Positive definite Generalized Toeplitz Kernels (GTK) appear in a natural way in some problems of harmonic analysis (Hilbert Transformation, Theorem of Helson and Szege), in scattering theory, theory of generalized stochastic processes. Many classical interpolation problems can be reduced to the study of positive definite GTK.
We discuss matrix-valued GTK. We prove that for kernel K(t,s) to be positive definite GTK it is necessary and sufficient to admit Bochner-type integral representation. From this representation it follows that any positive definite GTK which was initially defined either on a finite interval or on positive semi-axis of the real axis extends to the whole axis with preserving its structure and positive definiteness. This theorem contains as particular cases theorems of S. Bochner and M. G. Krein.
We discuss conditions of uniqueness of extension and description of all extensions if these conditions are not fulfilled. We also discuss application of these results to the Generalized Nehari Problem.
In our consideration we use theory of selfadjoint extensions of symmetric operators and technique of rigged Hilbert Spaces.