Reproducing Kernels and Inner Product Spaces. Applications

Abstract

The theory of complete inner product spaces with reproducing kernel functions has received a growing interest partly because of its own beauty and partly because of numerous applications in the theory of functions of one or several variables and in the theory of partial differential equations. It seems worthy of notice that, perhaps, the space in which the existence of a reproducing kernel was first recognized was the space of harmonic functions on a complex domain and which are elements of L². This fact was stated by S. Zaremba in 1908. The theory of reproducing kernel functions was decisively influenced by the works of S. Bergman in which the so-called ‘Bergman kernel’ was introduced and studied intensively. The abstract theory of reproducing kernel functions was greatly influenced by the works of Aronszajn (1943, 1950). For more information about the applications of the theory of reproducing kernels we refer to Bergman (1950), Bergman and Schiffer (1951), Epstein (1965) as well as to the references quoted in our bibliography.