Smooth Wavelet Decompositions with Blocky Coecient Kernels

We describe bases of smooth wavelets where the coe cients are obtained by integration against ( nite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were rst developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approach emphasizes the idea of average-interpolation { synthesizing a smooth function on the line having prescribed boxcar averages { and the link between average-interpolation and Dubuc-Deslauriers interpolation. We also emphasize characterizations of smooth functions via their coe cients. We describe boundary-corrected expansions for the interval, which have a simple and revealing form. We use these results to re-interpret the empirical wavelet transform { i.e. nite, discrete wavelet transforms of data arising from boxcar integrators (e.g. CCD devices). x