# Analysis of Compactly Supported Nonstationary Biorthogonal

Authors: Yeon Ju Lee; Jungho Yoon
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper is concerned with analyzing the mathematical properties, such as the
regularity and stability of nonstationary biorthogonal wavelet systems based on exponential B-splines.
We first discuss the biorthogonality condition of the nonstationary refinable functions,
and then we show that the refinable functions based on exponential B-splines have the same
regularities as the ones based on the polynomial B-splines of the corresponding orders. In the
context of nonstationary wavelets, the stability of wavelet bases is not implied by the stability of
a refinable function. For this reason, we prove that the suggested nonstationary wavelets form
Riesz bases for the space that they generate.


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## Body

## 1. Introduction

For the last two decades, the wavelet transforms have become very useful tools in a variety of applications such as signal and image processing and numerical computation. The construction of classical wavelets is now well understood thanks to such pioneering works as [1–3]. Many properties, such as symmetry (or antisymmetry), vanishing moments, regularity, and short support, are required in a practical use for application areas. In particular, polynomial splines have been a common source for wavelet construction [1, 3–6]. A new class of compactly supported biorthogonal wavelet systems that are constructed from pseudosplines was introduced in [7].Exponential B-splines and polynomials have been found to be quite useful in a number of applications such as computer-aided geometric design, shape-preserving curve fitting, and signal interpolation [8–10]. Exponential B-splines were used as a key ingredient for the construction of wavelets [11, 12] and particularly used in wavelet construction on 𝕊2 and 𝕊3 [13]. In particular, in the approximation and sparse representation of acoustic signals, polynomial-based (stationary) wavelet systems have an important limitation because they do not consider the spectral features (e.g., band limited) of a given signal. However, (non-stationary) wavelet systems based on the exponential B-spline can be tuned to the specific trait of the given signal, yielding better approximations and sparser representations than classical wavelets at strictly the same computational costs. Details on exponential splines can be found in the selected references [10, 11, 14–16]. Related studies on non-stationary wavelets can be found in [11, 12, 15, 17–22].One natural and convenient way to introduce wavelets is to follow the notion of multiresolution analysis. However, because the refinement masks we are interested in are non-stationary (i.e., scale dependent), we use the structure of non-stationary multiresolution analysis as introduced in [17]. Given an integer j0∈ℤ and compactly supported refinable functions ϕj, j≥j0, in L2(ℝ), we say that a sequence of subspaces
(1.1)Vj=span⁡¯{ϕj,k∶=2j/2ϕj(2j⋅-k):k∈Z},j≥j0
forms a non-stationary multiresolution analysis (MRA) of L2(ℝ) if the following conditions are satisfied: (1)
Vj⊂Vj+1 for all j≥j0,(2)
⋃j≥j0Vj is dense in L2(ℝ);(3)
the set{ϕj,k:k∈ℤ} is a Riesz (or stable) basis for Vj for each j≥j0.The nested embedding of the spacesVj implies the existence of a sequence a[j]∶=(an[j])n∈ℤ∈ℓ2(ℤ) that satisfies the non-stationary refinement equation(1.2)ϕj(x)=∑n∈Zan[j]ϕj+1(2x-n),j≥j0,
where the sequence a[j] is usually called the refinement mask for ϕj. One should notice that the function ϕj(2j·)∈Vj is no longer a dilated version of ϕj0 in Vj0. A refinable function ϕ̃j∈L2(ℝ) with the mask ã[j]∶=(ãn[j])n∈ℤ is called the dual refinable function of ϕj∈L2(ℝ) (or just the dual of ϕj for simplicity) if it satisfies
(1.3)〈ϕj,ϕ̃j(⋅-l)〉=δ0,l,l∈Z.
Let (Vj)j≥j0 and (Ṽj)j≥j0 be a pair of MRAs generated by a pair of dual refinable functions ϕj and ϕ̃j, j≥j0, respectively. The concept of biorthogonal wavelets is to find complement spaces Wj and W̃j of Vj and Ṽj, respectively, satisfying Vj+1=Vj+̇Wj, Wj⊥Ṽj and Ṽj+1=Ṽj+̇W̃j, W̃j⊥Vj. The corresponding biorthogonal wavelets are given by
(1.4)ψj=∑n∈Z(-1)nã1-n[j]ϕj+1(2⋅-n),ψ̃j=∑n∈Z(-1)na1-n[j]ϕ̃j+1(2⋅-n).A generalization of the biorthogonal wavelets of Cohen-Daubechies-Feuveau [1] was introduced that was based on exponential B-splines [12]. By generalizing the Strang-Fix conditions, the authors discussed the relationship between the reproduction of exponential polynomials (by ϕj or ϕ̃j) and the zeros of the corresponding Laurent polynomials. They also proved that for each j≥j0, the proposed non-stationary refinable function ϕj generates a Riesz basis for Vj and that the corresponding Riesz (upper and lower) bounds are independent of j≥j0. However, the authors did not explicitly address the biorthogonality condition of the corresponding non-stationary refinable functions. Moreover, some fundamental questions concerning the global stability, and regularity, were left unanswered. Therefore, the primary goal of this paper is to address these issues. First, we provide a sufficient condition for the biorthogonality (1.3) of non-stationary refinable functions, and then we prove that the refinable functions based on exponential B-splines have the same regularities as the ones based on the polynomial B-splines of the corresponding orders. In the context of non-stationary wavelets, the stability of the wavelet bases {ψj,k:k∈ℤ,j≥j0} is not implied by the stability of a refinable function. Therefore, we prove that the set {ψj,k:k∈ℤ,j≥j0} forms a Riesz basis for the space +̇j≥j0Wj. Furthermore, we show that the set {ϕj0,k:k∈ℤ}∪{ψj,k:k∈ℤ,j≥j0} becomes a Riesz basis for the space L2(ℝ).This paper is organized as follows. In Section2, we provide basic notions of exponential B-splines. Section 3 discusses the biorthogonality condition of non-stationary refinable functions and then studies their regularities. In Section 4, we prove the (global) stability of the proposed non-stationary wavelet bases.


