On the eigenfunction expansions associated with Fredholm integral equations of first kind in the presence of noise (Q1916707)

The authors consider the problem of approximately solving an integral equation of the first kind with a selfadjoint integral operator of the Hilbert-Schmidt class by eigenfunction expansion, thereby assuming the noise in the data function to be perturbed in absolute value by at most a (small) number \(\varepsilon\). They construct a continuous mapping from the data space to the solution space, without using any a priori bound or constraint. The crucial point is where to truncate the expansion (gained from the perturbed data), i.e. to identify the ``order-disorder transition'' in the sequence of Fourier coefficients of the data function. Furthermore, they develop a probabilistic version of their result, using the theory of (weak) random variables in Hilbert space and concepts from information theory.