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)



The wavelet dimension function is the trace function of a shift-invariant system

Authors: Amos Ron and Zuowei Shen
Journal: Proc. Amer. Math. Soc. 131 (2003), 1385-1398
MSC (2000): Primary 42C15; Secondary 42C30
Published electronically: December 6, 2002
MathSciNet review: 1949868
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we observe that the dimension function associated with a wavelet system is the trace of the Gramian fibers of the shift-invariant system generated by the negative dilations of the mother wavelets. When this shift-invariant system is a tight frame, each of the Gramian fibers is an orthogonal projector, and its trace, then, coincides with its rank. This connection leads to simple proofs of several results concerning the dimension function, and the arguments extend to the bi-frame case.

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

  • [A1] Auscher, P., Toute base d'ondelettes régulières de $L^2({\mathbb R})$: Analyse Multi-Résolution régulière, CRAS Série 315, 1, 1992, 1227-1230. MR 94e:42041
  • [A2] Auscher, P., Solution of two problems on wavelets, J. Geom. Anal., 5, 1995, 181-236. MR 96g:42016
  • [B] Baggett, L. W., An abstract interpretation of wavelet dimension function using group representation, J. Funct. Anal., 173, 2000, 1-20. MR 2001j:42028
  • [BL] Benedetto, J., Li, S. D., The theory of multiresolution analysis frames and applications to filter banks, Appl. Comput. Harmonic Anal., 5(4), 1998, 389-427.MR 99k:42054
  • [BDR1] de Boor, C., DeVore, R., Ron, A., Approximation from shift-invariant subspaces of $L_2({\mathbb R}^d)$, Trans. Amer. Math. Soc., 341, 1994, 787-806. MR 94d:41028
  • [BDR2] de Boor, C., DeVore, R., Ron, A., The structure of finitely generated shift-invariant spaces in $L_2({\mathbb R}^d)$, J. Funct. Anal., 119(1), 1994, 37-78. MR 95g:46050
  • [Bo] J Bownik, M., The structure of shift invariant subspaces of $L^2({\mathbb R}^n)$, J. Funct. Anal., 177, 2000, 282-309. MR 2001k:42037
  • [BRS] J Bownik, M., Rzeszotnik, Z., Speegle, D., A characterization of dimension function of wavelets, Appl. Comput. Harmonic Anal., 10, 2001, 71-92. MR 2001m:42058
  • [CSS] Chui, C. K., Shi, X., Stockler, J., Affine frames, quasi-frames and there duals, Appl. Comput. Harmonic Anal., 8, 1998, 1-17. MR 99b:42037
  • [DHRS] Daubechies, I., Han, B., Ron, A., Shen, Z., Framelets: MRA-based constructions of wavelet frames, preprint, 2001, Ftp site: file
  • [F] B. Folland, G. B., A Course in Abstract Harmonic Analysis, CRC Press (Boca Raton), 1995. MR 98c:43001
  • [G] Gripenberg, G., A necessary and sufficient condition for the existence of father wavelet, Studia Math., 114, 1995, 207-226. MR 96d:42049
  • [Ha] Han, Bin, Some applications of projection operators in wavelets, Acta Math. Sinica, New Series, 11, 1995, 105-112. MR 97k:42061
  • [H] Helson, H., Lectures on Invariant Subspace, Academic Press (New York), 1964. MR 30:1409
  • [HW] Hernández, E., Weiss, G., A first Course on Wavelets, Studies in Advanced Mathematics, CRC Press (Boca Raton FL), 1996. MR 97i:42015
  • [La] Lawton, W., Tight frames of compactly supported affine wavelets, J. Math. Phys., 31, 1990, 1898-1901. MR 92a:81068
  • [L1] Lemarié-Rieusset, P. G., Existence de ``fonction-pére''pour les ondelettes á support compact, C.R. Acad. Sci. Paris, Ser. I., Math., 314, 1992, 17-19. MR 93c:42033
  • [L2] Lemarié-Rieusset, P. G., Ondelettes généralisées et fonctions d'échelle á support compact, Rev. Mat. Iberoamericana, 9, 1993, 333-371. MR 94i:42045
  • [M] Mallat, S., Multiresolution approximations and wavelet orthonormal bases of $L^2({\mathbb R})$, Trans. Amer. Math. Soc., 315, 1989, 69-87. MR 90e:42046
  • [PSWX1] Paluszynski, M., Sikic, H., Weiss, G., Xiao, X., Generalized low pass filters and MRA frame wavelets, J. Geom. Anal., 11, 2001, 311-342.
  • [PSWX2] Paluszynski, M., Sikic, H., Weiss, G., Xiao, X., Tight frame wavelets, their dimension functions, MRA tight frame wavelets and connectivity properties, preprint, 2001.
  • [P] Papadakis, M., On the dimension function of orthonormal wavelets, Proc. Amer. Math. Soc., 128, 1999, 2042-2049. MR 2000m:42031
  • [RS1] Ron, A., Shen, Z., Frames and stable bases for shift-invariant subspaces of $L_2({\mathbb R}^d)$, Canad. J. Math., 47(5), 1995, 1051-1094; Ftp site: file MR 96k:42049
  • [RS2] Ron, A., Shen, Z., Affine systems in $L_2({\mathbb R}^d)$: the analysis of the analysis operator, J. Funct. Anal., 148, 1997, 408-447; Ftp site: file MR 99g:42043
  • [RS3] Ron, A., Shen, Z., Affine systems in $L_2({\mathbb R}^d)$ II: dual systems, J. Fourier Anal. Appl., 3, 1997, 617-637; Ftp site: file MR 99g:42044
  • [RS4] Ron, A., Shen, Z., Compactly supported tight affine spline frames in $L_2({\mathbb R}^d)$, Math. Comp., 67, 1998, 191-207; Ftp site: file MR 98c:42035
  • [W] Wang, X., The study of wavelets from properties of their Fourier transforms, Ph.D. Thesis, Washington University, St. Louis, MO, 1995.
  • [We] Weber, E., Applications of the wavelet multiplicity function, Contemp. Math., 247, 1999, 297-306. MR 2001f:42064

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42C15, 42C30

Retrieve articles in all journals with MSC (2000): 42C15, 42C30

Additional Information

Amos Ron
Affiliation: Computer Sciences Department, University of Wisconsin-Madison, 1210 West Dayton, Madison, Wisconsin 53706

Zuowei Shen
Affiliation: Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260

Keywords: Dimension function, frames, multiresolution analysis, wavelets
Received by editor(s): June 8, 2001
Published electronically: December 6, 2002
Additional Notes: This work was supported by the US National Science Foundation under Grants DMS-9872890, DBI-9983114 and ANI-0085984, the U.S. Army Research Office under Contract DAAG55-98-1-0443, and the Strategic Wavelet Program Grant from the National University of Singapore
Communicated by: David R. Larson
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society