## Abstract

Tight wavelet frames and orthonormal wavelet bases with a general dilation matrix have applications in many areas. In this paper, for any $d\times d$ dilation matrix $M$, we demonstrate in a constructive way that we can construct compactly supported tight $M$-wavelet frames and orthonormal $M$-wavelet bases in $L_2(\RR^d)$ of exponential decay, which are derived from compactly supported $M$-refinable functions, such that they can have both arbitrarily high smoothness and any preassigned order of vanishing moments. This paper improves several results in \cite{Bat, B, GR, Lem, St}.

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