Extremal problems for matrix-valued polynomials on the unit circle and applications to multivariate stationary sequences Original Research Article

The paper is devoted to a matrix generalization of a problem studied by Grenander and Rosenblatt (Trans. Amer. Math. Soc. 76 (1954) 112–126) and deals with the computation of the infimum Δ of ∫T Q∗(z)M(dz)Q(z), where M is a non-negative Hermitian matrix-valued Borel measure on the unit circle T and Q runs through the set of matrix-valued polynomials with prescribed values of some of their derivatives at a finite set J of complex numbers. Under some additional assumptions on M and J, the value of Δ is computed and the results are applied to linear prediction problems of multivariate weakly stationary random sequences. A related truncated problem is studied and further extremal problems are briefly discussed.