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 convergent, in the sense of L2-norm, when the bound of the noise tends to zero. This result allows us to construct a continuous mapping from the data space to the solution space, without using any constraint or a priori bound. We can also show a probabilistic version of this result, which is based on the order]disorder transition in the Fourier coefficients of the noisy data. From these results one can derive algorithms and in particular statistical methods able to furnish approximations of the solution without any use of prior knowledge. Q 1996 Academic Press, Inc.