(x). With a=3, the symbol of N33(x) is (56)h33z=1+3z+6z2+7z3+6z4+3z5+z627.Take ϕ1(x)=N3(x),ϕ2(x)=N3(x),ϕ(x)=(ϕ1(x),ϕ2(x))T, and ϕ satisfies (57)ϕx=191001ϕ3x+3003ϕ3x-1+6006ϕ3x-2+7007ϕ3x-3+6006ϕ3x-4+3003ϕ3x-5+1001ϕ3x-6.
Certainly, we have other choices to get the refinable equations associated withϕ. The components of the multiscaling function are linear dependence. It does not satisfy the assumption in Section 2. The sufficient and necessary conditions for minimum-energy frames in Theorem 5 reduce to sufficient conditions. We can still use Theorem 5 to get MEMWFI.
Nowγ=3,γ1=(a-1)γ=6,N=γ/a=1,j0=logaγ=1. Take (58)C0,-2=C0,21=C0,22=02,C0,-1=2C0,11=2C0,12=13I2.Then, we can construct MEMWFI by Theorem 2 and Theorem 5 (see Figure 5). It is so lengthy that we do not write the expression of P,Q1,Q2,Q3,Q4,Q5,Q6,Q7,Q8 here. The graphs of ϕ, left (right) boundary multiscaling functions, and MEMWFI are showed in Figure 5.
The graphs of the first row in Figure5 are ϕ and its left (right) boundary multiscaling functions, and the rest of rows belong to MEMWFI associated with ϕ.Figure 5
Graphs of multiscaling function and left (right) boundary multiscaling functions and MEMWFI with dilation factor 3 and multiplicity 2.



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*Source: 262637-2015-05-26.xml*