The operator Fejer-Riesz theorem

TL;DR

This paper surveys various generalizations of the Fejer-Riesz theorem, focusing on Rosenblum's operator extension and its developments in multiple variables and matrix-valued functions.

Contribution

It provides a comprehensive overview of both historical and recent advancements related to Rosenblum's operator generalization of the classical theorem.

Findings

Survey of classical and modern generalizations

Highlights Rosenblum's operator extension

Discusses multivariable and matrix-valued function cases

Abstract

The Fejer-Riesz theorem has inspired numerous generalizations in one and several variables, and for matrix- and operator-valued functions. This paper is a survey of some old and recent topics that center around Rosenblum's operator generalization of the classical Fejer-Riesz theorem.