## 2. Preliminaries: Exponential B-Splines

Given a set of complex numbersG={γj∈ℂ:j=1,…,N}, the corresponding Nth-order exponential B-splines can be defined as successive convolutions of the first-order B-spline
(2.1)ϕj∶=ϕj[G]∶=τj(eγ12-jB1*⋯*eγN2-jB1),
with a normalization factor τj defined so that ∥ϕj∥L1(ℝ)=1 (see [10]), where B1 indicates the first-order B-spline, that is, B1∶=χ[0,1], and eγ:x↦eγx, γ∈ℂ, is the exponential function. For simplicity, we will omit G in ϕj[G]. Obviously, the function ϕj is a compactly supported piecewise exponential polynomial. The global regularity of ϕj is CN-2 (see [10, 15]). A convenient way to represent an (Nth-order) exponential B-spline is with the Laurent polynomial
(2.2)a[j](z)∶=∑n∈Zan[j]zn∶=2cj∏n=1N12(1+eγn2-j-1z),j≥j0,
where cj is the normalization factor defined by
(2.3)cj∶=cj,G∶=∏n=1N2(1+eγn2-j-1).
We call a[j] the symbol of ϕj. Throughout this paper, γn is assumed to be a real or a pure imaginary number, that is, γn∈ℝ or γn∈iℝ. Since we want the mask (an[j])n∈ℤ to be symmetric with respect to its center, it is reasonable to assume that if γn∈G, then -γn∈G. The relationship between the reproduction of exponential polynomials by ϕj and the zeros of the corresponding Laurent polynomial a[j] is discussed in [12].It is well known that the integer translatesϕ0(·-k), k∈ℤ, are linearly independent if and only if γℓ-γn∉2πiℤ for γℓ≠γn [10, 16]. From (2.1) and (2.2), we can easily deduce that for each j≥j0, the integer translates ϕj(·-k), k∈ℤ, are linearly independent if and only if
(2.4)2-j(γl-γn)∉2πiZ,γl≠γn.
A concept related to (but weaker than) the linear independence is the notion of the stability of ϕj. It is known (see, e.g., [23]) that the set {ϕj(·-k):k∈ℤ} forms a Riesz basis if and only if there exist constants Aj,Bj>0 such that
(2.5)Aj≤[ϕ̂j,ϕ̂j]≤Bj,
where the bracket product[f,g] for f,g∈L2(ℝ) is defined by
(2.6)[f,g](ξ)∶=∑n∈Zf(ξ+2πn)g(ξ+2πn)¯.
We say that the function ϕj∈L2(ℝ) is stable if (2.5) is satisfied. If the integer translates of ϕj are linearly independent, the stability of ϕj is immediate [24, Theorem  1.2], that is, {ϕj(2j·-k):k∈ℤ} is a Riesz basis for Vj. Finally, the basic requirement on the set G={γ1,…,γN} can be summarized as follows: (i) each γn is a real or a pure imaginary number; (ii) both γn and -γn belong to G, and (iii) for any j≥j0, 2-j(γn-γℓ)∉2πiℤ with γn≠γℓ.


