The best approximate solution of Fredholm integral equations of the first kind via Gaussian process regression

Abstract

In this paper, the best approximate solution of Fredholm integral equations of the first kind with some scattered observations is studied. An explicit approximate solution has been obtained by our proposed method and converges to the exact solution with minimum norm of the integral equations with probability 1, which is identical with the solution by means of the regularization method. In addition, based on the H-Hk formulation, the gap between the range space and its closure is fully characterized, in which the infinite reproducing kernel Hilbert spaces are strictly embedded. We prove that if the right-hand term of the integral equation lies in this gap, the H-norm of the best approximate solution will diverge to infinity, as the number of observations increases. Finally, an integral equation confirms our conclusions.