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

DOI:
https://doi.org/10.1090/S0002-9939-02-06703-5

Published electronically:
September 19, 2002

MathSciNet review:
1949883

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Abstract | References | Similar Articles | Additional Information

Abstract: Let generate a tight affine frame with dilation factor , where , and sampling constant (for the zeroth scale level). Then for , oversampling (or oversampling by ) means replacing the sampling constant by . The Second Oversampling Theorem asserts that oversampling of the given tight affine frame generated by preserves a tight affine frame, provided that is relatively prime to (i.e., ). In this paper, we discuss the preservation of tightness in oversampling, where (i.e., and ). We also show that tight affine frame preservation in oversampling is equivalent to the property of shift-invariance with respect to of the affine frame operator defined on the zeroth scale level.

**1.**A. Aldroubi,*Portraits of frames*, Proc. Amer. Math. Soc.**123**(1995), 1661-1668. MR**95g:46037****2.**A. Aldroubi, Q. Sun and W.-S. Tang,*-frames and shift invariant subspaces of*, J. Fourier Anal. Appl.**7**(2001), 1-21. MR**2002c:42046****3.**M. Bownik,*A characterization of affine dual frames in*, Appl. Comp. Harmonic Anal.**8**(2000), 203-221. MR**2001d:42019****4.**A. Calogero,*Wavelets on general lattices*, ERA Amer. Math. Soc.**5**(1999), 1-10. MR**99i:42042****5.**A. Calogero,*A characterization of wavelets on general lattices*, J. Geom. Anal.**10**(2000), 597-622. MR**2002b:42051****6.**C. K. Chui, W. Czaja, M. Maggioni and G. Weiss,*Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling*, J. Fourier Anal. Appl., To appear.**7.**C. K. Chui and X. L. Shi,*Bessel sequences and affine frames*, Appl. Comp. Harmonic Anal.**1**(1993), 29-49. MR**95b:42028****8.**C. K. Chui and X. L. Shi,*oversampling preserves any tight affine frame for odd*, Proc. Amer. Math. Soc.**121**(1994), 511-517. MR**94h:42052****9.**C. K. Chui and X. L. Shi,*Inequalities on matrix-dilated Littlewood-Paley energy functions and oversampled affine frames*, SIAM J. Math. Anal.**28**(1997), 213-232. MR**98b:42046****10.**C. K. Chui and X. L. Shi,*Orthonormal wavelets and tight frames with arbitrary real dilations*, Appl. Comp. Harmonic Anal.**9**(2000), 243-264. MR**2002a:42025****11.**R. G. Catalán,*Oversampling and preservation of tightness in affine frames*, Proc. Amer. Math. Soc.**130**(2002), 1031-1034.**12.**E. Hernandez and G. Weiss,*A First Course on Wavelets*, CRC Press, Boca Raton, FL, 1996. MR**97i:42015****13.**J. Stoeckler, ``*Multivariate Affine Frames*'', Habil.-Schr. Thesis, Duisburg Univ., 1996.**14.**E. Weber,*On the translation invariance of wavelet subspaces*, J. Fourier Anal. Appl.**6**(2000), 551-558. MR**2001h:42057**

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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

DOI:
https://doi.org/10.1090/S0002-9939-02-06703-5

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