## Stability of wavelet frames with matrix dilations

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- by Ole Christensen and Wenchang Sun PDF
- Proc. Amer. Math. Soc.
**134**(2006), 831-842 Request permission

## Abstract:

Under certain assumptions we show that a wavelet frame \[ \{\tau (A_j,b_{j,k})\psi \}_{j,k\in \mathbb {Z}}:= \{|\det A_j |^{-1/2} \psi (A_j^{-1}(x-b_{j,k}))\}_{j,k\in \mathbb {Z}}\] 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 $\|A^{-j}A’_j - I\|_2\le \varepsilon$ and $\|\lambda _n - n\|_{\infty } \le \eta$ for sufficiently small $\varepsilon , \eta >0$.## References

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## Additional Information

**Ole Christensen**- Affiliation: Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark
- MR Author ID: 339614
- 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
- ORCID: 0000-0002-5841-9950
- Email: sunwch@nankai.edu.cn
- 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
- © Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc.
**134**(2006), 831-842 - MSC (2000): Primary 42C40, 41A58
- DOI: https://doi.org/10.1090/S0002-9939-05-08134-7
- MathSciNet review: 2180901