Matrix Fejér–Riesz theorem with gaps

The matrix Fejér–Riesz theorem characterizes positive semidefinite matrix polynomials on the real line RR. We extend a characterization to arbitrary closed semialgebraic sets K⊆RK⊆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.