We obtain necessary and sufficient conditions for the solvability of the strong matrix Hamburger moment problem. We describe all solutions of the moment problem by using the fundamental results of A. V. Shtraus on generalized resolvents of symmetric operators.
Similar content being viewed by others
References
W. B. Jones, O. Njåstad, and W. J. Thron, “Continued fractions and strong Hamburger moment problems,” Proc. London Math. Soc., (3) 47, No. 2, 363–384 (1983).
W. B. Jones, W. J. Thron, and O. Njåstad, “Orthogonal Laurent polynomials and the strong Hamburger moment problem,” J. Math. Anal. Appl., 98, No. 2, 528–554 (1984).
O. Njåstad, “Solutions of the strong Hamburger moment problem,” J. Math. Anal. Appl., 197, 227–248 (1996).
W. B. Jones and O. Njåstad, “Orthogonal Laurent polynomials and strong moment theory: a survey,” J. Comput. Appl. Math., 105, No. 1–2, 51–91 (1999).
K. K. Simonov, “Strong matrix moment problem of Hamburger,” Meth. Funct. Anal. Top., 12, No. 2, 183–196 (2006).
S. M. Zagorodnyuk, “Positive-definite kernels satisfying difference equations,” Meth. Funct. Anal. Top., 16, No. 1, 83–100 (2010).
A. V. Shtraus, “Generalized resolvents of symmetric operators,” Izv. Akad. Nauk SSSR, 18, 51–86 (1954).
M. M. Malamud and S. M. Malamud, “Operator measures in a Hilbert space,” Alg. Analiz, 15, No. 3, 1–52 (2003).
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space [in Russian], Gostekhteorizdat, Moscow (1950).
Yu. M. Berezanskii, Expansion in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
M. Sh. Birman and M. Z. Solomyak, Spectral Theory of Self-Adjoint Operators in a Hilbert Space [in Russian], Leningrad University, Leningrad (1980).
M. H. Stone, Linear Transformations in Hilbert Space and Their Applications to Analysis, American Mathematical Society, Providence, RI (1932).
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis [in Russian], Fizmatgiz, Moscow (1961).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 4, pp. 471–482, April, 2010.
Rights and permissions
About this article
Cite this article
Zagorodnyuk, S.M. On the strong matrix Hamburger moment problem. Ukr Math J 62, 537–551 (2010). https://doi.org/10.1007/s11253-010-0370-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-010-0370-2