Reproducing Kernel Hilbert spaces of analytic functions

Abstract

This thesis focuses on the study of reproducing kernel Hilbert spaces (RKHS), which play a fundamental role in complex analysis as well as in other fields such as statistics and machine learning. It presents the general theory of RKHS, including key results that characterize these spaces and their properties, such as the representation of kernels in terms of orthonormal bases. The study then focuses on examples of RKHS consisting of holomorphic functions, such as the Hardy, Bergman, Dirichlet, and Fock spaces, for which the reproducing kernels are computed and their RKHS structure is established. One of the key aspects of the work is the study of multipliers, functions that preserve the structure of the space under pointwise multiplication. The thesis concludes with an analysis of the Nevanlinna-Pick interpolation problem in the unit disk, a classical topic in complex analysis, providing a proof based on analytic techniques and Blaschke products.