Theory of reproducing kernels for Hilbert spaces of vector valued functions

Abstract

The general theory of reproducing kernels developed by N. Aronszajn provides a unifying point of view for the study of an important class of Hilbert spaces of real or complex valued functions and for the appli­cation of the methods of Hilbert space theory to different problems in the theory of partial differential equations. With a view to applications to systems of such equations the form which the theory takes in the case of spaces of vector valued functions was inves­tigated, initially for finite dimensional and Hilbert range spaces. It was found that the natural setting for such a generalization of the theory is that in which the functions of the functional Hilbert space take their values in an arbitrary locally convex linear topological space, since all of the main re­sults are essentially preserved in that setting and a more special case would restrict unduly the applications. The present study is confined to the exposition of the general theory with a few illustrations and undertakes to extend the basic notions of proper functional space, reproducing kernel and positive matrix and their proper­ ties as they occur in the paper of N. Aronszajn.