Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Density of irregular wavelet frames

Authors: Wenchang Sun and Xingwei Zhou
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 2377-2387
MSC (2000): Primary 42C40, 41A58
Published electronically: February 26, 2004
MathSciNet review: 2052416
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We show that if an irregular multi-generated wavelet system forms a frame, then both the time parameters and the logarithms of scale parameters have finite upper Beurling densities, or equivalently, both are relatively uniformly discrete. Moreover, if generating functions are admissible, then the logarithms of scale parameters possess a positive lower Beurling density. However, the lower Beurling density of the time parameters may be zero. Additionally, we prove that there are no frames generated by dilations of a finite number of admissible functions.

References [Enhancements On Off] (What's this?)

  • 1. J. J. Benedetto, Irregular sampling and frames, in ``Wavelets: A Tutorial in Theory and Applications (C. K. Chui, editor), Academic Press, Boston, 1992, pp. 445-507. MR 93c:42030
  • 2. A. Beurling, Balayage of Fourier-Stieltjes transforms, The Collected Works of Arne Beurling, Vol. 2, Harmonic Analysis, L. Carleson, P. Malliavin, J. Neuberger, and J. Wermer, editors, Birkhäuser, Boston, 1989, pp. 341-365. MR 92k:01046b
  • 3. O. Christensen, Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bulletin (new series) of Amer. Math. Soc., 38 (2001), 273-291. MR 2002c:42040
  • 4. O. Christensen, B. Deng, and C. Heil, Density of Gabor frames, Appl. Comput. Harmon. Anal., 7 (1999), 292-304. MR 2000j:42043
  • 5. C. K. Chui and X. L. Shi, Orthonormal wavelets and tight frames with arbitrary real dilations, Appl. Comput. Harmon. Anal., 9 (2000), 243-264. MR 2002a:42025
  • 6. I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, 36 (1990), 961-1005. MR 91e:42038
  • 7. I. Daubechies, Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. MR 93e:42045
  • 8. H. Feichtinger and K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I, J. Funct. Anal., 86 (1989), 307-340. MR 91g:43011
  • 9. K. Gröchenig, Describing functions: atomic decompositions versus frames, Monatsh. Math., 112 (1991), 1-42. MR 92m:42035
  • 10. K. Gröchenig, Irregular sampling of wavelet and short-time Fourier transforms, Constr. Approx., 9 (1993), 283-297. MR 94m:42077
  • 11. G. H. Hardy, J. E. Littlewood, and G. Pólya, Inequalities, 2nd Ed., Cambridge Univ. Press, Cambridge, 1952. MR 13:727e
  • 12. C. Heil and G. Kutyniok, Density of weighted wavelet frames, J. Geom. Anal., 13 (2003), 479-493.
  • 13. C. Heil and D. Walnut, Continuous and discrete wavelet transforms, SIAM Review, 31 (1989), 628-666. MR 91c:42032
  • 14. S. Jaffard, A density criterion for frames of complex exponentials, Michigan Math. J., 38 (1991), 339-348. MR 92i:42001
  • 15. P. Olsen and K. Seip, A note on irregular discrete wavelet transforms, IEEE Trans. Inform. Theory, 38 (1992), 861-863.
  • 16. T. E. Olson and R. Z. Zalik, Nonexistence of a Riesz basis of translates, in ``Approximation Theory'', pp. 401-408, Lecture Notes in Pure and Applied Math., Vol. 138, Marcel Dekker, New York, 1992. MR 93b:41001
  • 17. W. Sun and X. Zhou, Irregular wavelet frames, Science in China, series A, 43 (2000), 122-127. MR 2000m:42035
  • 18. W. Sun and X. Zhou, Irregular wavelet/Gabor frames, Appl. Comput. Harmon. Anal., 13 (2002), 63-76. MR 2003i:42057
  • 19. W. Sun and X. Zhou, Density and stability of wavelet frames, Appl. Comput. Harmon. Anal., 15 (2003), 117-133.
  • 20. R. Young, An Introduction to Nonharmonic Fourier Series, Pure and Applied Mathematics, no. 93, Academic Press, New York, 1980. MR 81m:42027
  • 21. X. Zhou and Y. Li, A class of irregular wavelet frames, Chinese Science Bulletin, 42 (1997), 1420-1424.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42C40, 41A58

Retrieve articles in all journals with MSC (2000): 42C40, 41A58

Additional Information

Wenchang Sun
Affiliation: Department of Mathematics, Nankai University, Tianjin 300071, China

Xingwei Zhou
Affiliation: Department of Mathematics, Nankai University, Tianjin 300071, China

Keywords: Wavelet frames, density, Beurling density
Received by editor(s): February 3, 2003
Received by editor(s) in revised form: May 7, 2003
Published electronically: February 26, 2004
Additional Notes: This work was supported by the National Natural Science Foundation of China (10171050 and 10201014), the Mathematical Tianyuan Foundation (TY10126007), the Research Fund for the Doctoral Program of Higher Education, and the Liuhui Center for Applied Mathematics.
Communicated by: David R. Larson
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society