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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 $d×d$ 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 ${\psi }_{j,k}={|\text{det}\phantom{\rule{4.pt}{0ex}}M|}^{j/2}\psi \left({M}^{j}·-k\right),j\in ℤ,k\in {ℤ}^{d}$ for some functions $\psi \in {L}^{2}\left({ℝ}^{d}\right)$; they are derived from refinable functions $\phi$, in the sense that they have the form $\psi =|\text{det}\phantom{\rule{4.pt}{0ex}}M|{\sum }_{k\in {ℤ}^{d}}{b}_{k}\phi \left(M·-k\right)$ for some sequence ${\left\{{b}_{k}\right\}}_{k\in {ℤ}^{d}}$. One of the main results is as follows. Given any positive integer $r$, there exists a collection ${\Psi }$ of at most ${\left(3/2\right)}^{d}|\text{det}\phantom{\rule{4.pt}{0ex}}M|$ functions in ${C}^{r}\left({ℝ}^{d}\right)$, derived from a refinable function with compact support, such that ${\Psi }$ has vanishing moments of order $r$ and generates a tight wavelet frame for ${L}^{2}\left({ℝ}^{d}\right)$.
##### MSC:
 42C40 Wavelets and other special systems