Stability of wavelet frames and Riesz bases, with respect to dilations and translations

Abstract:

We consider the perturbation problem of wavelet frame (Riesz basis) $\{{\psi _{j,k,a_0,b_0}\}}=\{a_0^{nj/2}\psi (a_0^jx-kb_0)\}$ about dilation and translation parameters $a_0$ and $b_0$. For wavelet functions whose Fourier transforms have small supports, we give a method to determine whether the perturbation system $\{\psi _{j,k,a,b_0}\}$ is a frame (Riesz basis) and prove the stability about dilation parameter $a_0$ on Paley-Wiener space. For a great deal of wavelet functions, we give a definite answer to the stability about translation $b_0$. Moreover, if the Fourier transform $\hat {\psi }$ has small support, we can estimate the frame bounds about the perturbation of translation parameter $b_0$. Our methods can be used to handle nonhomogeneous frames (Riesz basis).