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

Abstract

Continuous analogs of orthogonal polynomials on the circle are solutions of a canonical system of differential equations, introduced and studied by Krein and recently generalized to matrix systems by Sakhnovich. We prove that the continuous analogs of the adjoint polynomials converge 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 Szegö–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.