A Characterization for reproducing kernel Hilbert spaces

Abstract

Let G(t, s) be the Green's functions associated with N, a differential operator restricted to certain boundary conditions. Define (u, v)N = (Nu, v)L2. It is shown that the reproducing kernel Hilbert space generated by G is the same as the Hilbert-space completion with respect to ∥ · ∥N of the set of real valued functions which are in C2n and satisfy the boundary conditions. The concept of Sobolev spaces is used in the proof and examples are given. © 1974.