On a conjecture about MRA Riesz wavelet bases

Abstract

Abstract. Let φ be a compactly supported refinable function in L 2 (R) such that the shifts of φ are stable andφ(2ξ) =â(ξ)φ(ξ) for a 2π-periodic trigonometric polynomialâ. A wavelet function ψ can be derived from φ byψ(2ξ) := e −iξâ (ξ + π)φ(ξ). If φ is an orthogonal refinable function, then it is well known that ψ generates an orthonormal wavelet basis in L 2 (R). Recently, it has been shown in the literature that if φ is a B-spline or pseudo-spline refinable function, then ψ always generates a Riesz wavelet basis in L 2 (R). It was an open problem whether ψ can always generate a Riesz wavelet basis in L 2 (R) for any compactly supported refinable function in L 2 (R) with stable shifts. In this paper, we settle this problem by proving that for a family of arbitrarily smooth refinable functions with stable shifts, the derived wavelet function ψ does not generate a Riesz wavelet basis in L 2 (R). Our proof is based on some necessary and sufficient conditions on the 2π-periodic functionsâ andb in C ∞ (R) such that the wavelet function ψ, defined byψ(2ξ) :=b(ξ)φ(ξ), generates a Riesz wavelet basis in L 2 (R).

Refinable Functions and Cascade Algorithms in Weighted Spaces with Hölder Continuous Masks

“…As an application of Theorems 2.1 and 3.1, we characterize MRA Riesz wavelet bases in L 2 (R) in the following result, which improves and generalizes [14,Theorem 6] and [16, Theorem 1.1] by taking a different approach.…”

“…A natural and important question here is when ψ generates a Riesz wavelet basis in L 2 (R). MRA Riesz wavelet bases with compact support have been investigated in [5,14,16,17,21,24], where some necessary and sufficient conditions have been obtained for trigonometric polynomial masks. Most approaches in these papers largely rely on an interesting result of Cohen and Daubechies in [5] saying that for a maskâ with exponential decay, the transition operator Tâ acting on some weighted subspaces of 2 (Z) is a compact operator.…”