The Segal-Bargmann Transform on Euclidean Space and Generalizations

The main topic of this lecture note is the interplay between real and complex analysis using the heat equation and the Segal-Bargmann transform. The Segal-Bargmann transform or the heat transform, maps a square integrable function on a Euclidean space to a solution to the heat equation and this solution can be extended holomorphically to the complexification. The main ideas are introduced in the settings of a Euclidean space where only basic knowledge of analysis is needed. Those ideas are then presented in a more general and abstract framework. The connection to representation theory and infinite dimensional analysis is also discussed. We discuss the heat equation and the Segal-Bargmann transform on Riemannian symmetric spaces and the more general framework of the Heckmann-Opdam theory of hypergeometric functions associated to root systems. This is the only available textbook where all of those aspects of the theory are discussed. The first part of the text is on the level of advanced undergraduate students and beginning graduate students.
The second half is more advanced and leads the reader into the theory of Hilbert spaces of holomorphic functions and abstract harmonic analysis.