Perturbation of wavelet frames and Riesz bases I

Abstract

Suppose that psiin L^2(mathbb{R}) generates a wavelet frame(resp. Riesz basis) with bounds A and B. If phiinL^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 Bleft ( 1+ lambda sqrt { M/B} right )^2, where M is aconstant depending only on alpha,gamma, the dilation step a,and the translation step b.