Reproducing kernel Hilbert spaces

Abstract

In this thesis we make a survey of the theory of reproducing kernel Hilbert spaces associated with positive definite kernels and we illustrate their applications for interpolation problems of Nevanlinna-Pick type. Firstly we focus on the properties of reproducing kernel Hilbert spaces, generation of new spaces and relationships between their kernels and some theorems on extensions of functions and kernels. One of the most useful reproducing kernel Hilbert spaces, the Bergman space, is studied in details in chapter 3. After giving a brief definition of Hardy spaces, we dedicate the last part for applications of interpolation problems of NevanlinnaPick type with three main theorems: interpolation with a finite number of points, interpolation with an infinite number of points and interpolation with points on the boundary. Finally we include an Appendix that contains a brief recall of the main results from functional analysis and operator theory.