On an operator approach to interpolation problems for Stieltjes functions

Abstract

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered. IntroductionClassical interpolation problems (Schur, Nevanlinna-Pick, Carathéodory-Féjer problems, the moment problem et.c.) and their various generalizations were studied using several different approaches and methods (see e.g., [8] for short historical survey). It turns out that the interpolation data of each such problem satisfies a certain operator identity. The structure of this identity turns to be similar (and often, the same) for quite different problems. This allows one to consider a whole circle of problems in a unified way [19], [23]. On the other hand, the operator identities themselves are useful for applications [23], [24].In [13] a general interpolation problem generated by an operator identity has been considered in the Nevanlinna class of matrix valued functions analytic and with the nonnegative imaginary part in the upper half-plane C + . It was shown that the classical NevanlinnaPick and Carathéodory-Féjer interpolation problems are particular cases of this problem as well as the Hamburger moment problem, the Krein extension problem for positive functions on the interval and so on. In this paper we consider a similar problem for the Stieltjes functions (see Definition 2.1) which form an important subclass of the Nevanlinna class. The Stieltjes functions and their role in applications have been studied in [14]. The interpolation problems for Stieltjes functions has been considered in [21]

On Distinguished Solutions of Truncated Matricial Hamburger Moment Problems

“…The FMI approach of V. P. Potapov was enriched by L. A. Sakhnovich who introduced a method of operator identities which serves to unify the particular instances of V. P. Potapov's procedure under one framework (see [18,56,88]). …”

“…18) Combining(3.15),(3.13),(3.17),(3.16),(3.11), and (3.18) we obtain−H + n−1 y n,2n−1 I q * H n T n −H + n−1 y n,2n−1 I q = (0 q×nq , L n ) 0 q×q −H + n−1 y n,2n−1 = (L n L + n−1 · 0 q×nq , L n L + n−1 L n−1 ) 0 q×q −H + n−1 y n,2n−1 = L n L + n−1 (0 q×nq , L n−1 ) 0 q×q −H + n−1 y n,2n−1…”

Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations

“…We will use the following definition of the Stieltjes matrix-valued functions introduced by Mark Krein for the scalar case, e.g. see [4,17,1].…”

On Resolvent Matrix, Dyukarev–Stieltjes Parameters and Orthogonal Matrix Polynomials via $$[0, \infty )$$ [ 0 , ∞ ) -Stieltjes Transformed Sequences

“…The matrix version of the classical Stieltjes moment problem was studied in Adamyan/Tkachenko [1,2], Andô [4], Bolotnikov [5][6][7], Bolotnikov/Sakhnovich [8], Chen/Hu [9], Chen/Li [10], Dyukarev [16,17], Dyukarev/Katsnel ′ son [22,23], Hu/Chen [32].…”