Nonstationary Biorthogonal Wavelet Systems Based on Exponential B-Splines

and Applied Analysis 3 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 ψj ∑ n∈Z −1 ã j 1−nφj 1 2 · −n , ψ̃j ∑ n∈Z −1 a j 1−nφ̃j 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 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 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 {γj ∈ C : j 1, . . . ,N}, the corresponding Nth-order exponential B-splines can be defined as successive convolutions of the first-order B-spline φj : φ G j : τj ( eγ12−j B1 ∗ · · · ∗ eγN2−j B1 ) , 2.1 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 → e, γ ∈ C, is the exponential function. For simplicity, we will omit G in φ G j . 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 a j z : ∑ n∈Z a j n z n : 2cj N ∏ n 1 1 2 ( 1 eγn2 −j−1 z ) , j ≥ j0, 2.2 4 Abstract and Applied Analysis where cj is the normalization factor defined by