PERTURBATION OF WAVELET FRAMES AND RIESZ BASES I

Title & Authors
PERTURBATION OF WAVELET FRAMES AND RIESZ BASES I
Lee, Jin; Ha, Young-Hwa;

Abstract
Suppose that $\small{\psi{\;}\in{\;}L^2(\mathbb{R})}$ generates a wavelet frame (resp. Riesz basis) with bounds A and B. If $\small{\phi{\;}\in{\;}L^2(\mathbb{R})}$ satisfies $$\small{\mid}$\^{\psi}(\xi)\;\^{\phi}(\xi)$\small{\mid}${\;}<{\;}{\lambda}\frac{$\small{\mid}$\xi$\small{\mid}$^{\alpha}}{(1+$\small{\mid}$\xi$\small{\mid}$)^{\gamma}}$ for some positive constants $\small{\alpha,{\;}\gamma,{\;}\lambda}$ such that $1{\;}<1{\;}+{\;}\alpha{\;}<{\;}\gamma{\;}and{\;}{\lambda}^2M{\;}<{\;}A$, then $\small{\phi}$ also generates a wavelet frame (resp. Riesz basis) with bounds $\small{A(1{\;}-{\;}{\lambda}\sqrt{M/A})^2{\;}and{\;}B(1{\;}+{\;}{\lambda}\sqrt{M/A})^2}$, where M is a constant depending only on $\small{\alpha,{\;}\gamma}$ the dilation step a, and the translation step b.
Keywords
wavelet;frame;Riesz basis;perturbation;stability;
Language
English
Cited by
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