Analysis of Compactly Supported Nonstationary Biorthogonal Wavelet Systems Based on Exponential B-Splines

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.

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 𝜙_𝑗, 𝑗≥𝑗₀, in 𝐿₂(ℝ), we say that a sequence of subspaces 𝑉_𝑗=span𝜙_𝑗,𝑘∶=2^𝑗/2𝜙_𝑗2^𝑗⋅−𝑘∶𝑘∈ℤ,𝑗≥𝑗₀ (1.1) forms a non-stationary multiresolution analysis (MRA) of 𝐿₂(ℝ) if the following conditions are satisfied: (1)𝑉_𝑗⊂𝑉_𝑗+1 for all 𝑗≥𝑗₀,(2)⋃_𝑗≥𝑗0𝑉_𝑗 is dense in 𝐿₂(ℝ); (3)the set {𝜙_𝑗,𝑘∶𝑘∈ℤ} is a Riesz (or stable) basis for 𝑉_𝑗 for each 𝑗≥𝑗₀.

The nested embedding of the spaces 𝑉_𝑗 implies the existence of a sequence 𝐚^[𝑗]∶=(𝑎_𝑛^[𝑗])_𝑛∈ℤ∈ℓ₂(ℤ) that satisfies the non-stationary refinement equation𝜙_𝑗(𝑥)=_𝑛∈ℤ𝑎_𝑛^[𝑗]𝜙_𝑗+1(2𝑥−𝑛),𝑗≥𝑗₀, (1.2) where the sequence 𝐚^[𝑗] is usually called the refinement mask for 𝜙_𝑗. One should notice that the function 𝜙_𝑗(2^𝑗⋅)∈𝑉_𝑗 is no longer a dilated version of 𝜙_𝑗0 in 𝑉_𝑗0. A refinable function 𝜙_𝑗∈𝐿₂(ℝ) with the mask 𝐚^[𝑗]∶=(𝑎_𝑛^[𝑗])_𝑛∈ℤ is called the dual refinable function of 𝜙_𝑗∈𝐿₂(ℝ) (or just the dual of 𝜙_𝑗 for simplicity) if it satisfies 𝜙_𝑗,𝜙_𝑗(⋅−ℓ)=𝛿_0,ℓ,ℓ∈ℤ. (1.3) Let (𝑉_𝑗)_𝑗≥𝑗0 and (𝑉_𝑗)_𝑗≥𝑗0 be a pair of MRAs generated by a pair of dual refinable functions 𝜙_𝑗 and 𝜙_𝑗, 𝑗≥𝑗₀, respectively. The concept of biorthogonal wavelets is to find complement spaces 𝑊_𝑗 and 𝑊_𝑗 of 𝑉_𝑗 and 𝑉_𝑗, respectively, satisfying 𝑉_𝑗+1=𝑉_𝑗+̇𝑊_𝑗, 𝑊_𝑗⟂𝑉_𝑗 and 𝑉_𝑗+1=𝑉_𝑗+̇𝑊_𝑗, 𝑊_𝑗⟂𝑉_𝑗. The corresponding biorthogonal wavelets are given by 𝜓_𝑗=_𝑛∈ℤ(−1)^𝑛𝑎^[𝑗]_1−𝑛𝜙_𝑗+1(),𝜓2⋅−𝑛_𝑗=_𝑛∈ℤ(−1)^𝑛𝑎^[𝑗]_1−𝑛𝜙_𝑗+1(2⋅−𝑛). (1.4)

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 𝜙_𝑗 or 𝜙_𝑗) and the zeros of the corresponding Laurent polynomials. They also proved that for each 𝑗≥𝑗₀, the proposed non-stationary refinable function 𝜙_𝑗 generates a Riesz basis for 𝑉_𝑗 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 {𝜓_𝑗,𝑘∶𝑘∈ℤ,𝑗≥𝑗₀} is not implied by the stability of a refinable function. Therefore, we prove that the set {𝜓_𝑗,𝑘∶𝑘∈ℤ,𝑗≥𝑗₀} forms a Riesz basis for the space +̇_𝑗≥𝑗0𝑊_𝑗. Furthermore, we show that the set {𝜙_𝑗0,𝑘∶𝑘∈ℤ}∪{𝜓_𝑗,𝑘∶𝑘∈ℤ,𝑗≥𝑗₀} becomes a Riesz basis for the space 𝐿₂(ℝ).

This paper is organized as follows. In Section 2, 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 numbers 𝐺={𝛾_𝑗∈ℂ∶𝑗=1,…,𝑁}, the corresponding 𝑁th-order exponential B-splines can be defined as successive convolutions of the first-order B-spline 𝜙_𝑗∶=𝜙_𝑗^[𝐺]∶=𝜏_𝑗𝑒_𝛾12^−𝑗𝐵₁∗⋯∗𝑒_𝛾𝑁2^−𝑗𝐵₁, (2.1) with a normalization factor 𝜏_𝑗 defined so that ‖𝜙_𝑗‖_𝐿1(ℝ)=1 (see [10]), where 𝐵₁ indicates the first-order 𝐵-spline, that is, 𝐵₁∶=𝜒_[0,1], and 𝑒_𝛾∶𝑥↦𝑒^𝛾𝑥, 𝛾∈ℂ, is the exponential function. For simplicity, we will omit 𝐺 in 𝜙_𝑗^[𝐺]. Obviously, the function 𝜙_𝑗 is a compactly supported piecewise exponential polynomial. The global regularity of 𝜙_𝑗 is 𝐶^𝑁−2 (see [10, 15]). A convenient way to represent an (𝑁th-order) exponential B-spline is with the Laurent polynomial 𝑎^[𝑗](𝑧)∶=_𝑛∈ℤ𝑎_𝑛^[𝑗]𝑧^𝑛∶=2𝑐_𝑗^𝑁_𝑛=1121+𝑒^{𝛾𝑛2−𝑗−1}𝑧,𝑗≥𝑗₀, (2.2) where 𝑐_𝑗 is the normalization factor defined by 𝑐_𝑗∶=𝑐_𝑗,𝐺∶=^𝑁_𝑛=121+𝑒^{𝛾𝑛2−𝑗−1}. (2.3) We call 𝑎^[𝑗] the symbol of 𝜙_𝑗. Throughout this paper, 𝛾_𝑛 is assumed to be a real or a pure imaginary number, that is, 𝛾_𝑛∈ℝ or 𝛾_𝑛∈𝑖ℝ. Since we want the mask (𝑎_𝑛^[𝑗])_𝑛∈ℤ to be symmetric with respect to its center, it is reasonable to assume that if 𝛾_𝑛∈𝐺, then −𝛾_𝑛∈𝐺. The relationship between the reproduction of exponential polynomials by 𝜙_𝑗 and the zeros of the corresponding Laurent polynomial 𝑎^[𝑗] is discussed in [12].

