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Abstract: On a conjecture about MRA Riesz wavelet bases

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Bin Han

## 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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