Tight frame oversampling and its equivalence to shift-invariance of affine frame operators
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- by Charles K. Chui and Qiyu Sun PDF
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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 $1\le m|M$ (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.References
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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
- Email: cchui@stat.stanford.edu
- Qiyu Sun
- Affiliation: Department of Mathematics, National University of Singapore, Singapore 119260, Republic of Singapore
- Email: matsunqy@nus.edu.sg
- 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
- © Copyright 2002 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 131 (2003), 1527-1538
- MSC (2000): Primary 42C40
- DOI: https://doi.org/10.1090/S0002-9939-02-06703-5
- MathSciNet review: 1949883