Lutz Klotz

A Duality Method in Prediction Theory of Multivariate Stationary Sequences

Let W be an integrable positive Hermitian q × q–matrix valued function on the dual group of a discrete abelian group G such that W–1 is integrable. Generalizing results of T. Nakazi [N] and of A. G. Miamee and M. Pourahmadi [MiP] for q = 1 we establish a correspondence between trigonometric approximation problems in L2(W) and certain approximation problems in L2(W–1). The result is applied to prediction problems for q–variate stationary processes over G, inparticular, to the case G = ℤ.

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

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.

An Addendum to a Paper by Li and Zhang

Recently C.-K. Li and F. Zhang gave a short proof of S.W. Drury’s result that the trace norm preserves positive semidefiniteness of 3 ×3 partitioned positive semidefinite matrices. Slightly generalizing Li and Zhang’s proof we obtain a more general result under the assumption that all matrices on the principal diagonal are of the same size. Also, we answer a question asked by Marcus and Watkins in 1971.

An Operator-theoretic Approach to a Multiple Point Nevanlinna-Pick Problem for Generalized Carathéodory Functions

We study a Nevanlinna-Pick type interpolation problem for matrix-valued generalized Caratheodory functions, where the values of the function and the values of its derivatives up to a certain order are prescribed at finitely many points of the open unit disk. Under the assumption that the generalized Schwarz-Pick-Potapov block matrix, which is associated to the given data, is non-singular we establish a correspondence between the set of solutions of the problem and the set of minimal unitary extensions of a certain isometry in a Pontryagin space, which is one-to-one modulo unitary equivalence.

Completion of Non-negative Block Operators in Banach Spaces

AbstractLet the Banach space

Gleitende-Mittel-Darstellungen für Banach-Raum-wertige homogene verallgemeinerte zufällige Felder

\mathcal{B}

Lieb functions and m-positivity of norms

be the direct sum of three Banach spaces