## 3. Dual Refinable Functions

### 3.1. Construction of Dual Refinable Functions

Given refinable functionsϕj with j≥j0 the first step in the construction of a biorthogonal wavelet system is to find their dual refinable functions ϕ̃j∈L2(ℝ), j≥j0 (whose symbol is denoted by ã[j]). A necessary condition for ϕj and ϕ̃j to satisfy (1.3) is
(3.1)a[j](z)ã[j](z)¯+a[j](-z)ã[j](-z)¯=4,
where z̅ is the conjugate of the complex number z. Thus, the construction of ϕ̃j starts with the construction of a dual symbol ã[j](z)∶=∑n∈ℤãn[j]zn such that (3.1) is satisfied. The algorithm to find ã[j] is analogous to the classical method [1], except for the main difference that ã[j], j≥j0, are scale dependent. However, in order to ensure the existence of ã[j] satisfying (3.1), we need to prove that a[j] has no roots of opposite signs. For this purpose, denoting
(3.2)N=2L+τ,τ∈{0,1},
and setting z=e-iξ and y=sin2ξ/2 (i.e., y=-(1-z)2/4z), we represent a[j] as follows:
(3.3)a[j](e-iξ)=2e-iLξ(1+e-iξ2)τ∏n=1L(1-νj,ny),
where
(3.4)νj,n=4eγn2-j-1+e-γn2-j-1+2.
Indeed, we are looking for a Laurent polynomial ã[j] of the form ã[j](e-iξ)=a[j](e-iξ)Q[j](y) such that (3.1) holds. Therefore, if we define P[j](y)∶=a[j](e-iξ)a[j](e-iξ)¯/4, that is,
(3.5)P[j](y)∶=(1-y)τ∏n=1L(1-νj,ny)2,
then the problem of finding ã[j](z) is reduced to constructing Q[j](y) which satisfies the equation
(3.6)P[j](y)Q[j](y)+P[j](1-y)Q[j](1-y)=1,∀y∈[0,1].
Let us now prove that there is no common zero of P[j] and P[j](1-·) on [0,1].Proposition 3.1.
Let the polynomialsP[j],  j≥j0, be given as in (3.5).  Assume that for any j≥j0,  2-j(γn-γℓ)∉2πiℤ for any γn,γℓ∈G with γn≠γℓ, then P[j] and P[j](1-·) have no common roots on [0,1].Proof.
Assume thatP[j] and P[j](1-·) have a common root on [0,1]. This is equivalent to the existence of a number z0∈ℂ∖{0} such that
(3.7)a[j](z0)=a[j](-z0)=0.
From (2.2), we can deduce that for some γℓ≠γn, 1+e-2-j-1γℓz0=0 and 1+e-2-j-1γn(-z0)=0. It follows that 2-j(γn-γℓ)∈2πiℤ, which contradicts the initial assumption.By virtue of Proposition3.1, the Bezout theorem guarantees the existence of a unique polynomial Q[j] of degree N-1 that satisfies (3.6). One may look for a polynomial Q[j] of degree higher than N-1. However, if the corresponding dual refinable functions are to have the shortest possible support, the degree of Q[j] must be constrained to be N-1. On the other hand, it is of interest to see that by regrouping the simple fractions of Q[j] into two groups, that is,
(3.8)Q[j](y)=b[j](z)b̃[j](z),y=-(1-z)24z,
then we can define the adjusted Laurent polynomials a[j] and ã[j] by
(3.9)a[j](z)∶=2cjb[j](z)∏n=1N12(1+eγn2-j-1z),ã[j](z)∶=2cjb̃[j](z)∏n=1N12(1+eγn2-j-1z),
so that the lengths of a[j] and ã[j] are very close. This allows us to construct generalized non-stationary refinable functions ϕj and ϕ̃j. If b[j]=1, the resulting function ϕj becomes an exponential B-spline of order N. On the other hand, the classical counterparts of a[j] and ã[j] (which are obtained by setting γn=0 for all γn∈G) can be written as
(3.10)a(z)=2-N+1(1+z)Nb(z),ã(z)=2-N+1(1+z)Nb̃(z).
For notational simplicity, we will write
(3.11)b[j](z)=∑n∈Zbn[j]zn,b(z)=∑n∈Zbnzn.Lemma 3.2.
Letb[j]∶=(bn[j]:n∈ℤ) and b∶=(bn:n∈ℤ) with j≥j0,  then, as j tends to ∞,  one has ∥b[j]-b∥1=O(2-j).Proof.
This is a direct consequence of [12, Lemma  2].For the given Laurent polynomialsã[j] with j≥j0, there corresponds a potential candidate for the refinable function ϕ̃j which is defined in terms of the Fourier transform as
(3.12)ϕ̃̂j(ξ)=∏n=0∞12Ã[j+n](ξ2n+1),
where
(3.13)Ã[j](ξ)=ã[j](e-iξ),
provided that ϕ̃̂j(ξ/2n)→1 as n→∞. In fact, ϕ̃j in (3.12) is the only candidate for the refinable function associated with ã[j] such that ϕ̃j is a dual of ϕj. Although the infinite product in (3.12) converges pointwise and ϕ̃j is in L2(ℝ) whenever ϕ̃ is in L2(ℝ) [12], it still needs to be ensured that the function ϕ̃j in (3.12) is indeed a dual of ϕj. The following results address this issue. For simplicity, using b[j] and b̃[j] in (3.9), we introduce the notation
(3.14)Bk,j∶=sup⁡n≥jmax⁡ξ|∏l=1kb[n+l-1](e-iξ/2l)|1/k,B̃k̃,j∶=sup⁡n≥jmax⁡ξ|∏l=1k̃b̃[n+l-1](e-iξ/2l)|1/k̃.Lemma 3.3.
