Factorization of Positive Definite Kernels

Abstract

In this chapter we extend the Cholesky factorization for positive block matrices to the case of positive definite kernels. The basic construction exploits the Kolmogorov decomposition and the role of the Schur parameters is also emphasized. Several particular cases are discussed in some details. The connections between Szegö polynomials and factorization are presented in Section 3, together with the asymptotic properties of the orthogonal polynomials and properties of the associated Schur parameters. In the last section we explore the connection between the spectral factorization and the maximum entropy principle.