REPRODUCING KERNEL HILBERT SPACES: The Basics, Bergman Spaces, and Interpolation Problems

This is a series of lectures we have held during the academic year 2004-2005 at the Department of Mathematics of the Bilkent University in the seminar of operator theory. The theory of reproducing kernel Hilbert spaces has important applications to boundary value problems, integral operators, harmonic and analytic functions, in conformal mappings of simply- and multiply-connected domains, in pseudo-conformal mappings, in the study of invariant Riemann metrics, in probability theory, interpolation of functions, and in many other subjects. In this short presentation, we consider an introduction to this subject by emphasizing first the abstract theory, the Bergman kernels, and some of their applications to interpolation of functions in the unit disc. The book is aimed to a broader audience of graduate students, mathematicians, physicists, and engineers, and all those having an interest in getting a quick, but carefully presented, mathematically sound basic knowledge on this domain.