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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.

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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