Problem 1.7

Schur extremal problems

Lev Sakhnovich

Courant Institute of Mathematical Science

New York, NY 11223

USA

[email protected]

1 DESCRIPTION OF THE PROBLEM

In this paper we consider the well-known Schur problem the solution of which satisfy in addition the extremal condition

where w(z) and min are m × m matrices and min > 0. Here the matrix min is defined by a certain minimal-rank condition (see Definition 1). We remark that the extremal Schur problem is a particular case. The general case is considered in book [1] and paper [2]. Our approach to the extremal problems does not coincide with the superoptimal approach [3], [4]. In paper [2] we compare our approach to the extremal problems with the superoptimal approach. Interpolation has found great applications in control theory [5], [6].

Schur Extremal Problem: The m×m matrices a0, a1, ..., an are given.

Describe the set of m×m matrix functions w(z) holomorphic in the circle |z| < 1 and satisfying the relation