It is well known that the integer translates 𝜙₀(⋅−𝑘), 𝑘∈ℤ, are linearly independent if and only if 𝛾_ℓ−𝛾_𝑛∉2𝜋𝑖ℤ for 𝛾_ℓ≠𝛾_𝑛 [10, 16]. From (2.1) and (2.2), we can easily deduce that for each 𝑗≥𝑗₀, the integer translates 𝜙_𝑗(⋅−𝑘), 𝑘∈ℤ, are linearly independent if and only if 2^−𝑗𝛾_ℓ−𝛾_𝑛∉2𝜋𝑖ℤ,𝛾_ℓ≠𝛾_𝑛. (2.4) A concept related to (but weaker than) the linear independence is the notion of the stability of 𝜙_𝑗. It is known (see, e.g., [23]) that the set {𝜙_𝑗(⋅−𝑘)∶𝑘∈ℤ} forms a Riesz basis if and only if there exist constants 𝐴_𝑗,𝐵_𝑗>0 such that 𝐴_𝑗≤𝜙_𝑗,𝜙_𝑗≤𝐵_𝑗, (2.5) where the bracket product [𝑓,𝑔] for 𝑓,𝑔∈𝐿₂(ℝ) is defined by 𝑓,𝑔(𝜉)∶=_𝑛∈ℤ𝑓(𝜉+2𝜋𝑛)𝑔(𝜉+2𝜋𝑛). (2.6) We say that the function 𝜙_𝑗∈𝐿₂(ℝ) is stable if (2.5) is satisfied. If the integer translates of 𝜙_𝑗 are linearly independent, the stability of 𝜙_𝑗 is immediate [24, Theorem  1.2], that is, {𝜙_𝑗(2^𝑗⋅−𝑘)∶𝑘∈ℤ} is a Riesz basis for 𝑉_𝑗. Finally, the basic requirement on the set 𝐺={𝛾₁,…,𝛾_𝑁} can be summarized as follows: (i) each 𝛾_𝑛 is a real or a pure imaginary number; (ii) both 𝛾_𝑛 and −𝛾_𝑛 belong to 𝐺, and (iii) for any 𝑗≥𝑗₀, 2^−𝑗(𝛾_𝑛−𝛾_ℓ)∉2𝜋𝑖ℤ with 𝛾_𝑛≠𝛾_ℓ.

3. Dual Refinable Functions

3.1. Construction of Dual Refinable Functions

Given refinable functions 𝜙_𝑗 with 𝑗≥𝑗₀ the first step in the construction of a biorthogonal wavelet system is to find their dual refinable functions 𝜙_𝑗∈𝐿₂(ℝ), 𝑗≥𝑗₀ (whose symbol is denoted by 𝑎^[𝑗]). A necessary condition for 𝜙_𝑗 and 𝜙_𝑗 to satisfy (1.3) is 𝑎^[𝑗](𝑧)𝑎^[𝑗](𝑧)+𝑎^[𝑗](−𝑧)𝑎^[𝑗](−𝑧)=4, (3.1) where 𝑧 is the conjugate of the complex number 𝑧. Thus, the construction of 𝜙_𝑗 starts with the construction of a dual symbol 𝑎^[𝑗](𝑧)∶=∑_𝑛∈ℤ𝑎_𝑛^[𝑗]𝑧^𝑛 such that (3.1) is satisfied. The algorithm to find 𝑎^[𝑗] is analogous to the classical method [1], except for the main difference that 𝑎^[𝑗], 𝑗≥𝑗₀, are scale dependent. However, in order to ensure the existence of 𝑎^[𝑗] satisfying (3.1), we need to prove that 𝑎^[𝑗] has no roots of opposite signs. For this purpose, denoting 𝑁=2𝐿+𝜏,𝜏∈{0,1}, (3.2) and setting 𝑧=𝑒^−𝑖𝜉 and 𝑦=sin²𝜉/2 (i.e., 𝑦=−(1−𝑧)²/4𝑧), we represent 𝑎^[𝑗] as follows: 𝑎^[𝑗]𝑒^−𝑖𝜉=2𝑒^{−𝑖𝐿𝜉}1+𝑒^−𝑖𝜉2^𝜏𝐿_𝑛=11−𝜈_𝑗,𝑛𝑦, (3.3) where 𝜈_𝑗,𝑛=4𝑒^{𝛾𝑛2−𝑗−1}+𝑒^{−𝛾𝑛2−𝑗−1}+2. (3.4) Indeed, we are looking for a Laurent polynomial 𝑎^[𝑗] of the form 𝑎^[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, 𝑃^[𝑗]𝑦∶=1−𝑦^𝜏𝐿_𝑛=11−𝜈_𝑗,𝑛𝑦², (3.5) then the problem of finding 𝑎^[𝑗](𝑧) is reduced to constructing 𝑄^[𝑗](𝑦) which satisfies the equation 𝑃^[𝑗]𝑦𝑄^[𝑗]𝑦+𝑃^[𝑗]1−𝑦𝑄^[𝑗]1−𝑦=1,∀𝑦∈[0,1]. (3.6) 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 polynomials 𝑃^[𝑗], 𝑗≥𝑗₀, be given as in (3.5). Assume that for any 𝑗≥𝑗₀, 2^−𝑗(𝛾_𝑛−𝛾_ℓ)∉2𝜋𝑖ℤ for any 𝛾_𝑛,𝛾_ℓ∈𝐺 with 𝛾_𝑛≠𝛾_ℓ, then 𝑃^[𝑗] and 𝑃^[𝑗](1−⋅) have no common roots on [0,1].

