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Compactly supported tight wavelet frames and orthonormal wavelets of exponential decay with a general dilation matrix. (English) Zbl 1021.42020
For any $d\times d$ dilation matrix $M$ it is proved how to construct compactly supported tight wavelet frames and orthonormal wavelet bases having exponential decay; the bases have the form $\psi_{j,k}= |\text{det } M|^{j/2}\psi(M^j \cdot -k), j\in \bbfZ, k\in \bbfZ^d$ for some functions $\psi \in L^2(\bbfR^d)$; they are derived from refinable functions $\phi$, in the sense that they have the form $\psi = |\text{det } M|\sum_{k\in \bbfZ^d}b_k \phi (M \cdot -k)$ for some sequence $\{b_k\}_{k\in \bbfZ^d}$. One of the main results is as follows. Given any positive integer $r$, there exists a collection $\Psi$ of at most $(3/2)^d|\text{det } M|$ functions in $C^r(\bbfR^d)$, derived from a refinable function with compact support, such that $\Psi$ has vanishing moments of order $r$ and generates a tight wavelet frame for $L^2(\bbfR^d)$.

42C40Wavelets and other special systems
Full Text: DOI
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