We study the truncated matrix trigonometric problem of moments. A parametrization of all solutions of this problem (both in the nondegenerate and degenerate cases) is obtained by using the operator approach. This parametrization establishes the one-to-one correspondence between a certain class of analytic functions and all solutions of the problem. We use the important Chumakin results on the generalized resolvents of isometric operators.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N. I. Akhiezer, Classical Problem of Moments and Some Related Problems of Analysis [in Russian], Fizmatgiz, Moscow (1961).
G.-N. Chen and Y.-J. Hu, “On the multiple Nevanlinna–Pick matrix interpolation in the class φ p and the Carathéodory matrix coefficient problem,” Linear Algebra Appl., 283, 179–203 (1998).
B. Fritzsche and B. Kirstein, “The matricial Carathéodory problem in both nondegenerate and degenerate cases,” in: Operator Theory: Advances and Applications, Vol. 165, Birkhäuser, Basel (2006), pp. 251–290.
M. G. Krein and A. A. Nudel’man, Markov Problem of Moments and Extremal Problems. P. L. Chebyshev’s and A. A. Markov’s Ideas and Problems and Their Subsequent Development [in Russian], Nauka, Moscow (1973).
M. E. Chumakin, “Solutions of the truncated trigonometric problem of moments,” Uchen. Zap. Ul’yan. Ped. Inst., 20, Issue 4, 311–355 (1966).
M. E. Chumakin, “On the generalized resolvents of an isometric operator,” Dokl. Akad. Nauk SSSR, 154, No. 4, 791–794 (1964).
M. E. Chumakin, “Generalized resolvents of isometric operators,” Sib. Mat. Zh., 8, No. 4, 876–892 (1967).
T. Ando, “ Truncated moment problems for operators,” Acta Sci. Math. (Szeged), 31, No. 4, 319–334 (1970).
O. T. Inin, “Truncated matrix trigonometric problem of moments,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 84, No. 5, 49–57 (1969).
Yu. M. Berezanskii, Expansions in Eigenfunctions of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1965).
S. M. Zagorodnyuk, “Positive-definite kernels satisfying difference equations,” Meth. Funct. Anal. Top., 16, No. 1, 83–100 (2010).
S. M. Zagorodnyuk, “On the strong matrix Hamburger moment problem,” Ukr. Mat. Zh., 62, No. 4, 471–482 (2010); English translation: Ukr. Math. J., 62, No. 4, 537–551 (2010).
S. M. Zagorodnyuk, “A description of all solutions of the matrix Hamburger moment problem in a general case,” Meth. Funct. Anal. Topol., 16, No. 3, 271–288 (2010).
Yu. M. Berezansky, “Some generalizations of the classical moment problem,” Integr. Equat. Operator Theory, 44, 255–289 (2002).
M. M. Malamud and S. M. Malamud, “Operator measures in Hilbert spaces,” Alg. Anal., 15, No. 3, 1–52 (2003).
M. Rosenberg, “The square-integrability of matrix-valued functions with respect to a nonnegative Hermitian measure,” Duke Math. J., 31, 291–298 (1964).
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Spaces [in Russian], Gostekhteorizdat, Moscow (1950).
N. Akhiezer and M. Krein, On Some Problems of the Theory of Moments [in Russian], GONTI, Kharkov (1938).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 63, No. 6, pp. 786–797, June, 2011.
Rights and permissions
About this article
Cite this article
Zagorodnyuk, S.M. Truncated matrix trigonometric problem of moments: operator approach. Ukr Math J 63, 914–926 (2011). https://doi.org/10.1007/s11253-011-0552-6
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-011-0552-6