Proof. Assume that 𝑃^[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 𝑎^[𝑗]𝑧₀=𝑎^[𝑗]−𝑧₀=0. (3.7) From (2.2), we can deduce that for some 𝛾_ℓ≠𝛾_𝑛, 1+𝑒^{−2−𝑗−1𝛾ℓ}𝑧₀=0 and 1+𝑒^{−2−𝑗−1𝛾𝑛}(−𝑧₀)=0. It follows that 2^−𝑗(𝛾_𝑛−𝛾_ℓ)∈2𝜋𝑖ℤ, which contradicts the initial assumption.

By virtue of Proposition 3.1, 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, 𝑄^[𝑗]𝑦=𝑏^[𝑗](𝑧)𝑏̃^[𝑗](𝑧)(),𝑦=−1−𝑧²4𝑧, (3.8) then we can define the adjusted Laurent polynomials 𝑎^[𝑗] and 𝑎^[𝑗] by 𝑎^[𝑗](𝑧)∶=2𝑐_𝑗𝑏^[𝑗](𝑧)^𝑁_𝑛=1121+𝑒^{𝛾𝑛2−𝑗−1}𝑧,𝑎^[𝑗](𝑧)∶=2𝑐_𝑗𝑏̃^[𝑗](𝑧)^𝑁_𝑛=1121+𝑒^{𝛾𝑛2−𝑗−1}𝑧, (3.9) so that the lengths of 𝑎^[𝑗] and 𝑎^[𝑗] are very close. This allows us to construct generalized non-stationary refinable functions 𝜙_𝑗 and 𝜙_𝑗. If 𝑏^[𝑗]=1, the resulting function 𝜙_𝑗 becomes an exponential B-spline of order 𝑁. On the other hand, the classical counterparts of 𝑎^[𝑗] and 𝑎^[𝑗] (which are obtained by setting 𝛾_𝑛=0 for all 𝛾_𝑛∈𝐺) can be written as 𝑎(𝑧)=2^−𝑁+1(1+𝑧)^𝑁𝑏(𝑧),𝑎(𝑧)=2^−𝑁+1(1+𝑧)^𝑁𝑏̃(𝑧). (3.10) For notational simplicity, we will write 𝑏^[𝑗](𝑧)=_𝑛∈ℤ𝑏_𝑛^[𝑗]𝑧^𝑛,𝑏(𝑧)=_𝑛∈ℤ𝑏_𝑛𝑧^𝑛. (3.11)

Lemma 3.2. Let 𝐛^[𝑗]∶=(𝑏_𝑛^[𝑗]∶𝑛∈ℤ) and 𝐛∶=(𝑏_𝑛∶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 polynomials 𝑎^[𝑗] with 𝑗≥𝑗₀, there corresponds a potential candidate for the refinable function 𝜙_𝑗 which is defined in terms of the Fourier transform as 𝜙_𝑗(𝜉)=^∞_𝑛=012𝐴^[𝑗+𝑛]𝜉2^𝑛+1, (3.12) where 𝐴^[𝑗](𝜉)=𝑎^[𝑗]𝑒^−𝑖𝜉, (3.13) provided that 𝜙_𝑗(𝜉/2^𝑛)→1 as 𝑛→∞. In fact, 𝜙_𝑗 in (3.12) is the only candidate for the refinable function associated with 𝑎^[𝑗] such that 𝜙_𝑗 is a dual of 𝜙_𝑗. Although the infinite product in (3.12) converges pointwise and 𝜙_𝑗 is in 𝐿₂(ℝ) whenever 𝜙 is in 𝐿₂(ℝ) [12], it still needs to be ensured that the function 𝜙_𝑗 in (3.12) is indeed a dual of 𝜙_𝑗. The following results address this issue. For simplicity, using 𝑏^[𝑗] and 𝑏̃^[𝑗] in (3.9), we introduce the notation 𝐵_𝑘,𝑗∶=sup_𝑛≥𝑗max_𝜉^𝑘_ℓ=1𝑏^{[𝑛+ℓ−1]}(𝑒^{−𝑖𝜉/2ℓ})^1/𝑘,𝐵_𝑘,𝑗∶=sup_𝑛≥𝑗max_𝜉ℓ𝑘₌₁𝑏̃^{[𝑛+ℓ−1]}(𝑒^{−𝑖𝜉/2ℓ})^1/𝑘. (3.14)

Lemma 3.3. Let 𝜙_𝑗, 𝑗≥𝑗₀, be given as in (3.12) with the symbol 𝑎^[𝑗] in (3.9). Suppose that for some integer 𝑘>0, 𝐵_𝑘,𝑗<2^𝑁−1/2, then 𝜙_𝑗≤𝑐(1+|⋅|)^−𝑁+log2𝐵_𝑘,𝑗, (3.15) which implies that 𝜙_𝑗∈𝐿₂(ℝ). Moreover, the function 𝜙_𝑗 can be defined as in (3.12), then this lemma also holds for 𝜙_𝑗.

