Aljaž Zalar: Matrix Fejér-Riesz theorem with gaps

The classical Fejér-Riesz theorem states that any nonnegative real-valued trigonometric polynomial can be represented as a single Hermitian square of a holomorphic trigonometric polynomial of the same degree. Due to its importance in control theory, the theorem was generalized to matrix-valued trigonometric polynomials in the 1970s. In 2005, the scalar version was further extended to arbitrary closed semialgebraic subsets of the real line using preorderings from real algebraic geometry. In this talk, we present a matrix-valued generalization of this result. Interestingly, in certain cases, working on the dual side, by solving the associated truncated matrix-valued moment problem, yields optimal degree bounds for positivity certificates. These bounds cannot be obtained using the original algebraic techniques.