Markov theorem for weight functions on the unit circle

TL;DR

This paper extends Markov's theorem, originally for real-line orthogonal polynomials, to paraorthogonal polynomials on the unit circle, showing the theorem's broader applicability in complex analysis.

Contribution

It proves that Markov's theorem on zero variation applies to paraorthogonal polynomials on the unit circle, expanding its scope beyond real-line cases.

Findings

Markov's theorem is valid for paraorthogonal polynomials on the unit circle.

Zero variation properties hold in the complex domain.

The result bridges real-line and complex orthogonal polynomial theories.

Abstract

The aim of this paper is to prove that Markov's theorem on variation of zeros of orthogonal polynomials on the real line [Math. Ann., 27:177-182,1886] remains essentially valid in the case of paraorthogonal polynomials on the unit circle.