Proof. Let 𝑚∈ℕ and define 𝑢_𝑗,𝑚 by 𝑢_𝑗,𝑚(𝜉)∶=^𝑚_ℓ=112𝐴^{[𝑗+ℓ−1]}𝜉2^−ℓ𝜒_{[−𝜋,𝜋]}𝜉2^−𝑚. (3.16) For the given 𝛾_𝑛∈ℂ with 𝑛=1,…,𝑁, set 𝜉_{𝛾𝑛,𝑗}∶=𝜉+𝑖𝛾_𝑛2^−𝑗−1. Then, we get the identity ^𝑚_ℓ=11+𝑒^{𝛾2−𝑗−ℓ−1}𝑒^{−𝑖𝜉2−ℓ}2=^𝑚_ℓ=1sin𝜉_𝑛,𝑗2^−ℓ2sin𝜉_𝑛,𝑗2^−ℓ−1=sin𝜉_𝑛,𝑗2⁻¹2^𝑚sin𝜉_𝑛,𝑗2^−𝑚−1. (3.17) Here, we can see that there exist 𝜂>0 and 𝑀∈ℕ such that if 𝑚≥𝑀 and |𝜉2^−𝑚−1|≤𝜂, |sin(𝜉_𝑛,𝑗2^−𝑚−1)|≥2⁻¹|𝜉_𝑛,𝑗2^−𝑚−1|. Also, it is obvious that |sin(𝜉_𝑛,𝑗2⁻¹)|≤𝑐₁|𝜉_𝑛,𝑗2⁻¹| for some constant 𝑐₁>0 independent of 𝜉 but dependent on 𝛾_𝑛. Therefore, we have sin𝜉_𝑛,𝑗2⁻¹2^𝑚sin𝜉_𝑛,𝑗2^−𝑚−1≤𝑐₁𝜉_𝑛,𝑗2^−𝑚−1sin𝜉_𝑛,𝑗2^−𝑚−1≤𝑐₂, (3.18) with a constant 𝑐₂>0 depending on 𝛾_𝑛. 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, 2^𝑚sin𝜉_𝑛,𝑗2^−𝑚−1≥2^𝑚2⁻¹sin𝜉2^−𝑚−1−sin𝛾_𝑛2^{−𝑗−𝑚−2}≥𝑐₃𝜉−𝜖≥𝑐₄𝜉, (3.19) then we obtain that sin𝜉_𝑛,𝑗2⁻¹2^𝑚sin𝜉_𝑛,𝑗2^−𝑚−1≤𝑐1+𝜉, (3.20) for a constant 𝑐>0 depending on 𝛾_𝑛. Consequently, invoking the fact that 𝐴^[𝑗](𝜉)=𝑎^[𝑗](𝑒^−𝑖𝜉) with 𝑎^[𝑗] in (3.9), we obtain 𝑢_𝑗,𝑚(𝜉≤𝑐)1+𝜉^𝑁𝑚_ℓ=1𝑏̃^{[𝑗+ℓ−1]}𝑒^{−𝑖𝜉2−ℓ}𝜒_{[−𝜋,𝜋]}2^−𝑚𝜉, (3.21) for any 𝑗≥𝐽. Here, using the same argument in [1, Proposition  4.8] (see point 3), we can get 𝑏̃^𝑚_ℓ=1^{[𝑗+ℓ−1]}𝑒^{−𝑖𝜉2−ℓ}≤𝑐(1+|𝜉|)^log2𝐵_𝑘,𝑗. (3.22) This together with (3.21) implies that |𝑢_𝑗,𝑚(𝜉)|≤𝑐(1+|𝜉|)^−𝑁+log2𝐵_𝑘,𝑗 for any 𝑗≥𝐽, where 𝑐>0 is independent of 𝑚. Since 𝑢_𝑗,𝑚(𝜉)→𝜙_𝑗(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 𝜙_𝑗=𝐴^[𝑗](⋅/2)𝜙_𝑗+1(⋅/2), we obtain the required result. The case of 𝜙_𝑗 can be done similarly.

Lemma 3.4. Let 𝑎^[𝑗] and 𝑎^[𝑗], 𝑗≥𝑗₀, be given as in (3.9). Suppose that for some integers 𝑘,𝑘>0, 𝐵_𝑘,𝑗,𝐵_𝑘,𝑗<2^𝑁−1/2, (3.23) with 𝐵_𝑘,𝑗 and 𝐵_𝑘,𝑗 in (3.14), then, for any 𝑗≥𝑗₀, one has ⟨𝜙_𝑗,ℓ,𝜙_𝑗,𝑛⟩=𝛿_ℓ,𝑛 for all ℓ,𝑛∈ℤ.

Proof. Recalling the definition of 𝑢_𝑗,𝑚 in (3.16), define 𝑢_𝑗,𝑚 by 𝑢_𝑗,𝑚(𝜉)∶=^𝑚_ℓ=112𝐴^{[𝑗+ℓ−1]}𝜉2^−ℓ𝜒_{[−𝜋,𝜋]}𝜉2^−𝑚, (3.24) Then 𝑢_𝑗,𝑚 and 𝑢_𝑗,𝑚 converge pointwise to 𝜙_𝑗 and 𝜙_𝑗, respectively. Moreover, using (3.1), we can derive the relation 𝑢_𝑗,𝑚𝑢_𝑗,𝑚(ℓ)=^2𝑚+1𝜋₀^𝑚_ℓ=114𝐴^{[𝑗+ℓ−1]}𝜉2^−ℓ𝐴^{[𝑗+ℓ−1]}𝜉2^−ℓ𝑒^𝑖ℓ𝜉=𝑑𝜉_{2𝑚𝜋0𝑚−1ℓ=1}14𝐴^{[𝑗+ℓ−1]}𝜉2^−ℓ𝐴^{[𝑗+ℓ−1]}𝜉2^−ℓ𝑒^𝑖ℓ𝜉=𝑢𝑑𝜉_{𝑗,𝑚−1}𝑢_{𝑗,𝑚−1}(ℓ). (3.25) Repeating this process yields the identity 𝑢_𝑗,𝑚𝑢_𝑗,𝑚(ℓ)=𝑢_𝑗,1𝑢_𝑗,1(ℓ)=2𝜋𝛿_0,ℓ. By Lemma 3.3, 𝜙_𝑗,𝜙_𝑗∈𝐿₂(ℝ), then it is immediate from the Lebesgue-dominated convergence theorem that 𝑢_𝑗,𝑚 and 𝑢_𝑗,𝑚 converge to 𝜙_𝑗 and 𝜙_𝑗, respectively, in 𝐿₂. This in turn implies that 𝑢_𝑗,𝑚𝑢_𝑗,𝑚→𝜙_𝑗𝜙_𝑗 in 𝐿₁, as 𝑚→∞. Applying Plancherel’s theorem, we arrive at the conclusion that ⟨𝜙_𝑗,𝜙_𝑗(⋅−ℓ)⟩=𝛿_0,ℓ for any ℓ∈ℤ.

