Square Integrable and Holomorphic Kernels

Abstract

In this chapter, we study two special types of reproducing kernel Hilbert spaces, which are probably the most widely occurring types in the physical literature. While the very general reproducing kernel Hilbert spaces, constructed in the last chapter, were spaces of vector-valued functions, they were not assumed to be Hilbert spaces of square integrable functions, with respect to any measure. Most reproducing kernel Hilbert spaces that arise in physics and in group representation theory do, on the other hand, turn out to be spaces of square integrable fuctions. Another widely occurring variety of reproducing kernel Hilbert spaces are spaces of holomorphic or square integrable holomorphic functions. We look at these two situations more closely in this chapter. Recall from the discussion in Chapter 2 that the family of canonical CS arise from a reproducing kernel Hilbert space of square integrable functions, and, indeed, they may also be associated to a space of analytic functions (the Bargmann space).KeywordsHilbert SpaceHolomorphic FunctionCoherent StateMoment ProblemIntegrable KernelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.