Letϕ̃j,  j≥j0,  be given as in (3.12) with the symbol ã[j] in (3.9).  Suppose that for some integer k̃>0,  B̃k̃,j<2N-1/2,  then
(3.15)|ϕ̃̂j|≤c(1+|⋅|)-N+log⁡2B̃k̃,j,
which implies that ϕ̃j∈L2(ℝ). Moreover, the function ϕ̂j can be defined as in (3.12), then this lemma also holds for ϕj.Proof.
Letm∈ℕ and define ũj,m by
(3.16)ũj,m(ξ)∶=[∏l=1m12Ã[j+l-1](ξ2-l)]χ[-π,π](ξ2-m).
For the given γn∈ℂ with n=1,…,N, set ξγn,j∶=ξ+iγn2-j-1. Then, we get the identity
(3.17)∏l=1m|1+eγ2-j-l-1e-iξ2-l2|=∏l=1m|sin⁡(ξn,j2-l)2sin⁡(ξn,j2-l-1)|=|sin⁡(ξn,j2-1)2msin⁡(ξn,j2-m-1)|.
Here, we can see that there exist η>0 and M∈ℕ such that if m≥M and |ξ2-m-1|≤η, |sin(ξn,j2-m-1)|≥2-1|ξn,j2-m-1|. Also, it is obvious that |sin(ξn,j2-1)|≤c1|ξn,j2-1| for some constant c1>0 independent of ξ but dependent on γn. Therefore, we have
(3.18)|sin⁡(ξn,j2-1)2msin⁡(ξn,j2-m-1)|≤c1|ξn,j2-m-1sin⁡(ξn,j2-m-1)|≤c2,
with a constant c2>0 depending on γn. Next, consider the case of η<|ξ2-m-1|≤π/2 and let ϵ>0 be a sufficiently small number so that ϵ<η/2. Noting that |sinξ|≥(2/π)|ξ| for all η<|ξ|≤π/2, it is not difficult to see that for a sufficiently large j≥J,
(3.19)2m|sin⁡(ξn,j2-m-1)|≥2m(2-1|sin⁡(ξ2-m-1)|-|sin⁡(γn2-j-m-2)|)≥c3(|ξ|-ϵ)≥c4|ξ|,
then we obtain that
(3.20)|sin⁡(ξn,j2-1)2msin⁡(ξn,j2-m-1)|≤c1+|ξ|,
for a constant c>0 depending on γn. Consequently, invoking the fact that Ã[j](ξ)=ã[j](e-iξ) with ã[j] in (3.9), we obtain
(3.21)|ũj,m(ξ)|≤c(1+|ξ|)N∏l=1m|b̃[j+l-1](e-iξ2-l)|χ[-π,π](2-mξ),
for any j≥J. Here, using the same argument in [1, Proposition  4.8] (see point 3), we can get
(3.22)∏l=1mb̃[j+l-1](e-iξ2-l)≤c(1+|ξ|)log⁡2B̃k̃,j.
This together with (3.21) implies that |ũj,m(ξ)|≤c(1+|ξ|)-N+log2B̃k̃,j for any j≥J, where c>0 is independent of m. Since ũj,m(ξ)→ϕ̃̂j(ξ) pointwise as m→∞, we get the relation in (3.15) with j≥J. For j<J, applying an inductive argument based on the refinement equation ϕ̃̂j=Ã[j](·/2)ϕ̃̂j+1(·/2), we obtain the required result. The case of ϕ̂j can be done similarly.Lemma 3.4.
Leta[j] and ã[j],  j≥j0,  be given as in (3.9).  Suppose that for some integers k,k̃>0,
(3.23)Bk,j,B̃k̃,j<2N-1/2,
with Bk,j and B̃k,j in (3.14), then, for any j≥j0, one has 〈ϕj,ℓ,ϕ̃j,n〉=δℓ,n for all ℓ,n∈ℤ.Proof.
Recalling the definition ofũj,m in (3.16), define uj,m by
(3.24)uj,m(ξ)∶=[∏l=1m12A[j+l-1](ξ2-l)]χ[-π,π](ξ2-m),
Then uj,m and ũj,m converge pointwise to ϕ̂j and ϕ̃̂j, respectively. Moreover, using (3.1), we can derive the relation
(3.25)uj,mũ¯j,m̂(l)=∫02m+1π(∏l=1m14A[j+l-1](ξ2-l)Ã̅[j+l-1](ξ2-l))eilξdξ=∫02mπ(∏l=1m-114A[j+l-1](ξ2-l)Ã̅[j+l-1](ξ2-l))eilξdξ=uj,m-1ũ¯j,m-1̂(l).
Repeating this process yields the identity uj,mũ¯j,m̂(ℓ)=uj,1ũ¯j,1̂(ℓ)=2πδ0,ℓ. By Lemma 3.3, ϕj,ϕ̃j∈L2(ℝ), then it is immediate from the Lebesgue-dominated convergence theorem that uj,m and ũj,m converge to ϕ̂j and ϕ̃̂j, respectively, in L2. This in turn implies that uj,mũ¯j,m→ϕ̂jϕ̃̂¯j in L1, as m→∞. Applying Plancherel’s theorem, we arrive at the conclusion that 〈ϕj,ϕ̃j(·-ℓ)〉=δ0,ℓ for any ℓ∈ℤ.This result proves that〈ϕj,0,ϕ̃j,n〉=δ0,n, n∈ℤ, for some j (in fact, for a sufficiently large j) which guarantees the condition (3.23). But the following proposition indeed proves that this duality condition holds for any j≥j0, under some suitable condition on the symbols a and ã in (3.10). In the following analysis, it is useful to use the notation
(3.26)Bk∶=max⁡ξ|∏l=1kb(e-iξ/2l)|1/k,B̃k̃∶=max⁡ξ|∏l=1k̃b̃(e-iξ/2l)|1/k̃,
with b and b̃ in (3.10).Proposition 3.5.
Leta[j] and ã[j],  j≥j0,  be given as in (3.9). Assume that for some integers k,k̃>0,
(3.27)Bk,B̃k̃<2N-1/2,
then, for any j≥j0, one has 〈ϕj,0,ϕ̃j,n〉=δ0,n for all n∈ℤ.Proof.
Due to Lemma3.2, we find that as j→∞, b[j] and b̃[j] converge uniformly on |z|=1 to b and b̃, respectively. Thus, we can deduce that there exists a large J∈ℕ such that
(3.28)Bk,j,B̃k̃,j<2N-1/2,∀j≥J.
It follows from Lemma 3.4 that for any j≥J, 〈ϕj,0,ϕ̃j,n〉=δ0,n for all n∈ℤ. For the case j<J, this property can be derived by using an inductive argument based on the non-stationary refinement equation. Specifically, applying (1.2), we get
(3.29)〈ϕj,0,ϕ̃j,l〉=12∑n∈Zan[j]∑k∈Zãk-2l[j]〈ϕj+1,n,ϕ̃j+1,k〉=12∑n∈Zan[j]∑k∈Zãk-2l[j]δn,k=12∑n∈Zan[j]ãn-2l[j]=δ0,l,
for any ℓ∈ℤ. This completes the proof.


