The Nevanlinna matrix of the truncated Hausdorff matrix moment problem via orthogonal matrix polynomials on [a,b] for the case of an even number of moments

Abstract

The scalar moment problem was first introduced by T. J. Stieltjes in his work ``Recherches sur les fractions continues'' Annals of the Faculty of Sciences of Toulouse 8, 1--122, (1895). He formulated it as follows: Given the moments of order $k$ ($k=0,1,2,\dots$), find a positive mass distribution on the half-line $[0,+\infty)$. The study of matrix and operator moment problems was initiated by M. G. Krein in his seminal paper ``Fundamental aspects of the representation theory of Hermitian operators with deficiency index $(m,m)$'' Translations of the American Mathematical Society, Series II, 97, 75--143, (1949). This paper is related to the truncated Hausdorff matrix moment (THMM) problem: the truncated moment problem on a compact interval $[a,b]$ in contrast to the Stieltjes moment problem on $[0,+\infty)$ and the Hamburger moment problem on $(-\infty,+\infty)$. Our approach relies on V. P. Potapov’s method, which reformulates interpolation and moment problems as equivalent matrix inequalities and introduces auxiliary matrices that satisfy the $\widetilde{J}_q$--inner function property of the Potapov class, together with a system of column pairs.
The method begins by constructing Hankel matrices from the prescribed moments. If these matrices are positive semidefinite, the THMM problem is solvable. In the strictly positive definite case, known as the non-degenerate case, we transform the associated matrix inequalities to derive the Nevanlinna (or resolvent) matrix of the THMM problem, which characterizes its solutions. This framework has been extensively applied, for instance in A. E. Choque Rivero, Yu. M. Dyukarev, B. Fritzsche, and B. Kirstein, ``A truncated matricial moment problem on a finite interval'', in Interpolation, Schur Functions and Moment Problems, Operator Theory: Advances and Applications, Birkh\"{a}user, Basel, 165, 121--173, (2006). The main contribution of the present work is to represent the Nevanlinna matrix of the THMM problem in terms of orthogonal matrix polynomials (OMP) and their associated polynomials of the second kind at point $b$. Note that the representation at point $a$ was obtained earlier in A. E. Choque Rivero, ``From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem'' Bulletin of the Mexican Mathematical Society, 21(2), 233--259 (2015). In addition, we establish new identities involving OMP and reformulate an explicit relationship between the Nevanlinna matrices of the THMM problem at points $a$ and $b$, through OMP.