A New Discretization Principle in Analysis

Abstract

First, we give the basic concepts and a short history on the reproducing kernels in connection with new discretization methods. We state the main principles of a reproducing kernel theory and its basic theory: inversions of linear mappings, identification of linear systems, ill-posed problems, Kolmogorov factorization theorem, nonlinear mappings, identification of nonlinear systems. Furthermore, we refer to stochastic theory, support vector machine and feature space. Next, we introduce its basic general applications to bounded linear operator equations on Hilbert spaces combined with the Tikhonov regularization method. Here, we introduce a historically difficult problem on numerical and real inversion formula of the Laplace transform as a typical concrete result. In the last part of the paper, we describe a very recent new idea and method for a global discretization for analytical problems, together with very concrete results.