### 3.2. Smoothness of Refinable Functions

For a givenκ=n+s with n∈ℕ and s∈[0,1], the Hölder space Hκ (e.g., see [3]) is defined to be the space of n-times continuously differentiable functions f whose nth derivative f(n) satisfies the Lipschitz condition
(3.30)sup⁡x,h∈R|f(n)(x+h)-f(n)(x)||h|s≤C.
In particular, the regularity of f also can be analyzed by estimating the decay of f̂ around ∞. If |f̂(ξ)|≤c(1+|ξ|)-1-κ-ϵ, ξ∈ℝ, for an arbitrary small ϵ>0, then f belongs to the space Hκ. In the following theorem, under some more stringent condition on Bk (than (3.27)), we derive the smoothness of non-stationary refinable functions ϕ̃j, j≥j0, in (3.12). The smoothness of ϕj can be shown in a similar way.Theorem 3.6.
Assume that the Laurent polynomialã in (3.10) can be rewritten as
(3.31)ã(z)=2-K̃(1+z)K̃+1b̃(z),K̃<N,
such that B̃k̃<1 for some k̃∈ℕ. Let ϕ̃j, j≥j0, be given as in (3.12) with the symbol ã[j] in (3.9), then ϕ̃j∈HK̃+ν̃ for some ν̃∈(0,1).Proof.
The Laurent polynomialã[j] in (3.9) can be written as
(3.32)ã[j](z)=2cjb̃[j](z)∏n=1K̃+112(1+eγn2-j-1z),K̃<N.
From Lemma 3.2, we can deduce that b̃[j](z)→b̃(z) uniformly on |z|=1 as j→∞. This implies that B̃k̃,j in (3.14) converges to B̃k̃ (<1) as j→∞. Thus, there exists a sufficiently large J∈ℕ such that B̃k̃,j<1 for any j≥J, which means that log2B̃k̃,j<0. Hence, recalling from (3.15) that
(3.33)|ϕ̃̂j|≤c(1+|⋅|)-K̃-1+log⁡2B̃k̃,j,
we see that for any j≥J, ϕ̃j∈HK̃+ν̃ with ν̃∈(0,1). Next, consider the case of j<J. By applying an inductive argument based on the refinement equation
(3.34)ϕ̃j=∑n∈Zan[j]ϕ̃j+1(⋅-n),
we show that ϕ̃j∈HK̃+ν̃ for any j<J.Remark 3.7.
It is known (e.g., see [1]) that the Fourier transform of ϕ̃ with the symbol ã in (3.31) has the decay rate
(3.35)|ϕ̃̂|≤c(1+|⋅|)-K̃-1+log⁡2B̃k̃.
Since B̃k̃,j converges to B̃k̃ as j tends to ∞, we can deduce that the functions ϕ̃j, j≥j0, have the same regularity as ϕ̃. For the details about the (optimal) regularity of ϕ̃, the reader is referred to [1]. One may investigate the regularity of ϕ̃j by using the concept of asymptotical equivalent subdivision schemes (see [15, 19]). However, the methods in [15, 19] are mainly concerned with the integer smoothness of the refinable functions.


