REPRESENTATION OF MEASURABLE POSITIVE DEFINITE GENERALIZED TOEPLITZ KERNELS IN R

Abstract

We prove that every measurable positive definite generalized Toeplitz Kernel, defined in an (finite or infinite) interval (-a,a), is the sum of a positive definite generalized Toeplitz kernel given by continuous functions and a positive definite generalized Toeplitz kernel which vanishes almost everywhere. The proof is based on the theory of local semigroups of contractions developed in former works. In the case of ordinary Topeplitz kernels this result gives theorems of F. Riesz, M. Krein and M. Crum and a special case of a theorem of Z. Sasvari