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

AbstractThe 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 ∫TQ∗(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

Orthogonal Rational Functions on the Unit Circle with Prescribed Poles not on the Unit Circle

Orthogonal rational functions (ORF) on the unit circle generalize orthogonal polynomials (poles at infinity) and Laurent polynomials (poles at zero and infinity). In this paper we investigate the properties of and the relation between these ORF when the poles are all outside or all inside the unit disk, or when they can be anywhere in the extended complex plane outside the unit circle. Some properties of matrices that are the product of elementary unitary transformations will be proved and some connections with related algorithms for direct and inverse eigenvalue problems will be explained

functions

We study certain sequences of rational functions with poles outside the unit circle. Such kind of sequences are recursively constructed based on sequences of complex numbers with norm less than one. In fact, such sequences are closely related to the Schur-Nevanlinna algorithm for Schur functions on the one hand and on the other hand to orthogonal rational functions on the unit circle. We shall see that rational functions belonging to a Schur-Nevanlinna sequence can be used to parameterize the set of all solutions of an interpolation problem of Nevanlinna-Pick type for Schur functions. Keywords: Nevanlinna-Pick interpolation problem, Schur functions, rationa

Solution of a multiple Nevanlinna-Pick problem for Schur functions via orthogonal rational functions

An interpolation problem of Nevanlinna-Pick type for complexvalued\ud Schur functions in the open unit disk is considered. We prescribe\ud the values of the function and its derivatives up to a certain\ud order at infinitely many points. Primarily, we study the case that\ud there exist many Schur functions fullling the required conditions.\ud For this situation, an application of the theory of orthogonal rational\ud functions is used to characterize the set of all solutions of the problem\ud in question. Moreover, we treat briefl\ud y the case of exactly one\ud solution and present an explicit description of the unique solution in\ud that case