Semidefinite descriptions of cones defining spectral mask constraints

We discuss in detail an additive structure of cones of trigonometric polynomials nonnegative on the union of a finite number of pairwise disjoint segments of the unit circle. We derive new descriptions of these cones in terms of semidefinite constraints. We explain the results of M. Krein and A. Nudelman [M. Krein, A. Nudelman, The Markov Moment Problem and Extremal Problems, American Mathematical Society, Providence, 1977] providing a description of dual cones in terms of the same type of constraints.