On degenerate interpolation, entropy and extremal problems for matrix Schur functions

Abstract

We consider a general bitangential interpolation problem for matrix Schur functions and focus mainly on the case when the associated Pick matrix is singular (and positive semidefinite). Descriptions of the set of all solutions are given in terms of special linear fractional transformations which are obtained using two quite different approaches. As applications of the obtained results we consider the maximum entropy and the maximum determinant extension problems suitably adapted to the degenerate situation.

On boundary interpolation for matrix valued Schur functions

“…Theorem 5.1 was proved in [15] by adapting Potapov's method of Fundamental Matrix Inequalities (FMI's) to the aBIP framework. Additional analysis of the FMI corresponding to the aBIP is given in Section 3.…”

“…In Section 5 we present an alternative parametrization of the set S(M, N, P, C) in terms of a Redheffer linear fractional transformation (see Theorem 5.9 below) which sometimes (especially in the degenerate case) turns to be more convenient for applications (see, e.g., Sections 11 and 12 in [15]). This approach has been adapted in [27] and [15] from the work of Katsnelson, Kheifets and Yuditskii [36] on the abstract interpolation problem and the work of Arov and Grossman [9], [10] on the coupling of unitary colligations.…”

Notes on interpolation in the generalized Schur class. II. Nudel’man’s problem

“…Interpolation of interior data: [1], [4], [10], [11], [12], [16], [17], [19], [20], [22], [26], [31] Loewner theory and boundary data: [5], [7], [11] Parametrization of solutions: [1], [10], [11], [19] Interpolation in the Stieltjes class, including parametrizations: [2], [8] Moment problems: [13], [14], [24] Monotone operator functions: [3] Our generalization of Nudel man's problem and Main Theorem are formulated in Section 2. Sections 3 and 4 contain the applications of the Main Theorem to classical interpolation problems in the disk and half-plane cases, respectively; many of these results parallel the definite case [29].…”

Abstract Interpolation Problem in Nevanlinna Classes

“…Regular bitangential interpolation problems in the Schur and Nevanlinna classes were studied in [36], [9], [20], [28], [8], [6], [3]. Singular tangential and bitangential interpolation problems considered in [24], [36], [20], [14], [12], [21] can be included in the above consideration by imposing the assumption (A2) on the data set.…”

“…The results of the paper are illustrated in Section 5 with an example of bitangential interpolation problems in the classes N d×d and N d×d , reduced there to the AIP with appropriately chosen data set. These problems have been studied earlier in [36], [6], [28], [20], [21], [12]. Mention, that the Arov and Grossman's description of scattering matrices of unitary extensions of an isometry V in [7] used in the Schur type AIP is an analog of M.G.…”