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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Stability of wavelet frames with matrix dilations


Authors: Ole Christensen and Wenchang Sun
Journal: Proc. Amer. Math. Soc. 134 (2006), 831-842
MSC (2000): Primary 42C40, 41A58
Published electronically: July 20, 2005
MathSciNet review: 2180901
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Abstract: Under certain assumptions we show that a wavelet frame

\begin{displaymath}\{\tau(A_j,b_{j,k})\psi\}_{j,k\in \mathbb{Z} }:= \{\vert\det A_j \vert^{-1/2} \psi(A_j^{-1}(x-b_{j,k}))\}_{j,k\in \mathbb{Z} }\end{displaymath}

in $L^2(\mathbb{R} ^d)$ remains a frame when the dilation matrices $A_j$ and the translation parameters $b_{j,k}$ are perturbed. As a special case of our result, we obtain that if $ \{\tau(A^j,A^jBn)\psi\}_{j\in \mathbb{Z} ,n\in \mathbb{Z} ^d}$ is a frame for an expansive matrix $A$ and an invertible matrix $B$, then $\{\tau(A_j^\prime,A^jB\lambda_n)\psi\}_{j\in \mathbb{Z} , n\in\mathbb{Z} ^d}$ is a frame if $\Vert A^{-j}A'_j - I\Vert _2\le \varepsilon$and $ \Vert\lambda_n - n\Vert _{\infty} \le \eta$ for sufficiently small $\varepsilon, \eta>0$.


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Additional Information

Ole Christensen
Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
Email: Ole.Christensen@mat.dtu.dk

Wenchang Sun
Affiliation: Department of Mathematics and LPMC, Nankai University, Tianjin 300071, People’s Republic of China – and – NUHAG, Department of Mathematics, University of Vienna, Nordbergstrasse 15, A-1090 Vienna, Austria
Email: sunwch@nankai.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08134-7
PII: S 0002-9939(05)08134-7
Keywords: Wavelet frames, stability, matrix dilation
Received by editor(s): June 21, 2004
Received by editor(s) in revised form: October 21, 2004
Published electronically: July 20, 2005
Additional Notes: This work was done while the second author was visiting the Department of Mathematics, Technical University of Denmark. He thanks the Department for hospitality and support. He is supported partially by the National Natural Science Foundation of China (10201014), Program for New Century Excellent Talents in University, and the Research Fund for the Doctoral Program of Higher Education.
Communicated by: David R. Larson
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.