On deconvolution methods

Several methods for solving efficiently the one-dimensional deconvolution problem are proposed. The problem is to solve the Volterra equation ku := ∫ t 0 k(t − s)u(s)ds = g(t), 0 ≤ t ≤ T . The data, g(t), are noisy. Of special practical interest is the case when the data are noisy and known at a discrete set of times. A general approach to the deconvolution problem is proposed: represent k = A(I + S), where a method for a stable inversion of A is known, S is a compact operator, and I + S is injective. This method is illustrated by examples: smooth kernels k(t), and weakly singular kernels, corresponding to Abel-type of integral equations, are considered. A recursive estimation scheme for solving deconvolution problem with noisy discrete data is justified mathematically, its convergence is proved, and error estimates are obtained for the proposed deconvolution method.