  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 $\psi$ 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.
