Convergence estimates for the wavelet-Galerkin method: superconvergence at the node points

In this paper we consider a regular 1-periodic initial value problem and Galerkin approximate solutions in subspaces ν _t spanned by scaled translates of a basic function?. Our goal is to estimate the error when? is in a class of functions which we name . Herer is a regularity parameter andm is related with a property (the Strang and Fix condition) which determines the best order of accuracy in theL ²-norm of approximations from ν _h . Whenm=r, includes all scaling functions corresponding tor-regular multiresolution analyses ofL ²(?). We get the exact node values of the given initial condition as coefficients for the approximate initial data. With this procedure, the coefficients of the resulting Galerkin solution can give a very accurate approximation of the exact solution at the node points, provided that? has many vanishing moments. Since this property is not satisfied in general, we work with another modified basic function?* constructed from the integer translates of?. GlobalL ²-estimates are also obtained.