A note on the theorems of M. G. Krein and L. A. Sakhnovich on continuous analogs of orthogonal polynomials on the circle

TL;DR

This paper examines continuous analogs of orthogonal polynomials on the circle, proving convergence properties and identifying inaccuracies in foundational theorems by Krein and Sakhnovich.

Contribution

It clarifies the convergence behavior of continuous orthogonal polynomial analogs and corrects inaccuracies in Krein's and Sakhnovich's original theorems.

Findings

Convergence of the continuous analog of adjoint polynomials in the upper half-plane for L^2 coefficients.

Limit of the analog can only be defined up to a constant multiple in general.

Identifies and clarifies an inaccuracy in Krein's and Sakhnovich's original results.

Abstract

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by M.G.Krein and recently generalized to matrix systems by L.A.Sakhnovich. We prove that the continuous analog of the adjoint polynomials converges in the upper half-plane in the case of L^2 coefficients, but in general the limit can be defined only up to a constant multiple even when the coefficients are in L^p for any p>2, the spectral measure is absolutely continuous and the Szego-Kolmogorov-Krein condition is satisfied. Thus we point out that Krein's and Sakhnovich's papers contain an inaccuracy, which does not undermine known implications from these results.