Pointwise convergence of wavelet expansions

Abstract

In this note we announce that under general hypotheses, wavelet-type expansions (of functions in $L^p,\ 1\leq p \leq \infty$, in one or more dimensions) converge pointwise almost everywhere, and identify the Lebesgue set of a function as a set of full measure on which they converge. It is shown that unlike the Fourier summation kernel, wavelet summation kernels $P_j$ are bounded by radial decreasing $L^1$ convolution kernels. As a corollary it follows that best $L^2$ spline approximations on uniform meshes converge pointwise almost everywhere. Moreover, summation of wavelet expansions is partially insensitive to order of summation. \footnote We also give necessary and sufficient conditions for given rates of convergence of wavelet expansions in the sup norm. Such expansions have order of convergence $s$ if and only if the basic wavelet $ψ$ is in the homogeneous Sobolev space $H^{-s-d/2}_h$. We also present equivalent necessary and sufficient conditions on the scaling function. The above results hold in one and in multiple dimensions.