This book grew out of notes for seminars and courses on reproducing kernel Hilbert spaces taught at the University of Houston beginning in the 1990s.

The study of reproducing kernel Hilbert spaces grew out of work on integral operators by J. Mercer in 1909 and S. Bergman's [4] work in complex analysis on various domains. It was an idea that quickly grew and found applications in many areas.

The basic theory of reproducing kernel Hilbert spaces (RKHS) goes back to the seminal paper of Aronszajn [2]. In his paper, Aronszajn laid out the fundamental results in the general theory of RKHS. Much of the early part of this book is an expansion of his work.

The fascination with the subject of RKHS stems from the intrinsic beauty of the field together with the remarkable number of areas in which they play a role. The theory of RKHS appears in complex analysis, group representation theory, metric embedding theory, statistics, probability, the study of integral operators, and many other areas of analysis. It is for this reason that in our book the theory is complemented by numerous and varied examples and applications.

In this book we attempt to present this beautiful theory to as wide an audience as possible. For this reason we have tried to keep much of the book as self-contained as we could. This led us to rewrite considerable parts of the theory, and experts will recognize that many proofs that appear here are novel.

Our book is composed of two parts.

In Part I we present the fundamental theory of RKHS. We have attempted to make the theory accessible to anyone with a basic knowledge of Hilbert spaces. However, many interesting examples require some background in complex variables and measure theory, so the reader might find that they can follow the theory, but then need further knowledge for the details of some examples.

In Part II we present a variety of applications of RKHS theory. We hope that these applications will be interesting to a broad group of readers, and for this reason we have again tried to make the presentation as accessible as possible. For example, our chapter on integral operators gives a proof of Mercer's theorem that assumes no prior knowledge of compact operators.