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Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling. (English) Zbl 1005.42020
Let $$A$$ be a $$d\times d$$ dilation matrix (i.e., a matrix for which all eigenvalues have absolute value greater than one) and $$B$$ a nonsingular $$d\times d$$ matrix. Given a finite collection of functions $$\psi_1,\dots, \psi_n\in L^2(R^d)$$ the authors consider the wavelet system consisting of all functions $\psi_{l,j,k}(x)= |\text{det } A|^{j/2}\psi_l(A^jx-Bk), \;j\in Z, k\in Z^d, 1=1, \dots, L.$ The main results characterize dual wavelet frames of this type and tight frames. As an application the Second Oversampling Theorem by Chui and Shi is generalized.

##### MSC:
 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
##### Keywords:
dual wavelet frames; dilation matrix; oversampling; tight frames
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