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In the past ** Gong-ning Chen** has collaborated on articles with

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AbstractPrevious work (Gong-ning Chen, J. Math. Anal. Appl. 98 (1984), 305–313) on iteration of holomorphic maps of Cn is continued. The purpose of this note is to extend results given in the above mentioned reference to the case of complex Hilbert spaces. Other comments are appended.

AbstractThe concept of a G-function introduced by Nowosad and Hoffman is used to characterize classes of complex square matrices, resulting from various degrees of diagonal dominance associated with G-functions. Their relationship to the set of M-matrices is established.

AbstractThe study of Loewner matrices is continued in a direction close to the recent work by Fiedler, Pták, and Vavřín. All results are stated for the general case of multiple interpolation nodes, i.e., for generalized Loewner matrices. Both the “Loewner-Bézoutian” connection of Antoulas and Anderson and the “Loewner- Hankel” connection of Vavřín in the general case, which are re-proved in a simple way, provide a good opportunity for finding various properties of Loewner matrices, some of them being generalizations of results published by other authors for simple nodes only. There are also results which are new, especially the Barnett-like formulas and an intertwining relation characteristic for Loewner matrices, as well as the “Loewner matrix-Hankel vector” connection. In order to handle Loewner matrices further on, the authors introduce and deal with a class of so-called R-matrices. It turns out that some known results concerning Bezoutians, Hankel matrices, and their products can in a natural way be lifted to the level of R-matrices and of the products of Loewner and R-matrices.

AbstractThe so-called Hankel vector approach is used to handle three boundary versions of the multiple Nevanlinna–Pick interpolation problem in the Nevanlinna class N involving both interior and boundary data. It turns out that each of these boundary interpolation problems can be reduced to what amounts to a certain truncated (standard or nonstandard) Hamburger moment problem, associated with the Hankel vector of the former, with some possible constraint on distribution functions that assign no mass to any of the real nodes. In particular, this leads to solvability criteria for each of these interpolation problems and the description of solutions by using results from theory of moments.

AbstractThis paper presents a unified approach to the solution of the truncated matrix Stieltjes moment problem: Sk=∫0∞ukdσ(u) (k=0,…,m), in both the cases m=2n and m=2n−1, based on the use of the iterative Schur algorithm.

AbstractA matrix version of the boundary Nevanlinna-Pick interpolation problem in the class of Carathéodory matrix functions is considered. This matrix interpolation problem is reduced to a certain matrix trigonometric moment problem with specified constraints that the nonnegative matrix-valued measure has no mass distributions at a finite number of boundary points. Basing on the use of recent results due to Bolotnikov and Dym and this reduction, we obtain solvability criteria for both the boundary Nevanlinna-Pick interpolation problem and the moment problem. A parameterized description of all the solutions of each of these two problems under consideration in the nondegenerate case is given as well.

AbstractA significant extremal question is considered within the solution sets of two different nondegenerate truncated matricial Hamburger moment problems: Taking an arbitrary α∈R whether there exists one and only one solution of each of those two moment problems, which has a concentrated matrix mass at the point α equal to the maximum mass. The main concern of this paper is to indicate that for the first moment problem (Problem (THM′)2n), the answer is “yes”; however, for the second moment problem (Problem (THM)m) the existence is not generally fulfilled, and it holds if and only if α∈R has to satisfy some additional condition. These observations further serve as starting point to look for the corresponding extremal feature for three (nondegenerate) matrix interpolation problems of Nevanlinna–Pick type based on the so-called block Hankel vector approach.

AbstractThe present paper deals simultaneously with the nondegenerate and degenerate truncated Hamburger matrix moment problems in a unified way based on the use of the Schur algorithm involving matrix continued fractions. A full analysis of them together with a relative matrix moment problem on the real axis is given. With the help of the correspondence between the moment problem on the real axis and the Nevanlinna-Pick (NP) interpolation, the solutions of the nontangential NP interpolation in the Nevanlinna class are derived as an application.

AbstractIn the present paper an intrinsic one-to-one correspondence between the (finite) multiple Nevanlinna-Pick matrix interpolation (NP) problem in the Carathéodory class and the Carathéodory matrix coefficient (CC) problem (or, equivalently, the truncated trigonometric matrix moment (TM) problem) is established. Thanks to this correspondence, the NP problem is reduced to what amounts to solving the CC problem (or the TM problem) associated with the so-called Toeplitz block-vector of the former in both the nondegenerate and degenerate cases simultaneously.

AbstractInversion theorems for generalized block Loewner matrices with non-square blocks are presented. Connections are made to the general (i.e. Hermite) matrix rational interpolation, theory of partial realization and its extensions, theoretical and computational aspects of many other special types of matrices such as block Hankel, block Toeplitz, Vandermonde, block Bezout, etc.

AbstractThe objective of this paper is to establish an intrinsic and simple correspondence between a multiple Nevanlinna–Pick matrix interpolation (NP) problem in the Nevanlinna class Np with a denumerable set of nodes and a certain full Hamburger matrix moment problem. With the help of the indicated correspondence, the solution of an infinite NP problem in Np can be reduced to what amounts to the study of the associated full Hamburger matrix moment problem with the so-called block Hankel vector of the former, and vice versa. These investigations can be regarded as a natural extension of our previous work on both the NP problems with a finite set of interpolation data and the truncated power matrix moment problems.

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