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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 \(\psi_{j,k}= |\text{det } M|^{j/2}\psi(M^j \cdot -k), j\in \mathbb{Z}, k\in \mathbb{Z}^d\) for some functions \(\psi \in L^2(\mathbb{R}^d)\); they are derived from refinable functions \(\phi\), in the sense that they have the form \(\psi = |\text{det } M|\sum_{k\in \mathbb{Z}^d}b_k \phi (M \cdot -k)\) for some sequence \(\{b_k\}_{k\in \mathbb{Z}^d}\). 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 \(C^r(\mathbb{R}^d)\), derived from a refinable function with compact support, such that \(\Psi\) has vanishing moments of order \(r\) and generates a tight wavelet frame for \(L^2(\mathbb{R}^d)\).

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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