Convergence Estimates for the Wavelet Galerkin Method

Abstract

This paper presents an analysis of the Galerkin approximation of a time dependent initial value problem correctly posed in the Petrovskii sense. The approximating spaces $\mathcal {V}_h $ are spanned by translations and dilations of a single function $\Phi $, and Fourier techniques are used to analyze the accuracy of the method. This kind of procedure has already been applied in the literature for spline approximations. Our purpose here is to point out that the same methodology can be used for wavelet-based methods since the hypotheses required are automatically satisfied in the context of wavelet analysis. For instance, the basic function $\Phi $ is supposed to be regular, which means that $\Phi $ and all its derivatives up to a certain order should have fast decay at infinity. $\Phi $ also must satisfy the so-called Strang and Fix condition, which guarantees that smooth functions can be approximated from $\mathcal {V}_h $ with good accuracy. This class of functions includes not only the B-splines but all regular scaling functions related to orthogonal wavelet basis. We also analyze here two cases of initial approximate schemes: $L^2 $ orthogonal projection and interpolation on the mesh points.