Translational averaging for completeness, characterization and oversampling of wavelets

Abstract

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 $a$ and $b$ and a function $\psi\in L^2(\mathbb{R}^d)$, and assume either $a$ is expanding or else $a$ is amplifying for $\psi$. We prove that the system $\{\vert$det a$\vert^{j/2}\psi(a^jx-bk) : j\in\mathbb{Z},k\in\mathbb{Z}^d\}$ is an orthonormal basis for $L^2(\mathbb{R}^d)$ if and only if it is orthonormal and $\sum_{j\in\mathbb{Z}}\vert\hat\psi(\xi a^j)\vert^2 = \vert$det b$\vert$ for almost every row vectcor $\xi\in\mathbb{R}^d