This result proves that ⟨𝜙_𝑗,0,𝜙_𝑗,𝑛⟩=𝛿_0,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 𝑎 in (3.10). In the following analysis, it is useful to use the notation 𝐵_𝑘∶=max_𝜉^𝑘_ℓ=1𝑏𝑒^{−𝑖𝜉/2ℓ1/𝑘},𝐵_𝑘∶=max_𝜉ℓ𝑘₌₁𝑏̃𝑒^{−𝑖𝜉/2ℓ1/}𝑘, (3.26) with 𝑏 and 𝑏̃ in (3.10).

Proposition 3.5. Let 𝑎^[𝑗] and 𝑎^[𝑗], 𝑗≥𝑗₀, be given as in (3.9). Assume that for some integers 𝑘,𝑘>0, 𝐵_𝑘,𝐵_𝑘<2^𝑁−1/2, (3.27) then, for any 𝑗≥𝑗₀, one has ⟨𝜙_𝑗,0,𝜙_𝑗,𝑛⟩=𝛿_0,𝑛 for all 𝑛∈ℤ.

Proof. Due to Lemma 3.2, 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 𝐵_𝑘,𝑗,𝐵_𝑘,𝑗<2^𝑁−1/2,∀𝑗≥𝐽. (3.28) It follows from Lemma 3.4 that for any 𝑗≥𝐽, ⟨𝜙_𝑗,0,𝜙_𝑗,𝑛⟩=𝛿_0,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 𝜙_𝑗,0,𝜙_𝑗,ℓ=12_𝑛∈ℤ𝑎_𝑛^[𝑗]_𝑘∈ℤ𝑎^[𝑗]_𝑘−2ℓ𝜙_𝑗+1,𝑛,𝜙_𝑗+1,𝑘=12_𝑛∈ℤ𝑎_𝑛^[𝑗]_𝑘∈ℤ𝑎^[𝑗]_𝑘−2ℓ𝛿_𝑛,𝑘=12_𝑛∈ℤ𝑎_𝑛^[𝑗]𝑎^[𝑗]_𝑛−2ℓ=𝛿_0,ℓ, (3.29) for any ℓ∈ℤ. This completes the proof.

3.2. Smoothness of Refinable Functions

For a given 𝜅=𝑛+𝑠 with 𝑛∈ℕ and 𝑠∈[0,1], the Hölder space 𝐻^𝜅 (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 sup_{𝑥,ℎ∈ℝ}𝑓^(𝑛)(𝑥+ℎ)−𝑓^(𝑛)(𝑥)|ℎ|^𝑠≤𝐶. (3.30) In particular, the regularity of 𝑓 also can be analyzed by estimating the decay of 𝑓 around ∞. If |𝑓(𝜉)|≤𝑐(1+|𝜉|)^{−1−𝜅−𝜖}, 𝜉∈ℝ, for an arbitrary small 𝜖>0, then 𝑓 belongs to the space 𝐻^𝜅. In the following theorem, under some more stringent condition on 𝐵_𝑘 (than (3.27)), we derive the smoothness of non-stationary refinable functions 𝜙_𝑗, 𝑗≥𝑗₀, in (3.12). The smoothness of 𝜙_𝑗 can be shown in a similar way.

Theorem 3.6. Assume that the Laurent polynomial 𝑎 in (3.10) can be rewritten as 𝑎(𝑧)=2⁻𝐾(1+𝑧)𝐾₊₁𝑏̃(𝑧),𝐾<𝑁, (3.31) such that 𝐵_𝑘<1 for some 𝑘∈ℕ. Let 𝜙_𝑗, 𝑗≥𝑗₀, be given as in (3.12) with the symbol 𝑎^[𝑗] in (3.9), then 𝜙_𝑗∈𝐻𝐾₊𝜈 for some 𝜈∈(0,1).

Proof. The Laurent polynomial 𝑎^[𝑗] in (3.9) can be written as 𝑎^[𝑗](𝑧)=2𝑐_𝑗𝑏̃^[𝑗](𝑧)𝐾_+1𝑛=1121+𝑒^{𝛾𝑛2−𝑗−1}𝑧,𝐾<𝑁. (3.32) From Lemma 3.2, we can deduce that 𝑏̃^[𝑗](𝑧)→𝑏̃(𝑧) uniformly on |𝑧|=1 as 𝑗→∞. This implies that 𝐵_𝑘,𝑗 in (3.14) converges to 𝐵_𝑘 (<1) as 𝑗→∞. Thus, there exists a sufficiently large 𝐽∈ℕ such that 𝐵_𝑘,𝑗<1 for any 𝑗≥𝐽, which means that log₂𝐵_𝑘,𝑗<0. Hence, recalling from (3.15) that 𝜙_𝑗≤𝑐(1+|⋅|)⁻𝐾^−1+log₂𝐵_𝑘,𝑗, (3.33) we see that for any 𝑗≥𝐽, 𝜙_𝑗∈𝐻𝐾₊𝜈 with 𝜈∈(0,1). Next, consider the case of 𝑗<𝐽. By applying an inductive argument based on the refinement equation 𝜙_𝑗=_𝑛∈ℤ𝑎_𝑛^[𝑗]𝜙_𝑗+1(⋅−𝑛), (3.34) we show that 𝜙_𝑗∈𝐻𝐾₊𝜈 for any 𝑗<𝐽.