## 3.1. Construction of Dual Refinable Functions

Given refinable functionsϕj with j≥j0 the first step in the construction of a biorthogonal wavelet system is to find their dual refinable functions ϕ̃j∈L2(ℝ), j≥j0 (whose symbol is denoted by ã[j]). A necessary condition for ϕj and ϕ̃j to satisfy (1.3) is
(3.1)a[j](z)ã[j](z)¯+a[j](-z)ã[j](-z)¯=4,
where z̅ is the conjugate of the complex number z. Thus, the construction of ϕ̃j starts with the construction of a dual symbol ã[j](z)∶=∑n∈ℤãn[j]zn such that (3.1) is satisfied. The algorithm to find ã[j] is analogous to the classical method [1], except for the main difference that ã[j], j≥j0, are scale dependent. However, in order to ensure the existence of ã[j] satisfying (3.1), we need to prove that a[j] has no roots of opposite signs. For this purpose, denoting
(3.2)N=2L+τ,τ∈{0,1},
and setting z=e-iξ and y=sin2ξ/2 (i.e., y=-(1-z)2/4z), we represent a[j] as follows:
(3.3)a[j](e-iξ)=2e-iLξ(1+e-iξ2)τ∏n=1L(1-νj,ny),
where
(3.4)νj,n=4eγn2-j-1+e-γn2-j-1+2.
Indeed, we are looking for a Laurent polynomial ã[j] of the form ã[j](e-iξ)=a[j](e-iξ)Q[j](y) such that (3.1) holds. Therefore, if we define P[j](y)∶=a[j](e-iξ)a[j](e-iξ)¯/4, that is,
(3.5)P[j](y)∶=(1-y)τ∏n=1L(1-νj,ny)2,
then the problem of finding ã[j](z) is reduced to constructing Q[j](y) which satisfies the equation
(3.6)P[j](y)Q[j](y)+P[j](1-y)Q[j](1-y)=1,∀y∈[0,1].
Let us now prove that there is no common zero of P[j] and P[j](1-·) on [0,1].Proposition 3.1.
Let the polynomialsP[j],  j≥j0, be given as in (3.5).  Assume that for any j≥j0,  2-j(γn-γℓ)∉2πiℤ for any γn,γℓ∈G with γn≠γℓ, then P[j] and P[j](1-·) have no common roots on [0,1].Proof.
Assume thatP[j] and P[j](1-·) have a common root on [0,1]. This is equivalent to the existence of a number z0∈ℂ∖{0} such that
(3.7)a[j](z0)=a[j](-z0)=0.
From (2.2), we can deduce that for some γℓ≠γn, 1+e-2-j-1γℓz0=0 and 1+e-2-j-1γn(-z0)=0. It follows that 2-j(γn-γℓ)∈2πiℤ, which contradicts the initial assumption.By virtue of Proposition3.1, the Bezout theorem guarantees the existence of a unique polynomial Q[j] of degree N-1 that satisfies (3.6). One may look for a polynomial Q[j] of degree higher than N-1. However, if the corresponding dual refinable functions are to have the shortest possible support, the degree of Q[j] must be constrained to be N-1. On the other hand, it is of interest to see that by regrouping the simple fractions of Q[j] into two groups, that is,
(3.8)Q[j](y)=b[j](z)b̃[j](z),y=-(1-z)24z,
then we can define the adjusted Laurent polynomials a[j] and ã[j] by
(3.9)a[j](z)∶=2cjb[j](z)∏n=1N12(1+eγn2-j-1z),ã[j](z)∶=2cjb̃[j](z)∏n=1N12(1+eγn2-j-1z),
so that the lengths of a[j] and ã[j] are very close. This allows us to construct generalized non-stationary refinable functions ϕj and ϕ̃j. If b[j]=1, the resulting function ϕj becomes an exponential B-spline of order N. On the other hand, the classical counterparts of a[j] and ã[j] (which are obtained by setting γn=0 for all γn∈G) can be written as
(3.10)a(z)=2-N+1(1+z)Nb(z),ã(z)=2-N+1(1+z)Nb̃(z).
For notational simplicity, we will write
(3.11)b[j](z)=∑n∈Zbn[j]zn,b(z)=∑n∈Zbnzn.Lemma 3.2.
Letb[j]∶=(bn[j]:n∈ℤ) and b∶=(bn:n∈ℤ) with j≥j0,  then, as j tends to ∞,  one has ∥b[j]-b∥1=O(2-j).Proof.
This is a direct consequence of [12, Lemma  2].For the given Laurent polynomialsã[j] with j≥j0, there corresponds a potential candidate for the refinable function ϕ̃j which is defined in terms of the Fourier transform as
(3.12)ϕ̃̂j(ξ)=∏n=0∞12Ã[j+n](ξ2n+1),
where
(3.13)Ã[j](ξ)=ã[j](e-iξ),
provided that ϕ̃̂j(ξ/2n)→1 as n→∞. In fact, ϕ̃j in (3.12) is the only candidate for the refinable function associated with ã[j] such that ϕ̃j is a dual of ϕj. Although the infinite product in (3.12) converges pointwise and ϕ̃j is in L2(ℝ) whenever ϕ̃ is in L2(ℝ) [12], it still needs to be ensured that the function ϕ̃j in (3.12) is indeed a dual of ϕj. The following results address this issue. For simplicity, using b[j] and b̃[j] in (3.9), we introduce the notation
(3.14)Bk,j∶=sup⁡n≥jmax⁡ξ|∏l=1kb[n+l-1](e-iξ/2l)|1/k,B̃k̃,j∶=sup⁡n≥jmax⁡ξ|∏l=1k̃b̃[n+l-1](e-iξ/2l)|1/k̃.Lemma 3.3.
Letϕ̃j,  j≥j0,  be given as in (3.12) with the symbol ã[j] in (3.9).  Suppose that for some integer k̃>0,  B̃k̃,j<2N-1/2,  then
(3.15)|ϕ̃̂j|≤c(1+|⋅|)-N+log⁡2B̃k̃,j,
which implies that ϕ̃j∈L2(ℝ). Moreover, the function ϕ̂j can be defined as in (3.12), then this lemma also holds for ϕj.Proof.
Letm∈ℕ and define ũj,m by
(3.16)ũj,m(ξ)∶=[∏l=1m12Ã[j+l-1](ξ2-l)]χ[-π,π](ξ2-m).
For the given γn∈ℂ with n=1,…,N, set ξγn,j∶=ξ+iγn2-j-1. Then, we get the identity
(3.17)∏l=1m|1+eγ2-j-l-1e-iξ2-l2|=∏l=1m|sin⁡(ξn,j2-l)2sin⁡(ξn,j2-l-1)|=|sin⁡(ξn,j2-1)2msin⁡(ξn,j2-m-1)|.
Here, we can see that there exist η>0 and M∈ℕ such that if m≥M and |ξ2-m-1|≤η, |sin(ξn,j2-m-1)|≥2-1|ξn,j2-m-1|. Also, it is obvious that |sin(ξn,j2-1)|≤c1|ξn,j2-1| for some constant c1>0 independent of ξ but dependent on γn. Therefore, we have
(3.18)|sin⁡(ξn,j2-1)2msin⁡(ξn,j2-m-1)|≤c1|ξn,j2-m-1sin⁡(ξn,j2-m-1)|≤c2,
with a constant c2>0 depending on γn. Next, consider the case of η<|ξ2-m-1|≤π/2 and let ϵ>0 be a sufficiently small number so that ϵ<η/2. Noting that |sinξ|≥(2/π)|ξ| for all η<|ξ|≤π/2, it is not difficult to see that for a sufficiently large j≥J,
(3.19)2m|sin⁡(ξn,j2-m-1)|≥2m(2-1|sin⁡(ξ2-m-1)|-|sin⁡(γn2-j-m-2)|)≥c3(|ξ|-ϵ)≥c4|ξ|,
then we obtain that
(3.20)|sin⁡(ξn,j2-1)2msin⁡(ξn,j2-m-1)|≤c1+|ξ|,
for a constant c>0 depending on γn. Consequently, invoking the fact that Ã[j](ξ)=ã[j](e-iξ) with ã[j] in (3.9), we obtain
(3.21)|ũj,m(ξ)|≤c(1+|ξ|)N∏l=1m|b̃[j+l-1](e-iξ2-l)|χ[-π,π](2-mξ),
for any j≥J. Here, using the same argument in [1, Proposition  4.8] (see point 3), we can get
(3.22)∏l=1mb̃[j+l-1](e-iξ2-l)≤c(1+|ξ|)log⁡2B̃k̃,j.
This together with (3.21) implies that |ũj,m(ξ)|≤c(1+|ξ|)-N+log2B̃k̃,j for any j≥J, where c>0 is independent of m. Since ũj,m(ξ)→ϕ̃̂j(ξ) pointwise as m→∞, we get the relation in (3.15) with j≥J. For j<J, applying an inductive argument based on the refinement equation ϕ̃̂j=Ã[j](·/2)ϕ̃̂j+1(·/2), we obtain the required result. The case of ϕ̂j can be done similarly.Lemma 3.4.
Leta[j] and ã[j],  j≥j0,  be given as in (3.9).  Suppose that for some integers k,k̃>0,
(3.23)Bk,j,B̃k̃,j<2N-1/2,
with Bk,j and B̃k,j in (3.14), then, for any j≥j0, one has 〈ϕj,ℓ,ϕ̃j,n〉=δℓ,n for all ℓ,n∈ℤ.Proof.
Recalling the definition ofũj,m in (3.16), define uj,m by
(3.24)uj,m(ξ)∶=[∏l=1m12A[j+l-1](ξ2-l)]χ[-π,π](ξ2-m),
Then uj,m and ũj,m converge pointwise to ϕ̂j and ϕ̃̂j, respectively. Moreover, using (3.1), we can derive the relation
(3.25)uj,mũ¯j,m̂(l)=∫02m+1π(∏l=1m14A[j+l-1](ξ2-l)Ã̅[j+l-1](ξ2-l))eilξdξ=∫02mπ(∏l=1m-114A[j+l-1](ξ2-l)Ã̅[j+l-1](ξ2-l))eilξdξ=uj,m-1ũ¯j,m-1̂(l).
Repeating this process yields the identity uj,mũ¯j,m̂(ℓ)=uj,1ũ¯j,1̂(ℓ)=2πδ0,ℓ. By Lemma 3.3, ϕj,ϕ̃j∈L2(ℝ), then it is immediate from the Lebesgue-dominated convergence theorem that uj,m and ũj,m converge to ϕ̂j and ϕ̃̂j, respectively, in L2. This in turn implies that uj,mũ¯j,m→ϕ̂jϕ̃̂¯j in L1, as m→∞. Applying Plancherel’s theorem, we arrive at the conclusion that 〈ϕj,ϕ̃j(·-ℓ)〉=δ0,ℓ for any ℓ∈ℤ.This result proves that〈ϕj,0,ϕ̃j,n〉=δ0,n, n∈ℤ, for some j (in fact, for a sufficiently large j) which guarantees the condition (3.23). But the following proposition indeed proves that this duality condition holds for any j≥j0, under some suitable condition on the symbols a and ã in (3.10). In the following analysis, it is useful to use the notation
(3.26)Bk∶=max⁡ξ|∏l=1kb(e-iξ/2l)|1/k,B̃k̃∶=max⁡ξ|∏l=1k̃b̃(e-iξ/2l)|1/k̃,
with b and b̃ in (3.10).Proposition 3.5.
Leta[j] and ã[j],  j≥j0,  be given as in (3.9). Assume that for some integers k,k̃>0,
(3.27)Bk,B̃k̃<2N-1/2,
then, for any j≥j0, one has 〈ϕj,0,ϕ̃j,n〉=δ0,n for all n∈ℤ.Proof.
Due to Lemma3.2, we find that as j→∞, b[j] and b̃[j] converge uniformly on |z|=1 to b and b̃, respectively. Thus, we can deduce that there exists a large J∈ℕ such that
(3.28)Bk,j,B̃k̃,j<2N-1/2,∀j≥J.
It follows from Lemma 3.4 that for any j≥J, 〈ϕj,0,ϕ̃j,n〉=δ0,n for all n∈ℤ. For the case j<J, this property can be derived by using an inductive argument based on the non-stationary refinement equation. Specifically, applying (1.2), we get
(3.29)〈ϕj,0,ϕ̃j,l〉=12∑n∈Zan[j]∑k∈Zãk-2l[j]〈ϕj+1,n,ϕ̃j+1,k〉=12∑n∈Zan[j]∑k∈Zãk-2l[j]δn,k=12∑n∈Zan[j]ãn-2l[j]=δ0,l,
for any ℓ∈ℤ. This completes the proof.


