Ukrainian Mathematical Bulletin

Volume 4, Issue 2, 2007 pp. 235-263.

Rational Approximations and Strong Moment Problem

Authors: Kirill K. Simonov
Author institution: Donetsk National University, ul. Universitetskaya 24, 83055, Donetsk, Ukraine xi@gamma.dn.ua

Summary: The \emph{Hamburger moment problem} is a problem where, for a given moment sequence $\seq{S_k}$, one needs to find a measure $d\Sigma$ satisfying $S_k=\infint t^k\,d\Sigma(t)$. If the indices $k$ take not only positive but also negative values, then the moment problem is called \emph{strong}. A strong moment problem is closely related to \emph{two-point Pad\'e approximants} that are rational approximations that approximate, simultaneously, two given series in points $\lambda=\infty$ and $\lambda=0$, correspondingly. In this paper, we consider a \emph{strong truncated matrix Hamburger moment problem}, which means that the indices $k$ range in the set $-2\mu_-\le k \le 2\mu_+$, and the moments $S_k$ are self-adjoint matrices. We will find conditions for solvability and uniqueness of a solution of this problem, and give a description of all solutions in terms of self-adjoint extensions of a certain model symmetric operator. Moreover, we construct a sequence of two-point diagonal Pad\'e approximants corresponding to the strong moment problem, and study convergence of this sequence. Finally, we factorize the resolvent matrix of the strong truncated moment problem.

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