## Abstract

Let $\phi$ be a compactly supported refinable function in $L_2(\RR)$ such that the shifts of $\phi$ are stable and $\hat\phi(2\xi)=\hat a(\xi)\hat \phi(\xi)$ for a $2\pi$-periodic trigonometric polynomial $\hat a$. A wavelet function $\psi$ can be derived from $\phi$ by $\hat \psi(2\xi):=e^{-i\xi}\ol{\hat a(\xi+\pi)} \hat \phi(\xi)$. If $\phi$ is an orthogonal refinable function, then it is well known that $\psi$ generates an orthonormal wavelet basis in $L_2(\RR)$. Recently, it has been shown in the literature (\cite{DS, HS:rw1d}) that if $\phi$ is a $B$-spline or pseudo-spline refinable function, then $\psi$ always generates a Riesz wavelet basis in $L_2(\RR)$. It was an open problem whether $\psi$ can always generate a Riesz wavelet basis in $L_2(\RR)$ for any compactly supported refinable function in $L_2(\RR)$ 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 $\psi$ does not generate a Riesz wavelet basis in $L_2(\RR)$. Our proof is based on some necessary and sufficient conditions on the $2\pi$-periodic functions $\hat a$ and $\hat b$ in $C^{\infty}(\RR)$ such that the wavelet function $\psi$, defined by $\hat \psi(2\xi):=\hat b(\xi)\hat \phi(\xi)$, generates a Riesz wavelet basis in $L_2(\RR)$.

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