Generalized Inverses in Reproducing Kernel Spaces: An Approach to Regularization of Linear Operator Equations

Abstract

In this paper a study of generalized inverses of linear operators in reproducing kernel Hilbert spaces (RKHS) is initiated. Explicit expressions for generalized inverses and minimal-norm solutions of linear operator equations in RKHS are obtained in several forms. The relation between the regularization operator of the equation $Af = g$ and the generalized inverse of the operator A in RKHS is demonstrated. In particular, it is shown that they are the same if the range of the operator is closed in an appropriate RKHS. Finally, properties of the regularized pseudosolutions in this setting are studied.