Regularization in a functional reproducing kernel Hilbert space

Abstract

Abstract We consider solving a system of semi-discrete first kind integral equations with a right-hand-side being a finite dimensional vector of sampling values and propose a regularization method for the system in a functional reproducing kernel Hilbert space (FRKHS), where the linear functionals that define the semi-discrete integral operator are continuous. A representer theorem for the regularization method is established, which reduces the infinite dimensional problem to a finite dimensional linear system and expresses its solution as a linear combination of the FRKHS kernel sessions. We construct specific FRKHSs and their associated kernels for reconstruction of a function from Radon data, and develop related regularization methods for the reconstruction. We present numerical results which demonstrate that the proposed regularization method outperforms either the traditional Tikhonov regularization in the L 2 space or the regularization in the classical reproducing kernel Hilbert space.