Remark 3.7. It is known (e.g., see [1]) that the Fourier transform of 𝜙 with the symbol 𝑎 in (3.31) has the decay rate 𝜙≤𝑐(1+|⋅|)⁻𝐾^−1+log₂𝐵_𝑘. (3.35) Since 𝐵_𝑘,𝑗 converges to 𝐵_𝑘 as 𝑗 tends to ∞, we can deduce that the functions 𝜙_𝑗, 𝑗≥𝑗₀, have the same regularity as 𝜙. For the details about the (optimal) regularity of 𝜙, the reader is referred to [1]. One may investigate the regularity of 𝜙_𝑗 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 each 𝑗≥𝑗₀, if the condition (2.4) holds, the integer translates 𝜙_𝑗(⋅−𝑘), 𝑘∈ℤ, are linearly independent, and in particular, the set {𝜙_𝑗(⋅−𝑘)∶𝑘∈ℤ} forms a Riesz basis for 𝑉_𝑗, that is, there exist constants 𝐴_𝑗,𝐵_𝑗>0 such that 𝐴_𝑗≤𝜙_𝑗,𝜙_𝑗≤𝐵_𝑗. (4.1) In [12], it was proved that there exists some constants 0<𝐴,𝐵<∞ independent of 𝑗≥𝑗₀ (but dependent on 𝑗₀) such that 𝐴≤𝜙_𝑗,𝜙_𝑗≤𝐵,∀𝑗≥𝑗₀. (4.2) However, in the context of non-stationary wavelets, the stability of 𝜙_𝑗 does not imply the global stability of the wavelet bases {𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ}, where ℐ∶=𝑗,𝑘∈ℤ²∶𝑗≥𝑗₀,𝑘∈ℤ. (4.3) The results in this section fill the gap. Specifically, we prove that the set {𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ} is a Riesz basis for +̇_𝑗≥𝑗0𝑊_𝑗. Further, we show that the set ℬ∶=𝜙_𝑗0,𝑘∶𝑘∈ℤ∪𝜓_𝑗,𝑘∶𝑗,𝑘∈ℐ (4.4) becomes a Riesz basis for the space 𝐿₂(ℝ).

In the sequel, we will use the notation 𝜙^𝑎, 𝜙𝑎 when referring to the pair of dual refinable functions based on the 𝑁th-order polynomial B-spline, and 𝜓^𝑎, 𝜓𝑎 for their corresponding biorthogonal wavelets (i.e., 𝛾_𝑛=0 for all 𝑛=1,…,𝑁).

Lemma 4.1. Let 𝜓_𝑗 and 𝜓_𝑗 be a pair of biorthogonal wavelet functions associated with 𝜙_𝑗 and 𝜙_𝑗, 𝑗≥𝑗₀, then 𝜓_𝑗 (resp., 𝜓_𝑗) converges to 𝜓^𝑎 (resp., 𝜓𝑎) in 𝐿₂(ℝ), as 𝑗→∞, with the convergence rate 𝑂(2^−𝑗).

Proof. With the refinement masks 𝐚^[𝑗] and 𝐚 of 𝜙_𝑗 and 𝜙^𝑎, respectively, it is apparent from (1.4) that 𝜓_𝑗−𝜓^𝑎_𝐿2(ℝ)≤𝐚^[𝑗]−𝐚₁𝜙_{𝑗𝐿2(ℝ)}+‖𝐚‖₁𝜙_𝑗−𝜙^𝑎_𝐿2(ℝ). (4.5) It has been proved in the proof of Theorem  3 in [12] that ‖𝜙_𝑗−𝜙^𝑎‖_𝐿2(ℝ)=𝑂(2^−𝑗) as 𝑗→∞. Thus, by Lemma 3.2, it is obvious that ‖𝜓_𝑗−𝜓^𝑎‖_𝐿2(ℝ)=𝑂(2^−𝑗). Similarly, the convergence of 𝜓_𝑗 to 𝜓𝑎 (as 𝑗→∞) can be proved.

Proposition 4.2. Assume that 𝜓_𝑗 and 𝜓_𝑗 are a pair of biorthogonal wavelet functions associated with 𝜙_𝑗 and 𝜙_𝑗, 𝑗≥𝑗₀. Let 𝜓_𝑗,𝑘∶=2^𝑗/2𝜓_𝑗(2^𝑗⋅−𝑘) and 𝜓_𝑗,𝑘∶=2^𝑗/2𝜓_𝑗(2^𝑗⋅−𝑘), then, for any 𝑗≥𝑗₀, the sets {𝜓_𝑗,𝑘∶𝑘∈ℤ} and {𝜓_𝑗,𝑘∶𝑘∈ℤ} form Riesz bases for 𝑊_𝑗 and 𝑊_𝑗, respectively. Furthermore, there exist constants 𝐶,𝐷,𝐶,𝐷>0 independent of 𝑗≥𝑗₀ (but dependent on 𝑗₀) such that 𝐶≤𝜓_𝑗,𝜓_𝑗≤𝐷,𝐶≤𝜓_𝑗,𝜓_𝑗≤𝐷,∀𝑗≥𝑗₀. (4.6)

