Ukrainian Mathematical Bulletin
Volume 2, Issue 3, 2005 pp. 351-367.
On the Moment Problem of Discrete Sobolev TypeAuthors: Sergey M. Zagorodnyuk
Author institution: V.N.Karazin Kharkov National University 4 Svoboda Sq., 61077, Kharkov, Ukraine
Summary: In the present work, we study the following moment problem: to find a left-continuous function $\sigma (\lambda)$, $\sigma(0)=0$, nondecreasing on the real axis and a real symmetric nonnegative matrix $M$ such that $$ \int\limits_{\mathbb{R}} \lambda^{n+m} d\sigma (\lambda) + (\lambda^n, (\lambda^n)',\ldots, (\lambda^n)^{(N-1)} ) M \left(\begin{smallmatrix} \lambda^m\\ (\lambda^m)'\\ \vdots\\ (\lambda^m)^{(N-1)} \end{smallmatrix}\right)\Bigg|_{\lambda=0} = s_{n,m}, $$ $n,m \in {\mathbb{Z}}_+$, where $\{s_{n,m} \}_{n,m=0}^\infty$ is a given sequence of real numbers, $N\in {\mathbb N}$. We establish necessary and sufficient conditions of solvability of this problem and describe all solutions of the problem. For $N=2$, we obtain simpler conditions of solvability. In the case where the matrix $M$ is sought in the diagonal form, the necessary and sufficient conditions of solvability of the problem are established and all solutions of the problem are described for any $N$.
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