## 3.2. Smoothness of Refinable Functions

For a givenκ=n+s with n∈ℕ and s∈[0,1], the Hölder space Hκ (e.g., see [3]) is defined to be the space of n-times continuously differentiable functions f whose nth derivative f(n) satisfies the Lipschitz condition
(3.30)sup⁡x,h∈R|f(n)(x+h)-f(n)(x)||h|s≤C.
In particular, the regularity of f also can be analyzed by estimating the decay of f̂ around ∞. If |f̂(ξ)|≤c(1+|ξ|)-1-κ-ϵ, ξ∈ℝ, for an arbitrary small ϵ>0, then f belongs to the space Hκ. In the following theorem, under some more stringent condition on Bk (than (3.27)), we derive the smoothness of non-stationary refinable functions ϕ̃j, j≥j0, in (3.12). The smoothness of ϕj can be shown in a similar way.Theorem 3.6.
Assume that the Laurent polynomialã in (3.10) can be rewritten as
(3.31)ã(z)=2-K̃(1+z)K̃+1b̃(z),K̃<N,
such that B̃k̃<1 for some k̃∈ℕ. Let ϕ̃j, j≥j0, be given as in (3.12) with the symbol ã[j] in (3.9), then ϕ̃j∈HK̃+ν̃ for some ν̃∈(0,1).Proof.
The Laurent polynomialã[j] in (3.9) can be written as
(3.32)ã[j](z)=2cjb̃[j](z)∏n=1K̃+112(1+eγn2-j-1z),K̃<N.
From Lemma 3.2, we can deduce that b̃[j](z)→b̃(z) uniformly on |z|=1 as j→∞. This implies that B̃k̃,j in (3.14) converges to B̃k̃ (<1) as j→∞. Thus, there exists a sufficiently large J∈ℕ such that B̃k̃,j<1 for any j≥J, which means that log2B̃k̃,j<0. Hence, recalling from (3.15) that
(3.33)|ϕ̃̂j|≤c(1+|⋅|)-K̃-1+log⁡2B̃k̃,j,
we see that for any j≥J, ϕ̃j∈HK̃+ν̃ with ν̃∈(0,1). Next, consider the case of j<J. By applying an inductive argument based on the refinement equation
(3.34)ϕ̃j=∑n∈Zan[j]ϕ̃j+1(⋅-n),
we show that ϕ̃j∈HK̃+ν̃ for any j<J.Remark 3.7.
It is known (e.g., see [1]) that the Fourier transform of ϕ̃ with the symbol ã in (3.31) has the decay rate
(3.35)|ϕ̃̂|≤c(1+|⋅|)-K̃-1+log⁡2B̃k̃.
Since B̃k̃,j converges to B̃k̃ as j tends to ∞, we can deduce that the functions ϕ̃j, j≥j0, have the same regularity as ϕ̃. For the details about the (optimal) regularity of ϕ̃, the reader is referred to [1]. One may investigate the regularity of ϕ̃j by using the concept of asymptotical equivalent subdivision schemes (see [15, 19]). However, the methods in [15, 19] are mainly concerned with the integer smoothness of the refinable functions.


