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)

 
 

 

On characterizations of multiwavelets in $L^{2}(\mathbb{R}^n)$


Author: Marcin Bownik
Journal: Proc. Amer. Math. Soc. 129 (2001), 3265-3274
MSC (2000): Primary 42C40
DOI: https://doi.org/10.1090/S0002-9939-01-05942-1
Published electronically: March 29, 2001
MathSciNet review: 1845001
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

We present a new approach to characterizing (multi)wavelets by means of basic equations in the Fourier domain. Our method yields an uncomplicated proof of the two basic equations and a new characterization of orthonormality and completeness of (multi)wavelets.


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

  • [B1] Bownik, Marcin, A Characterization of Affine Dual Frames in $L^{2}({\mathbb{R}^n})$, Appl. Comput. Harmon. Anal. 8 (2000), 203-221. CMP 2000:09
  • [B2] -, The structure of shift invariant subspaces of $L^{2}({\mathbb{R}^n})$, J. Funct. Anal. 177 (2000), 282-309. CMP 2001:04
  • [B3] -, The anisotropic Hardy spaces and wavelets, Ph. D. Thesis, Washington University, 2000.
  • [C1] Calogero, A., Wavelets on general lattices, associated with general expanding maps of ${\mathbb{R}^n}$, Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 1-10 (electronic). MR 99i:42042
  • [C2] -, A characterization of wavelets on general lattices, J. Geom. Anal. (to appear).
  • [CSS] Chui, Charles K., Shi, Xianliang and Stöckler, Joachim, Affine frames, quasi-affine frames, and their duals, Adv. Comput. Math. 8 (1998), 1-17. MR 99b:42037
  • [FGWW] Frazier, Michael, Garrigós, Gustavo, Wang, Kunchuan and Weiss, Guido, A characterization of functions that generate wavelet and related expansion, Proceedings of the conference dedicated to Professor Miguel de Guzmán (El Escorial, 1996), J. Fourier Anal. Appl. 3 (1997), 883-906. MR 99c:42058
  • [GS] Garrigós, Gustavo and Speegle, Darrin, Completeness in the set of wavelets, Proc. Amer. Math. Soc. 128 (2000), 1157-1166. MR 2000i:42021
  • [GH] Gröchenig, Karlheinz and Haas, Andrew, Self-similar lattice tilings, J. Fourier Anal. Appl. 1 (1994), 131-170. MR 96j:52037
  • [HKLS] Ha, Young-Hwa, Kang, Hyeonbae, Lee, Jungseob and Seo, Jin Keun, Unimodular wavelets for ${L}\sp{2}$and the Hardy space ${H}\sp{2}$, Michigan Math. J. 41 (1994), 345-361. MR 95g:42050
  • [HW] Hernández, Eugenio and Weiss, Guido, A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. MR 97i:42015
  • [LR] Lemarié-Rieusset, Pierre-Gilles, Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions, Rev. Mat. Iberoamericana 10 (1994), 283-347. MR 95e:42039
  • [M] Madych, W. R., Orthogonal wavelet bases for ${L}\sp{2}({\mathbb{R} }\sp{n})$, Fourier analysis (Orono, ME, 1992), Dekker, New York, 1994, pp. 243-302. MR 95g:42052
  • [RS1] Ron, Amos and Shen, Zuowei, Frames and stable bases for shift-invariant subspaces of ${L}\sb 2({\mathbb{R}}\sp{d})$, Canad. J. Math. 47 (1995), 1051-1094. MR 96k:42049
  • [RS2] -, Affine systems in ${L}\sb 2({\mathbb{R}}\sp{d})$: the analysis of the analysis operator, J. Funct. Anal. 148 (1997), 408-447. MR 99g:42043
  • [Rz] Rzeszotnik, Ziemowit, Characterization theorems in the theory of wavelets, Ph. D. Thesis, Washington University, 2000.

Similar Articles

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

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


Additional Information

Marcin Bownik
Affiliation: Department of Mathematics, Washington University, Campus Box 1146, St. Louis, Missouri 63130
Address at time of publication: Department of Mathematics, University of Michigan, 525 East University Avenue, Ann Arbor, Michigan 48109-1109
Email: marbow@math.wustl.edu, marbow@math.lsa.umich.edu

DOI: https://doi.org/10.1090/S0002-9939-01-05942-1
Keywords: Bessel family, affine frame, quasi affine frame, (multi)wavelet
Received by editor(s): September 27, 1999
Received by editor(s) in revised form: March 10, 2000
Published electronically: March 29, 2001
Additional Notes: The author thanks Richard Rochberg, Ziemowit Rzeszotnik, and Darrin Speegle for helpful comments and the referee for apt questions leading to the improvement of the paper.
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society