The Transformation of Issai Schur and Related Topics in an Indefinite Setting

Abstract

We review our recent work on the Schur transformation for scalar generalized Schur and Nevanlinna functions. The Schur transformation is defined for these classes of functions in several situations, and it is used to solve corresponding basic interpolation problems and problems of factorization of rational J-unitary matrix functions into elementary factors. A key role is played by the theory of reproducing kernel Pontryagin spaces and linear relations in these spaces.

Boundary interpolation and rigidity for generalized Nevanlinna functions

“…It can be shown that any Θ ∈ U z1 can be written as a minimal product of elementary factors; see [2] and [20] for the case z 1 = ∞.…”

Boundary Triplets, Weyl Functions, and the Kreĭn Formula

“…Application of Theorem 10 yields a description of the setˆÄ.s/. This choice of the basis fP n g 1 nD0 is motivated by the step-by-step process of solving the indefinite moment problem studied in [6,27], which leads to the expansion of a solution ' 2ˆÄ.s/ in a continued P -fraction. Application of the operator approach to the indefinite moment problem allowed to prove the convergence of Pade approximants for a large class of N Ä -functions [28].…”

Indefinite Inner Product Spaces, Schur Analysis, and Differential Equations

“…Heinz proposed that Aad, Henk, their student Piet Bruinsma and this editor consider the interpolation problem using Krein's formula and the theory of resolvent matrices for the description of the self-adjoint extensions of a given Hermitian operator (see [22,33]); this led in particular to the publications [4,5]. A bit later, collaboration between Aad, Heinz and DA began (mainly on the Schur algorithm for generalized Schur functions) and lead to seventeen publications, some of them written in collaboration with Thomas Azizov, R. Buursema, Simeon Reich, David Shoikhet, Yuri Shondin, Dan Volok, and Gerald Wanjala; see for instance [2,3,8,9,10]. The encounter, and the subsequent collaboration with Heinz was fascinating on numerous grounds.…”