Constructing pairs of dual bandlimited frame wavelets in $L^2(\mathbb{R}^n)$

Details Constructing pairs of dual bandlimited frame wavelets in $L^2(\mathbb{R}^n)$ Constructing pairs of dual bandlimited frame wavelets in $L^2(\mathbb{R}^n)$

2010-09-22

arxiv.org/abs/1009.4351v1

Mathematics

Functional Analysis

21 pages, 6 figures

Scientific paper

10.1016/j.acha.2011.07.002

Given a real, expansive dilation matrix we prove that any bandlimited function $\psi \in L^2(\mathbb{R}^n)$, for which the dilations of its Fourier transform form a partition of unity, generates a wavelet frame for certain translation lattices. Moreover, there exists a dual wavelet frame generated by a finite linear combination of dilations of $\psi$ with explicitly given coefficients. The result allows a simple construction procedure for pairs of dual wavelet frames whose generators have compact support in the Fourier domain and desired time localization. The construction relies on a technical condition on $\psi$, and we exhibit a general class of function satisfying this condition.

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