Stability of wavelet frames with matrix dilations

Christensen, Ole; Sun, Wenchang

doi:10.1090/S0002-9939-05-08134-7

Stability of wavelet frames with matrix dilations
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by Ole Christensen and Wenchang Sun

Proc. Amer. Math. Soc. 134 (2006), 831-842

DOI: https://doi.org/10.1090/S0002-9939-05-08134-7

Published electronically: July 20, 2005

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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 $\varepsilon , \eta >0$.

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