ON THE EIGENFUNCTION-EXPANSIONS ASSOCIATED WITH FREDHOLM INTEGRAL-EQUATIONS OF FIRST KIND IN THE PRESENCE OF NOISE

In this paper we consider the eigenfunction expansions associated with Fredholm integral equations of first kind when the data are perturbed by noise. We prove that these expansions are asymptotically convergen t, in the sense of L(2)-norm, when the bound of the noise tends to zer o. This result allows us to construct a continuous mapping from the da ta space to the solution space, without using any constraint or a prio ri bound. We can also show a probabilistic version of this result, whi ch is based on the order-disorder transition in the Fourier coefficien ts of the noisy data. From these results one can derive algorithms and in particular statistical methods able to furnish approximations of t he solution without any use of prior knowledge. (C) 1996 Academic Pres s, Inc.