On a Conjecture about Mra Riesz Wavelet Bases

Let φ be a compactly supported refinable function in L2(R) such that the shifts of φ are stable and φ̂(2ξ) = â(ξ)φ̂(ξ) for a 2π-periodic trigonometric polynomial â. A wavelet function ψ can be derived from φ by ψ̂(2ξ) := e−iξ â(ξ + π)φ̂(ξ). If φ is an orthogonal refinable function, then it is well known that ψ generates an orthonormal wavelet basis in L2(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 L2(R). It was an open problem whether ψ can always generate a Riesz wavelet basis in L2(R) for any compactly supported refinable function in L2(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 L2(R). Our proof is based on some necessary and sufficient conditions on the 2π-periodic functions â and b̂ in C∞(R) such that the wavelet function ψ, defined by ψ̂(2ξ) := b̂(ξ)φ̂(ξ), generates a Riesz wavelet basis in L2(R).