Perturbation of Wavelet Frames and Riesz Bases I
Commun. Korean Math. Soc. 2004 Vol. 19, No. 1, 119-127
Printed March 1, 2004
Jin Lee, Young-Hwa Ha
Ajou University, Ajou University
Abstract : Suppose that $\psi\in L^2(\mathbb{R})$ generates a wavelet frame (resp. Riesz basis) with bounds $A$ and $B$. If $\phi\in L^2(\mathbb{R})$ satisfies $| \widehat{\psi}(\xi) - \widehat{\phi}(\xi) | < \lambda \frac{| \xi |^\alpha } { ( 1 + | \xi | )^\gamma} $ for some positive constants $\alpha , \gamma , \lambda$ such that $1< 1+\alpha < \gamma $ and $\lambda^2 M< A $, then $\phi$ also generates a wavelet frame (resp. Riesz basis) with bounds $A \left ( 1- \lambda \sqrt { M/A} \right )^2 $ and $B \left ( 1+ \lambda \sqrt { M/B} \right )^2,$ where $M$ is a constant depending only on $\alpha,\gamma,$ the dilation step $a$, and the translation step $b$.
Keywords : wavelet, frame, Riesz basis, perturbation, stability
MSC numbers : 42C15, 41A30
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