Proof. Since 𝜓_𝑗 and 𝜓_𝑗 are compactly supported functions, there exist constants 𝐷_𝑗,𝐷_𝑗>0 such that 𝜓_𝑗,𝜓_𝑗≤𝐷_𝑗,𝜓_𝑗,𝜓_𝑗≤𝐷_𝑗. (4.7) Notice that the duality condition ⟨𝜓_𝑗(⋅−𝑘),𝜓_𝑗⟩=𝛿_0,𝑘 is equivalent to [𝜓_𝑗,𝜓_𝑗]=1, then, by using the Cauchy-Schwartz inequality, we get 𝜓_𝑗,𝜓_𝑗𝜓_𝑗,𝜓_𝑗≥1. (4.8) Putting 𝐶_𝑗=𝐷_𝑗⁻¹, we obtain from (4.7) that 𝐶_𝑗≤[𝜓_𝑗,𝜓_𝑗]≤𝐷_𝑗. Next, we prove that the Riesz bounds 𝐶_𝑗 and 𝐷_𝑗, 𝑗≥𝑗₀, are in some interval [𝐶,𝐷] with 0<𝐶,𝐷<∞. For this, we see that the 2𝜋 periodic function [𝜓_𝑗,𝜓_𝑗] is of the form 𝜓_𝑗,𝜓_𝑗(𝜉)=∶_𝑛∈ℤ𝑐_𝑛^[𝑗]𝑒^𝑖𝑛𝜉,where𝑐_𝑛^[𝑗]=𝜓_𝑗,𝜓_𝑗(⋅−𝑛). (4.9) Here, by Lemma 4.1, 𝜓_𝑗 converges to 𝜓^𝑎 in 𝐿₂(ℝ) as 𝑗→∞. Thus, it follows that for any 𝑛∈ℤ, 𝑐_𝑛^[𝑗] converge to 𝑐_𝑛=⟨𝜓^𝑎,𝜓^𝑎(⋅−𝑛)⟩ as 𝑗→∞. Moreover, since 𝜓_𝑗 is compactly supported, only a finite number of 𝑐_𝑛^[𝑗] is nonzero. It yields that 𝜓_𝑗,𝜓_𝑗(𝜉)⟶_𝑛∈ℤ𝑐_𝑛𝑒^𝑖𝑛𝜉=𝜓^𝑎,𝜓^𝑎(𝜉),as𝑗⟶∞. (4.10) Therefore, we can deduce that the Riesz bounds 𝐶_𝑗 and 𝐷_𝑗 converge to 𝐶_𝜓𝑎 and 𝐷_𝜓𝑎 (resp.), as 𝑗→∞, which are the Riesz (upper and lower) bounds of 𝜓^𝑎. It concludes that 𝐶≤𝜓_𝑗,𝜓_𝑗≤𝐷,∀𝑗≥𝑗₀, (4.11) with some constants 𝐶,𝐷>0 independent of 𝑗≥𝑗₀. In a similar fashion, it can be proved that 𝐶≤[𝜓_𝑗,𝜓_𝑗]≤𝐷 with some constants 𝐶,𝐷>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 {𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ} (resp., {𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ}) with ℐ in (4.3). The following results treat this problem.

Lemma 4.3. Let 𝜓_𝑗 and 𝜓_𝑗 be a pair of biorthogonal wavelet functions associated with 𝜙_𝑗 and 𝜙_𝑗, 𝑗≥𝑗₀, then there exist two constants 𝐷,𝐷>0 depending on 𝑗₀ such that for all 𝑓∈𝐿₂(ℝ), _{(𝑗,𝑘)∈ℐ}𝑓,𝜓_𝑗,𝑘²≤𝐷𝑓²_𝐿2(ℝ),_{(𝑗,𝑘)∈ℐ}⟨𝑓,𝜓_𝑗,𝑘⟩²≤𝐷𝑓²_𝐿2(ℝ), (4.12) where ℐ is given in (4.3).

Proof. Let 𝜓^𝑎_𝑗,𝑘∶=2^𝑗/2𝜓^𝑎(2^𝑗−𝑘), and invoke that 𝜓^𝑎 has the same support as 𝜓_𝑗 for any 𝑗≥𝑗₀. Put 𝑆_𝑘∶=supp𝜓^𝑎(⋅−𝑘) with 𝑘∈ℤ, then, we observe that 𝑓,𝜓_𝑗,𝑘−𝜓^𝑎_𝑗,𝑘=_ℝ𝑓(𝑥)𝜓_𝑗,𝑘(𝑥)−𝜓^𝑎_𝑗,𝑘(𝑥)𝑑𝑥≤2^−𝑗/2_𝑆𝑘𝑓2^−𝑗𝑥𝜓_𝑗(𝑥−𝑘)−𝜓^𝑎_𝑗(𝑥−𝑘)𝑑𝑥≤2^−𝑗/2𝜓_𝑗−𝜓^𝑎_{𝐿2(ℝ)𝑆𝑘}𝑓2^−𝑗𝑥²𝑑𝑥^1/2≤𝑐2^−𝑗_{2−𝑗𝑆𝑘}𝑓(𝑥)²𝑑𝑥^1/2, (4.13) where the last bound is obtained by Lemma 4.1. Then, we get _{(𝑗,𝑘)∈ℐ}𝑓,𝜓_𝑗,𝑘−𝜓^𝑎_𝑗,𝑘²≤𝑐_{𝑗,𝑘∈ℐ}2^−2𝑗_{2−𝑗𝑆𝑘}𝑓(𝑥)²𝑑𝑥=𝑐_𝑗≥𝑗02^−2𝑗_{𝑘∈ℤ2−𝑗𝑆𝑘}𝑓(𝑥)²𝑑𝑥≤𝑐_𝑗≥𝑗02^−2𝑗𝑓²_𝐿2(ℝ)≤𝑐𝑓²_𝐿2(ℝ). (4.14) Putting all together, it is immediate that _{𝑗,𝑘∈ℐ}𝑓,𝜓_𝑗,𝑘^21/2≤_{(𝑗,𝑘)∈ℐ}𝑓,𝜓^𝑎_𝑗,𝑘^21/2+_{(𝑗,𝑘)∈ℐ}𝑓,𝜓_𝑗,𝑘−𝜓^𝑎_𝑗,𝑘^21/2≤𝐷_𝜓𝑎𝑓_𝐿2(ℝ)+𝑐𝑓_𝐿2(ℝ)=𝐷𝑓_𝐿2(ℝ), (4.15) for some constant 𝐷>0, where 𝐷_𝜓𝑎 is the Riesz upper bound of 𝜓^𝑎. Similarly, we can prove the second relation in (4.12).

We now arrive at the central results of this section.

Theorem 4.4. Let 𝜓_𝑗 and 𝜓_𝑗 be a pair of biorthogonal wavelet functions associated with 𝜙_𝑗 and 𝜙_𝑗, 𝑗≥𝑗₀, then the sets {𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ} and {𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ} form Riesz bases for the spaces +̇_𝑗≥𝑗0𝑊_𝑗 and +̇_𝑗≥𝑗0𝑊_𝑗, respectively, where ℐ is given in (4.3).

