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Stability of wavelet frames and Riesz bases, with respect to dilations and translations

Author: Jing Zhang
Journal: Proc. Amer. Math. Soc. 129 (2001), 1113-1121
MSC (2000): Primary 42C15; Secondary 41A30
Published electronically: December 7, 2000
MathSciNet review: 1814149
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Abstract: We consider the perturbation problem of wavelet frame (Riesz basis) $\{{\psi_{j,k,a_0,b_0}\}}=\{a_0^{nj/2}\psi(a_0^jx-kb_0)\}$ about dilation and translation parameters $a_0$ and $b_0$. For wavelet functions whose Fourier transforms have small supports, we give a method to determine whether the perturbation system $\{\psi_{j,k,a,b_0}\}$ is a frame (Riesz basis) and prove the stability about dilation parameter $a_0$ on Paley-Wiener space. For a great deal of wavelet functions, we give a definite answer to the stability about translation $b_0$. Moreover, if the Fourier transform $\hat{\psi}$ has small support, we can estimate the frame bounds about the perturbation of translation parameter $b_0$. Our methods can be used to handle nonhomogeneous frames (Riesz basis).

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Additional Information

Jing Zhang
Affiliation: Institute of Mathematics, Academia Sinica, Beijing, People’s Republic of China 100080
Address at time of publication: Department of Mathematics, Washington University, Campus Box 1146, One Brookings Drive, St. Louis, Missouri 63130-4899

Keywords: Frames, stability, wavelets
Received by editor(s): July 13, 1998
Received by editor(s) in revised form: July 1, 1999
Published electronically: December 7, 2000
Dedicated: Dedicated to the memory of Professor Long Ruilin
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society