Matrix Fejér-Riesz theorem with gaps

Abstract

The matrix Fejér-Riesz theorem characterizes positive semidefinite matrix polynomials on the real line $\mathbb{R}$. We extend a characterization to arbitrary closed semialgebraic sets $K\subseteq \mathbb{R}$ by the use of matrix preorderings from real algebraic geometry. In the compact case a denominator-free characterization exists, while in the non-compact case there are counterexamples. However, there is a weaker characterization with denominators in the non-compact case. At the end we extend the results to algebraic curves.

19 pages

References in corpus (2)

• Moment problems for operator polynomials
• A simple proof of the matrix-valued Fejér-Riesz theorem