Proof. Let 𝑓 be a function in the space +̇_𝑗≥𝑗0𝑊_𝑗. By the Cauchy-Schwartz inequality and Lemma 4.3, we obtain 𝑓²_𝐿2(ℝ)=lim^𝐽_{𝐽→∞𝑗=𝑗0𝑘∈ℤ}𝑓,𝜓_𝑗,𝑘𝑓,𝜓_𝑗,𝑘≤_{𝑗,𝑘∈ℐ}𝑓,𝜓_𝑗,𝑘^21/2_{(𝑗,𝑘)∈ℐ}𝑓,𝜓_𝑗,𝑘^21/2≤𝐷^1/2𝑓_{𝐿2(ℝ)(𝑗,𝑘)∈ℐ}𝑓,𝜓_𝑗,𝑘^21/2, (4.16) for some constant 𝐷>0. Putting 𝐶=𝐷⁻¹, we have 𝐶𝑓²_𝐿2(ℝ)≤_{(𝑗,𝑘)∈ℐ}𝑓,𝜓_𝑗,𝑘². (4.17) Since the upper bound is proved in Lemma 4.3, it concludes that 𝐶𝑓²_𝐿2(ℝ)≤_{(𝑗,𝑘)∈ℐ}𝑓,𝜓_𝑗,𝑘²≤𝐷𝑓²_𝐿2(ℝ), (4.18) for some constant 𝐷>0. Further, it is obvious that 𝜓_𝑗,𝑘, (𝑗,𝑘)∈ℐ, are linearly independent. It indeed proves that the set {𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ} constitutes a Riesz basis for the space +̇_𝑗≥𝑗0𝑊_𝑗 (see [4, Theorem  3.20]). Similarly, we can show that {𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ} forms a Riesz basis for the space +̇_𝑗≥𝑗0𝑊_𝑗.

Since 𝜙_𝑗→𝜙_𝑎 and 𝜙_𝑗→𝜙_𝑎 in 𝐿₂(ℝ) as 𝑗→∞, it is immediate that for any 𝑓∈𝐿₂(ℝ), the biorthogonal projection 𝑃_𝑗𝑓=_𝑘∈ℤ𝑓,𝜙_𝑗,𝑘𝜙_𝑗,𝑘∈𝑉_𝑗 (4.19) converges to 𝑓 in the 𝐿₂-norm as 𝑗→∞, then based on this observation, we get the following result.

Theorem 4.5. Let 𝜓_𝑗 and 𝜓_𝑗 be a pair of biorthogonal wavelet functions associated with 𝜙_𝑗 and 𝜙_𝑗, 𝑗≥𝑗₀, then the set ℬ={𝜙_𝑗0,𝑘∶𝑘∈ℤ}∪{𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ} forms a Riesz basis for 𝐿₂(ℝ). This result also applies to the set of dual functions ℬ∶={𝜙_𝑗0,𝑘∶𝑘∈ℤ}∪{𝜓_𝑗,𝑘∶(𝑗,𝑘)∈ℐ}.

Proof. For a given function 𝑓∈𝐿₂(ℝ), for notational simplicity, we define the following sequences: 𝐜∶=𝑓,𝜙_{𝑗0,𝑘𝑘∈ℤ},𝐝_𝑗∶=⟨𝑓,𝜓_𝑗,𝑘⟩_𝑘∈ℤ,𝑗≥𝑗₀. (4.20) Similarly, let 𝐜̃ and 𝐝_𝑗, 𝑗≥𝑗₀, be the sequences consisting of the terms ⟨𝑓,𝜙_𝑗0,𝑘⟩ and ⟨𝑓,𝜓_𝑗,𝑘⟩, 𝑘∈ℤ, respectively. It has been proven in [12] that there exist constants 𝐴,𝐵>0 such that 𝐴𝑓²_𝐿2(ℝ)≤‖𝐜̃‖²₂≤𝐵𝑓²_𝐿2(ℝ). (4.21) Also, by Theorem 4.4, ^∞_𝑗=𝑗0𝐝_𝑗²₂≤𝐷𝑓²_𝐿2(ℝ), (4.22) for some constant 𝐷>0, then, by the Cauchy-Schwartz inequality and Theorem 4.4, we get 𝑓²_𝐿2(ℝ)=lim_{𝐽→∞𝑘∈ℤ}𝑓,𝜙_𝑗0,𝑘𝑓,𝜙_𝑗0,𝑘+^𝐽_{𝑗=𝑗0𝑘∈ℤ}𝑓,𝜓_𝑗,𝑘𝑓,𝜓_𝑗,𝑘≤‖𝐜‖²₂+^∞_𝑗=𝑗0𝐝_𝑗²₂^1/2‖𝐜̃‖²₂+^∞_𝑗=𝑗0𝐝_𝑗²₂^1/2≤max𝐵,𝐷^1/2𝑓_𝐿2(ℝ)‖𝐜‖²₂+^∞_𝑗=𝑗0𝐝_𝑗²₂^1/2. (4.23) Putting 𝐸∶=max(𝐵,𝐷)⁻¹, we have 𝐸𝑓²_𝐿2(ℝ)≤‖𝐜‖²₂+^∞_𝑗=𝑗0𝐝_𝑗²₂. (4.24) Since the upper bound is clear from Lemma 4.3, it concludes that 𝐸𝑓²_𝐿2(ℝ)≤‖𝐜‖²₂+^∞_𝑗=𝑗0𝐝_𝑗²₂≤𝐹𝑓²_𝐿2(ℝ), (4.25) 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 ℬ forms a Riesz basis for 𝐿₂(ℝ).

Acknowledgments

The authors are grateful to the anonymous referees for their valuable suggestions on this paper. The work of Y. J. Lee was supported by Basic Science Research Program (2009-0068156) and J. Yoon was supported by Mid-Career Researcher Program (2009-0084583) and Basic Science Research Program (2010-0016257), through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology.