On a conjecture about MRA Riesz wavelet bases

Abstract

Let ϕ \phi be a compactly supported refinable function in L 2 ( R ) L_2(\mathbb {R}) such that the shifts of ϕ \phi are stable and ϕ ^ ( 2 ξ ) = a ^ ( ξ ) ϕ ^ ( ξ ) \hat \phi (2\xi )=\hat a(\xi )\hat \phi (\xi ) for a 2 π 2\pi -periodic trigonometric polynomial a ^ \hat a . A wavelet function ψ \psi can be derived from ϕ \phi by ψ ^ ( 2 ξ ) := e − i ξ a ^ ( ξ + π ) ¯ ϕ ^ ( ξ ) \hat \psi (2\xi ):=e^{-i\xi }\overline {\hat a(\xi +\pi )} \hat \phi (\xi ) . If ϕ \phi is an orthogonal refinable function, then it is well known that ψ \psi generates an orthonormal wavelet basis in L 2 ( R ) L_2(\mathbb {R}) . Recently, it has been shown in the literature that if ϕ \phi is a B B -spline or pseudo-spline refinable function, then ψ \psi always generates a Riesz wavelet basis in L 2 ( R ) L_2(\mathbb {R}) . It was an open problem whether ψ \psi can always generate a Riesz wavelet basis in L 2 ( R ) L_2(\mathbb {R}) for any compactly supported refinable function in L 2 ( R ) L_2(\mathbb {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 \psi does not generate a Riesz wavelet basis in L 2 ( R ) L_2(\mathbb {R}) . Our proof is based on some necessary and sufficient conditions on the 2 π 2\pi -periodic functions a ^ \hat a and b ^ \hat b in C ∞ ( R ) C^{\infty }(\mathbb {R}) such that the wavelet function ψ \psi , defined by ψ ^ ( 2 ξ ) := b ^ ( ξ ) ϕ ^ ( ξ ) \hat \psi (2\xi ):=\hat b(\xi )\hat \phi (\xi ) , generates a Riesz wavelet basis in L 2 ( R ) L_2(\mathbb {R}) .