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Stability of wavelet frames with matrix dilations

Authors:	Ole Christensen Wenchang Sun

Institution:	Department of Mathematics, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark ; 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

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 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 and an invertible matrix , 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 $\varepsilon, \eta>0$ .

Keywords:	Wavelet frames stability matrix dilation

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