LOCAL CONVERGENCE FOR WAVELET EXPANSIONS

Wavelets provide a new class of orthogonal expansions in L(2)(R(d)) with good time/frequency localization and regularity/approximation properties. They have been successfully applied to signal processing, numerical analysis, and quantum mechanics. We study pointwise convergence properties of wavelet expansions and show that such expansions (and more generally, multiscale expansions) of LP functions (1 less than or equal to p less than or equal to infinity) converge pointwise almost everywhere, and more precisely everywhere on the Lebesgue set of the function being expanded. We show that such convergence is partially insensitive to the order of summation of the expansion. It is shown that unlike Fourier series, a wavelet expansion has a summation kernel which is absolutely bounded by dilations of a radial decreasing L(1) convolution kernel H(\x - y\). This fact provides another proof of L(p) convergence. These results hold in all dimensions, and apply to related multiscale expansions, including best approximations using spline functions. (C) 1994 Academic Press, Inc.