Affine and quasi-affine frames for rational dilations
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- by Marcin Bownik and Jakob Lemvig PDF
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Abstract:
In this paper we extend the investigation of quasi-affine systems, which were originally introduced by Ron and Shen [J. Funct. Anal. 148 (1997), 408–447] for integer, expansive dilations, to the class of rational, expansive dilations. We show that an affine system is a frame if, and only if, the corresponding family of quasi-affine systems are frames with uniform frame bounds. We also prove a similar equivalence result between pairs of dual affine frames and dual quasi-affine frames. Finally, we uncover some fundamental differences between the integer and rational settings by exhibiting an example of a quasi-affine frame such that its affine counterpart is not a frame.References
- Marcin Bownik, The structure of shift-invariant subspaces of $L^2(\textbf {R}^n)$, J. Funct. Anal. 177 (2000), no. 2, 282–309. MR 1795633, DOI 10.1006/jfan.2000.3635
- Marcin Bownik, A characterization of affine dual frames in $L^2(\mathbf R^n)$, Appl. Comput. Harmon. Anal. 8 (2000), no. 2, 203–221. MR 1743536, DOI 10.1006/acha.2000.0284
- Marcin Bownik, Quasi-affine systems and the Calderón condition, Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001) Contemp. Math., vol. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 29–43. MR 1979930, DOI 10.1090/conm/320/05597
- Marcin Bownik and Ziemowit Rzeszotnik, The spectral function of shift-invariant spaces on general lattices, Wavelets, frames and operator theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 49–59. MR 2066821, DOI 10.1090/conm/345/06240
- Marcin Bownik and Eric Weber, Affine frames, GMRA’s, and the canonical dual, Studia Math. 159 (2003), no. 3, 453–479. Dedicated to Professor Aleksander Pełczyński on the occasion of his 70th birthday (Polish). MR 2052234, DOI 10.4064/sm159-3-8
- J. W. S. Cassels, An introduction to the geometry of numbers, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition. MR 1434478
- Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. MR 1946982, DOI 10.1007/978-0-8176-8224-8
- Charles K. Chui, Wojciech Czaja, Mauro Maggioni, and Guido Weiss, Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling, J. Fourier Anal. Appl. 8 (2002), no. 2, 173–200. MR 1891728, DOI 10.1007/s00041-002-0007-4
- Charles K. Chui, Xianliang Shi, and Joachim Stöckler, Affine frames, quasi-affine frames, and their duals, Adv. Comput. Math. 8 (1998), no. 1-2, 1–17. MR 1607452, DOI 10.1023/A:1018975725857
- A. Cohen, Ingrid Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), no. 5, 485–560. MR 1162365, DOI 10.1002/cpa.3160450502
- Michael Frazier, Gustavo Garrigós, Kunchuan Wang, and Guido Weiss, A characterization of functions that generate wavelet and related expansion, Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996), 1997, pp. 883–906. MR 1600215, DOI 10.1007/BF02656493
- Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94. MR 1686653, DOI 10.1090/memo/0697
- Eugenio Hernández, Demetrio Labate, and Guido Weiss, A unified characterization of reproducing systems generated by a finite family. II, J. Geom. Anal. 12 (2002), no. 4, 615–662. MR 1916862, DOI 10.1007/BF02930656
- Eugenio Hernández, Demetrio Labate, Guido Weiss, and Edward Wilson, Oversampling, quasi-affine frames, and wave packets, Appl. Comput. Harmon. Anal. 16 (2004), no. 2, 111–147. MR 2038268, DOI 10.1016/j.acha.2003.12.002
- E. Hewitt, K. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Academic Press Inc., Publishers, New York, 1963.
- M. Holschneider, Wavelets, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. An analysis tool; Oxford Science Publications. MR 1367088
- Richard S. Laugesen, Completeness of orthonormal wavelet systems for arbitrary real dilations, Appl. Comput. Harmon. Anal. 11 (2001), no. 3, 455–473. MR 1866351, DOI 10.1006/acha.2001.0365
- Richard S. Laugesen, Translational averaging for completeness, characterization and oversampling of wavelets, Collect. Math. 53 (2002), no. 3, 211–249. MR 1940326
- Amos Ron and Zuowei Shen, Frames and stable bases for shift-invariant subspaces of $L_2(\mathbf R^d)$, Canad. J. Math. 47 (1995), no. 5, 1051–1094. MR 1350650, DOI 10.4153/CJM-1995-056-1
- Amos Ron and Zuowei Shen, Affine systems in $L_2(\mathbf R^d)$: the analysis of the analysis operator, J. Funct. Anal. 148 (1997), no. 2, 408–447. MR 1469348, DOI 10.1006/jfan.1996.3079
- Amos Ron and Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in $L_2(\mathbf R^d)$, Duke Math. J. 89 (1997), no. 2, 237–282. MR 1460623, DOI 10.1215/S0012-7094-97-08913-4
- Amos Ron and Zuowei Shen, Affine systems in $L_2(\textbf {R}^d)$. II. Dual systems, J. Fourier Anal. Appl. 3 (1997), no. 5, 617–637. Dedicated to the memory of Richard J. Duffin. MR 1491938, DOI 10.1007/BF02648888
- Amos Ron and Zuowei Shen, Generalized shift-invariant systems, Constr. Approx. 22 (2005), no. 1, 1–45. MR 2132766, DOI 10.1007/s00365-004-0563-8
Additional Information
- Marcin Bownik
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403–1222
- MR Author ID: 629092
- Email: mbownik@uoregon.edu
- Jakob Lemvig
- Affiliation: Department of Mathematics, Technical University of Denmark, Matematiktorvet, Building 303S, DK-2800 Kgs. Lyngby, Denmark
- Address at time of publication: Institut für Mathematik, Universität Osnabrück, 49069 Osnabrück, Germany
- Email: J.Lemvig@mat.dtu.dk, jlemvig@uni-osnabrueck.de
- Received by editor(s): September 24, 2008
- Published electronically: November 5, 2010
- Additional Notes: The first author was partially supported by NSF grant DMS-0653881.
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 363 (2011), 1887-1924
- MSC (2010): Primary 42C40
- DOI: https://doi.org/10.1090/S0002-9947-2010-05200-6
- MathSciNet review: 2746669