On a conjecture about MRA Riesz wavelet bases

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