Asymptotic theory of filtering for linear operator equations with discrete noisy data

Abstract

We consider Fredholm integral equations of the first kind with continuous kernels in which the data is discretely sampled and contaminated by white noise. A sufficient condition for the convergence of general filtering methods applied to such equations is derived. The condition in essence relates the decay rate of the singular values of the integral operator to the shape of the filter function used in the regularization method. Specific illustrations of the condition are given for Tikhonov regularization, the truncated singular value decomposition, and Landweber iteration.