Smooth Wavelet Decompositions with Blocky Coeecient Kernels

We describe bases of smooth wavelets where the coeecients 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 coeecients. 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). x1. Introduction The now-classical orthogonal wavelet transform is based on the calculation of integrals j;k = R j;k (t)f(t)dt with j;k very special L 2 (IR) functions given by dyadic scaling and integer translation of a mother wavelet 43,12]. When the wavelet is smooth and has several vanishing moments, the wavelet coeecients (j;k) can decay rapidly with increasing resolution j, a property which is useful for data compression purposes 2,20]. 1.1. Transforms via digital retinae One can envision situations where technology ooers unique opportunities to eeciently calculate integrals with boxcar functions j;k = 2 j=2 1 k=2 j ;(k+1)=2 j ] ; it is interesting to consider wavelet-like transforms with coeecient functionals derived from those integrals. For a two-dimensional example, consider this caricature of CCD imaging: a \digital retina" is formed from a square array of square cells, each of which contains a miniature device integrating the ux of light falling inside the cell. The cells are connected to a pyramid of circuitry Recent Advances in Wavelet Analysis