Translational averaging for completeness, characterization and oversampling of wavelets


The single underlying method of {"}averaging the wavelet functional over translates{"} yields first a new completeness criterion for orthonormal wavelet systems, and then a unified treatment of known results on characterization of wavelets on the Fourier transform side, on preservation of frame bounds by oversampling, and on the equivalence of affine and quasiaffine frames. The method applies to multiwavelet systems in all dimensions, to dilation matrices that are in some cases not expanding, and to dual frame pairs. The completeness criterion we establish is precisely the discrete Calderón condition. In the single wavelet case this means we take invertible matrices aa and bb and a function ψL2(Rd)\psi\in L^2(\mathbb{R}^d), and assume either aa is expanding or else aa is amplifying for ψ\psi. We prove that the system {\{\vert det aj/2ψ(ajxbk):jZ,kZd}\vert^{j/2}\psi(a^jx-bk) : j\in\mathbb{Z},k\in\mathbb{Z}^d\} is an orthonormal basis for L2(Rd)L^2(\mathbb{R}^d) if and only if it is orthonormal and jZψ^(ξaj)2=\sum_{j\in\mathbb{Z}}\vert\hat\psi(\xi a^j)\vert^2 = \vert det b\vert for almost every row vectcor $\xi\in\mathbb{R}^d

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Revistes Catalanes amb Accés Obert

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oaioai:raco.cat:article/56537Last time updated on 6/13/2016

This paper was published in Revistes Catalanes amb Accés Obert.

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