On the extension for Toeplitz matrices of certain Markov inequalities

TL;DR

This paper extends classical Markov inequalities related to Hankel matrices and orthogonal polynomials to Toeplitz matrices, focusing on sequences with Pólya frequency properties and involving CD kernels and paraorthogonal polynomials.

Contribution

It introduces a novel extension of Markov inequalities to Toeplitz matrices using advanced tools like CD kernels and paraorthogonal polynomials, broadening the scope of classical results.

Findings

Extended Markov inequalities to Toeplitz matrices.

Established connections with Pólya frequency sequences.

Analyzed properties of orthogonal polynomials on the unit circle.

Abstract

Starting from a doubly infinite sequence of complex numbers, the aim of this paper is to extend certain Markov inequalities for the determinant of Hankel matrices and the zeros of the corresponding orthogonal polynomials on the real line (A. Markov in Notes of the Imperial Academy of Sciences, St. Petersburg, 74 (Appendix n. 2) (1894) 1-30. English translation, by J. Shohat, Duke Math. J. 7 (1940), 85-96) to the Toeplitz case, where the central role is played by CD kernels and paraorthogonal polynomials on the unit circle. In particular, we consider the case in which the starting sequence is a two-sided P\'olya frequency sequence.