Affine and quasi-affine frames for rational dilations. (English) Zbl 1219.42026
Quasi-affine systems were introduced by A. Ron and Z. Shen in [J. Funct. Anal. 148, No. 2, 408–447 (1997; Zbl 0891.42018)], where they proved the important result that an affine system is a frame if, and only if, its quasi-affine counterpart is a frame (with the same frame bounds). In [Contemp. Math. 320, 29–43 (2003; Zbl 1047.42024)], M. Bownik generalized the notion of a quasi-affine system for rational expansive dilations.
The goal of this work is to extend the study of quasi-affine systems to rational expansive dilations. The authors 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. They also prove a similar equivalence result between pairs of dual affine frames and dual quasi-affine frames. Moreover, they uncover a fundamental difference between the integer and rational settings by exhibiting an example of a quasi–affine frame such that its affine counterpart is not a frame.
The goal of this work is to extend the study of quasi-affine systems to rational expansive dilations. The authors 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. They also prove a similar equivalence result between pairs of dual affine frames and dual quasi-affine frames. Moreover, they uncover a fundamental difference between the integer and rational settings by exhibiting an example of a quasi–affine frame such that its affine counterpart is not a frame.
Reviewer: Richard A. Zalik (Auburn University)
MSC:
| 42C40 | Nontrigonometric harmonic analysis involving wavelets and other special systems |
Keywords:
wavelets; affine systems; quasi-affine systems; rational dilations; shift–invariant systems; oversamplingReferences:
| [1] | Marcin Bownik, The structure of shift-invariant subspaces of \?²(\?\(^{n}\)), J. Funct. Anal. 177 (2000), no. 2, 282 – 309. · Zbl 0986.46018 · doi:10.1006/jfan.2000.3635 |
| [2] | Marcin Bownik, A characterization of affine dual frames in \?²(\?\(^{n}\)), Appl. Comput. Harmon. Anal. 8 (2000), no. 2, 203 – 221. · Zbl 0961.42018 · doi:10.1006/acha.2000.0284 |
| [3] | 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. · Zbl 1047.42024 · doi:10.1090/conm/320/05597 |
| [4] | 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. · Zbl 1083.42025 · doi:10.1090/conm/345/06240 |
| [5] | 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). · Zbl 1063.42023 · doi:10.4064/sm159-3-8 |
| [6] | J. W. S. Cassels, An introduction to the geometry of numbers, Classics in Mathematics, Springer-Verlag, Berlin, 1997. Corrected reprint of the 1971 edition. · Zbl 0866.11041 |
| [7] | Ole Christensen, An introduction to frames and Riesz bases, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2003. · Zbl 1017.42022 |
| [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. · Zbl 1005.42020 · doi:10.1007/s00041-002-0007-4 |
| [9] | 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. · Zbl 0892.42019 · doi:10.1023/A:1018975725857 |
| [10] | A. Cohen, Ingrid Daubechies, and J.-C. Feauveau, Biorthogonal bases of compactly supported wavelets, Comm. Pure Appl. Math. 45 (1992), no. 5, 485 – 560. · Zbl 0776.42020 · doi:10.1002/cpa.3160450502 |
| [11] | 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. · Zbl 0896.42022 · doi:10.1007/BF02656493 |
| [12] | Deguang Han and David R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc. 147 (2000), no. 697, x+94. · Zbl 0971.42023 · doi:10.1090/memo/0697 |
| [13] | 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. · Zbl 1039.42032 · doi:10.1007/BF02930656 |
| [14] | 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. · Zbl 1046.42027 · doi:10.1016/j.acha.2003.12.002 |
| [15] | E. Hewitt, K. Ross, Abstract harmonic analysis. Vol. I: Structure of topological groups. Integration theory, group representations, Academic Press Inc., Publishers, New York, 1963. · Zbl 0115.10603 |
| [16] | M. Holschneider, Wavelets, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995. An analysis tool; Oxford Science Publications. · Zbl 0874.42020 |
| [17] | Richard S. Laugesen, Completeness of orthonormal wavelet systems for arbitrary real dilations, Appl. Comput. Harmon. Anal. 11 (2001), no. 3, 455 – 473. · Zbl 1014.42024 · doi:10.1006/acha.2001.0365 |
| [18] | Richard S. Laugesen, Translational averaging for completeness, characterization and oversampling of wavelets, Collect. Math. 53 (2002), no. 3, 211 – 249. · Zbl 1035.42035 |
| [19] | Amos Ron and Zuowei Shen, Frames and stable bases for shift-invariant subspaces of \?\(_{2}\)(\?^{\?}), Canad. J. Math. 47 (1995), no. 5, 1051 – 1094. · Zbl 0838.42016 · doi:10.4153/CJM-1995-056-1 |
| [20] | Amos Ron and Zuowei Shen, Affine systems in \?\(_{2}\)(\?^{\?}): the analysis of the analysis operator, J. Funct. Anal. 148 (1997), no. 2, 408 – 447. · Zbl 0891.42018 · doi:10.1006/jfan.1996.3079 |
| [21] | Amos Ron and Zuowei Shen, Weyl-Heisenberg frames and Riesz bases in \?\(_{2}\)(\?^{\?}), Duke Math. J. 89 (1997), no. 2, 237 – 282. · Zbl 0892.42017 · doi:10.1215/S0012-7094-97-08913-4 |
| [22] | Amos Ron and Zuowei Shen, Affine systems in \?\(_{2}\)(\?^{\?}). II. Dual systems, J. Fourier Anal. Appl. 3 (1997), no. 5, 617 – 637. Dedicated to the memory of Richard J. Duffin. · Zbl 0904.42025 · doi:10.1007/BF02648888 |
| [23] | Amos Ron and Zuowei Shen, Generalized shift-invariant systems, Constr. Approx. 22 (2005), no. 1, 1 – 45. · Zbl 1080.42025 · doi:10.1007/s00365-004-0563-8 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
