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

Authors: Charles K. Chui and Qiyu Sun
Journal: Proc. Amer. Math. Soc. 131 (2003), 1527-1538
MSC (2000): Primary 42C40
Published electronically: September 19, 2002
MathSciNet review: 1949883
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Abstract: Let $\Psi=\{\psi_1, \ldots, \psi_L\}\subset L^2:=L^2(-\infty, \infty)$ generate a tight affine frame with dilation factor $M$, where $2\le M\in \mathbf{Z}$, and sampling constant $b=1$ (for the zeroth scale level). Then for $1\le N\in \mathbf{Z}$, $N\times$oversampling (or oversampling by $N$) means replacing the sampling constant $1$ by $1/N$. 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 $N=N_0$ is relatively prime to $M$ (i.e., $gcd(N_0,M)=1$). 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 $gcd(m,M)=m$). 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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Additional Information

Charles K. Chui
Affiliation: 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

Qiyu Sun
Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260, Republic of Singapore

Received by editor(s): February 8, 2001
Received by editor(s) in revised form: December 16, 2001
Published electronically: September 19, 2002
Additional Notes: The research of the first author was partially supported by NSF Grant #CCR-9988289 and ARO Grant #DAAD 19-00-1-0512
The second author is also a visiting member of the Institute of Computational Harmonic Analysis, University of Missouri–St. Louis
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society