Tight frame oversampling and its equivalence to shift-invariance of affine frame operators期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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Tight frame oversampling and its equivalence to shift-invariance of affine frame operators

Authors:	Charles K Chui Qiyu Sun

Institution:	Department of Mathematics and Computer Science, University of Missouri--St. Louis, St. Louis, Missouri 63121-4499 -- and -- Department of Statistics, Stanford University, Stanford, California 94305 ; Department of Mathematics, National University of Singapore, Singapore 119260, Republic of Singapore

Abstract:	Let $\Psi=\{\psi_1, \ldots, \psi_L\}\subset L^2:=L^2(-\infty, \infty)$ generate a tight affine frame with dilation factor , where $2\le M\in \mathbf{Z}$ , and sampling constant (for the zeroth scale level). Then for $1\le N\in \mathbf{Z}$ , $N\times$ oversampling (or oversampling by ) means replacing the sampling constant by . The Second Oversampling Theorem asserts that $N\times$ oversampling of the given tight affine frame generated by $\Psi$ preserves a tight affine frame, provided that is relatively prime to (i.e., ). In this paper, we discuss the preservation of tightness in $mN_0\times$ oversampling, where $1\le m\vert M$ (i.e., $1\le m\le M$ and ). We also show that tight affine frame preservation in $mN_0\times$ oversampling is equivalent to the property of shift-invariance with respect to $\frac{1}{mN_0}\mathbf{Z}$ of the affine frame operator $Q_{0,N_0}$ defined on the zeroth scale level.

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