## 4. Stability of Nonstationary Biorthogonal Wavelets

For eachj≥j0, if the condition (2.4) holds, the integer translates ϕj(·-k), k∈ℤ, are linearly independent, and in particular, the set {ϕj(·-k):k∈ℤ} forms a Riesz basis for Vj, that is, there exist constants Aj,Bj>0 such that
(4.1)Aj≤[ϕ̂j,ϕ̂j]≤Bj.
In [12], it was proved that there exists some constants 0<A,B<∞ independent of j≥j0 (but dependent on j0) such that
(4.2)A≤[ϕ̂j,ϕ̂j]≤B,∀j≥j0.
However, in the context of non-stationary wavelets, the stability of ϕj does not imply the global stability of the wavelet bases {ψj,k:(j,k)∈ℐ}, where
(4.3)I∶={(j,k)∈Z2:j≥j0,k∈Z}.
The results in this section fill the gap. Specifically, we prove that the set {ψj,k:(j,k)∈ℐ} is a Riesz basis for +̇j≥j0Wj. Further, we show that the set
(4.4)B∶={ϕj0,k:k∈Z}∪{ψj,k:(j,k)∈I}
becomes a Riesz basis for the space L2(ℝ).In the sequel, we will use the notationϕa, ϕ̃ã when referring to the pair of dual refinable functions based on the Nth-order polynomial B-spline, and ψa, ψ̃ã for their corresponding biorthogonal wavelets (i.e., γn=0 for all n=1,…,N).Lemma 4.1.
Letψj and ψ̃j be a pair of biorthogonal wavelet functions associated with ϕj and ϕ̃j,  j≥j0,  then ψj (resp., ψ̃j) converges to ψa (resp., ψ̃ã) in L2(ℝ),  as j→∞,  with the convergence rate O(2-j).Proof.
With the refinement masksa[j] and a of ϕj and ϕa, respectively, it is apparent from (1.4) that
(4.5)‖ψj-ψa‖L2(R)≤‖a[j]-a‖1‖ϕj‖L2(R)+‖a‖1‖ϕj-ϕa‖L2(R).
It has been proved in the proof of Theorem  3 in [12] that ∥ϕj-ϕa∥L2(ℝ)=O(2-j) as j→∞. Thus, by Lemma 3.2, it is obvious that ∥ψj-ψa∥L2(ℝ)=O(2-j). Similarly, the convergence of ψ̃j to ψ̃ã (as j→∞) can be proved.Proposition 4.2.
Assume thatψj and ψ̃j are a pair of biorthogonal wavelet functions associated with ϕj and ϕ̃j,  j≥j0. Let ψj,k∶=2j/2ψj(2j·-k) and ψ̃j,k∶=2j/2ψ̃j(2j·-k),  then, for any j≥j0,  the sets {ψj,k:k∈ℤ} and {ψ̃j,k:k∈ℤ} form Riesz bases for Wj and W̃j, respectively.  Furthermore, there exist constants C,D,C̃,D̃>0 independent of j≥j0 (but dependent on j0) such that
(4.6)C≤[ψ̂j,ψ̂j]≤D,C̃≤[ψ̃̂j,ψ̃̂j]≤D̃,∀j≥j0.Proof.
Sinceψj and ψ̃j are compactly supported functions, there exist constants Dj,D̃j>0 such that
(4.7)[ψ̂j,ψ̂j]≤Dj,[ψ̃̂j,ψ̃̂j]≤D̃j.
Notice that the duality condition 〈ψj(·-k),ψ̃j〉=δ0,k is equivalent to [ψ̂j,ψ̃̂j]=1, then, by using the Cauchy-Schwartz inequality, we get
(4.8)[ψ̂j,ψ̂j][ψ̃̂j,ψ̃̂j]≥1.
Putting Cj=D̃j-1, we obtain from (4.7) that Cj≤[ψ̂j,ψ̂j]≤Dj. Next, we prove that the Riesz bounds Cj and Dj, j≥j0, are in some interval [C,D] with 0<C,D<∞. For this, we see that the 2π periodic function [ψ̂j,ψ̂j] is of the form
(4.9)[ψ̂j,ψ̂j](ξ)=:∑n∈Zcn[j]einξ,wherecn[j]=〈ψj,ψj(⋅-n)〉.
Here, by Lemma 4.1, ψj converges to ψa in L2(ℝ) as j→∞. Thus, it follows that for any n∈ℤ, cn[j] converge to cn=〈ψa,ψa(·-n)〉 as j→∞. Moreover, since ψj is compactly supported, only a finite number of cn[j] is nonzero. It yields that
(4.10)[ψ̂j,ψ̂j](ξ)⟶∑n∈Zcneinξ=[ψa,ψa](ξ),asj⟶∞.
Therefore, we can deduce that the Riesz bounds Cj and Dj converge to Cψa and Dψa (resp.), as j→∞, which are the Riesz (upper and lower) bounds of ψa. It concludes that
(4.11)C≤[ψ̂j,ψ̂j]≤D,∀j≥j0,
with some constants C,D>0 independent of j≥j0. In a similar fashion, it can be proved that C̃≤[ψ̃̂j,ψ̃̂j]≤D̃ with some constants C̃,D̃>0 independent of j≥j0.The above proposition discusses the stability of wavelet functions at each fixed level. The real problem is the global stability of the set{ψj,k:(j,k)∈ℐ} (resp., {ψ̃j,k:(j,k)∈ℐ}) with ℐ in (4.3). The following results treat this problem.Lemma 4.3.
Letψj and ψ̃j be a pair of biorthogonal wavelet functions associated with ϕj and ϕ̃j, j≥j0, then there exist two constants D,D̃>0 depending on j0 such that for all f∈L2(ℝ),
(4.12)∑(j,k)∈I|〈f,ψj,k〉|2≤D‖f‖L2(R)2,∑(j,k)∈I|〈f,ψ̃j,k〉|2≤D̃‖f‖L2(R)2,
where ℐ is given in (4.3).Proof.
Letψj,ka∶=2j/2ψa(2j-k), and invoke that ψa has the same support as ψj for any j≥j0. Put Sk∶=suppψa(·-k) with k∈ℤ, then, we observe that
(4.13)|〈f,ψj,k-ψj,ka〉|=|∫Rf(x)(ψj,k(x)-ψj,ka(x))dx|≤2-j/2∫Sk|f(2-jx)[ψj(x-k)-ψja(x-k)]|dx≤2-j/2‖ψj-ψa‖L2(R)(∫Sk|f(2-jx)|2dx)1/2≤c2-j(∫2-jSk|f(x)|2dx)1/2,
where the last bound is obtained by Lemma 4.1. Then, we get
(4.14)∑(j,k)∈I|〈f,ψj,k-ψj,ka〉|2≤c∑(j,k)∈I2-2j∫2-jSk|f(x)|2dx=c∑j≥j02-2j∑k∈Z∫2-jSk|f(x)|2dx≤c∑j≥j02-2j‖f‖L2(R)2≤c‖f‖L2(R)2.
Putting all together, it is immediate that
(4.15)(∑(j,k)∈I|〈f,ψj,k〉|2)1/2≤(∑(j,k)∈I|〈f,ψj,ka〉|2)1/2+(∑(j,k)∈I|〈f,ψj,k-ψj,ka〉|2)1/2≤Dψa‖f‖L2(R)+c‖f‖L2(R)=D‖f‖L2(R),
for some constant D>0, where Dψa is the Riesz upper bound of ψa. Similarly, we can prove the second relation in (4.12).We now arrive at the central results of this section.Theorem 4.4.
Letψj and ψ̃j be a pair of biorthogonal wavelet functions associated with ϕj and ϕ̃j,  j≥j0,  then the sets {ψj,k:(j,k)∈ℐ} and {ψ̃j,k:(j,k)∈ℐ} form Riesz bases for the spaces +̇j≥j0Wj and +̇j≥j0W̃j, respectively, where ℐ is given in (4.3).Proof.
Letf be a function in the space +̇j≥j0Wj. By the Cauchy-Schwartz inequality and Lemma 4.3, we obtain
(4.16)‖f‖L2(R)2=lim⁡J→∞∑j=j0J∑k∈Z〈f,ψ̃j,k〉〈f,ψj,k〉¯≤(∑(j,k)∈I|〈f,ψ̃j,k〉|2)1/2(∑(j,k)∈I|〈f,ψj,k〉|2)1/2≤D̃1/2‖f‖L2(R)(∑(j,k)∈I|〈f,ψj,k〉|2)1/2,
for some constant D̃>0. Putting C=D̃-1, we have
(4.17)C‖f‖L2(R)2≤∑(j,k)∈I|〈f,ψj,k〉|2.
Since the upper bound is proved in Lemma 4.3, it concludes that
(4.18)C‖f‖L2(R)2≤∑(j,k)∈I|〈f,ψj,k〉|2≤D‖f‖L2(R)2,
for some constant D>0. Further, it is obvious that ψj,k, (j,k)∈ℐ, are linearly independent. It indeed proves that the set {ψj,k:(j,k)∈ℐ} constitutes a Riesz basis for the space +̇j≥j0Wj (see [4, Theorem  3.20]). Similarly, we can show that {ψ̃j,k:(j,k)∈ℐ} forms a Riesz basis for the space +̇j≥j0W̃j.Sinceϕj→ϕa and ϕ̃j→ϕ̃a in L2(ℝ) as j→∞, it is immediate that for any f∈L2(ℝ), the biorthogonal projection
(4.19)Pjf=∑k∈Z〈f,ϕ̃j,k〉ϕj,k∈Vj
converges to f in the L2-norm as j→∞, then based on this observation, we get the following result.Theorem 4.5.
Letψj and ψ̃j be a pair of biorthogonal wavelet functions associated with ϕj and ϕ̃j,  j≥j0,  then the set ℬ={ϕj0,k:k∈ℤ}∪{ψj,k:(j,k)∈ℐ} forms a Riesz basis for L2(ℝ). This result also applies to the set of dual functions ℬ̃∶={ϕ̃j0,k:k∈ℤ}∪{ψ̃j,k:(j,k)∈ℐ}.Proof.
For a given functionf∈L2(ℝ), for notational simplicity, we define the following sequences:
(4.20)c∶=(〈f,ϕj0,k〉)k∈Z,dj∶=(〈f,ψj,k〉)k∈Z,j≥j0.
Similarly, let c̃ and d̃j, j≥j0, be the sequences consisting of the terms 〈f,ϕ̃j0,k〉 and 〈f,ψ̃j,k〉, k∈ℤ, respectively. It has been proven in [12] that there exist constants Ã,B̃>0 such that
(4.21)Ã‖f‖L2(R)2≤‖c̃‖22≤B̃‖f‖L2(R)2.
Also, by Theorem 4.4,
(4.22)∑j=j0∞‖d̃j‖22≤D̃‖f‖L2(R)2,
for some constant D̃>0, then, by the Cauchy-Schwartz inequality and Theorem 4.4, we get
(4.23)‖f‖L2(R)2=lim⁡J→∞(∑k∈Z〈f,ϕ̃j0,k〉〈f,ϕj0,k〉+∑j=j0J∑k∈Z〈f,ψ̃j,k〉〈f,ψj,k〉)≤(‖c‖22+∑j=j0∞‖dj‖22)1/2(‖c̃‖22+∑j=j0∞‖d̃j‖22)1/2≤max⁡(B̃,D̃)1/2‖f‖L2(R)(‖c‖22+∑j=j0∞‖dj‖22)1/2.
Putting E∶=max(B̃,D̃)-1, we have
(4.24)E‖f‖L2(R)2≤‖c‖22+∑j=j0∞‖dj‖22.
Since the upper bound is clear from Lemma 4.3, it concludes that
(4.25)E‖f‖L2(R)2≤‖c‖22+∑j=j0∞‖dj‖22≤F‖f‖L2(R)2,
for some constant F>0. It proves that the set ℬ constitutes a Riesz basis for L2(ℝ) (see [25, Theorem  6.1.1]). Similarly, we can show that ℬ̃ forms a Riesz basis for L2(ℝ).



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*Source: 593436-2011-11-17.xml*