The truncated matrix trigonometric moment problem with an open gap

Sergey Zagorodnyuk

Concrete Operators (2015)

  • Volume: 2, Issue: 1, page 37-46, electronic only
  • ISSN: 2299-3282

Abstract

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This paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, 63, no. 6, 786-797, and Ukrainian Math. J., 2013, 64, no. 8, 1199- 1214. In this paper we shall study the truncated matrix trigonometric moment problem with an additional constraint posed on the matrix measure MT(δ), δ ∈ B(T), generated by the seeked function M(x): MT(∆) = 0, where ∆ is a given open subset of T (called a gap). We present necessary and sufficient conditions for the solvability of the moment problem with a gap. All solutions of the moment problem with a gap can be constructed by a Nevanlinna-type formula.

How to cite

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Sergey Zagorodnyuk. "The truncated matrix trigonometric moment problem with an open gap." Concrete Operators 2.1 (2015): 37-46, electronic only. <http://eudml.org/doc/268869>.

@article{SergeyZagorodnyuk2015,
abstract = {This paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, 63, no. 6, 786-797, and Ukrainian Math. J., 2013, 64, no. 8, 1199- 1214. In this paper we shall study the truncated matrix trigonometric moment problem with an additional constraint posed on the matrix measure MT(δ), δ ∈ B(T), generated by the seeked function M(x): MT(∆) = 0, where ∆ is a given open subset of T (called a gap). We present necessary and sufficient conditions for the solvability of the moment problem with a gap. All solutions of the moment problem with a gap can be constructed by a Nevanlinna-type formula.},
author = {Sergey Zagorodnyuk},
journal = {Concrete Operators},
keywords = {moment problem; generalized resolvent; spectral function; isometric operator; trigonometric moment problem},
language = {eng},
number = {1},
pages = {37-46, electronic only},
title = {The truncated matrix trigonometric moment problem with an open gap},
url = {http://eudml.org/doc/268869},
volume = {2},
year = {2015},
}

TY - JOUR
AU - Sergey Zagorodnyuk
TI - The truncated matrix trigonometric moment problem with an open gap
JO - Concrete Operators
PY - 2015
VL - 2
IS - 1
SP - 37
EP - 46, electronic only
AB - This paper is a continuation of our previous investigations on the truncated matrix trigonometric moment problem in Ukrainian Math. J., 2011, 63, no. 6, 786-797, and Ukrainian Math. J., 2013, 64, no. 8, 1199- 1214. In this paper we shall study the truncated matrix trigonometric moment problem with an additional constraint posed on the matrix measure MT(δ), δ ∈ B(T), generated by the seeked function M(x): MT(∆) = 0, where ∆ is a given open subset of T (called a gap). We present necessary and sufficient conditions for the solvability of the moment problem with a gap. All solutions of the moment problem with a gap can be constructed by a Nevanlinna-type formula.
LA - eng
KW - moment problem; generalized resolvent; spectral function; isometric operator; trigonometric moment problem
UR - http://eudml.org/doc/268869
ER -

References

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  2. [1] Ando T., Truncated moment problems for operators, Acta Sci. Math. (Szeged), 1970, 31, no. 4, 319-334 Zbl0202.13403
  3. [2] Berezanskii Ju.M., Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence, RI, 1968 (Russian edition: Naukova Dumka, Kiev, 1965) Zbl0142.37203
  4. [3] Chen G.-N., Hu Y.-J., On the multiple Nevanlinna-Pick matrix interpolation in the class 'p and the Carathéodory matrix coefficient problem, Linear Algebra Appl., 1998, 283, 179-203 Zbl0948.47018
  5. [4] Chumakin M.E., Solutions of the truncated trigonometric moment problem, Uchen. zapiski Ulyanovskogo pedag. instituta, 1966, 20, issue 4, 311-355, (in Russian) 
  6. [5] Fritzsche B., Kirstein B., Thematricial Carathéodory problem in both nondegenerate and degenerate cases, Oper. Theory Adv. Appl., 2006, 165, 251-290 Zbl1117.44002
  7. [6] Ilmushkin G.M., Turitsyn A.B., A truncated operator trigonometric moment problem, Izv. Vyssh. Uchebn. Zaved. Mat., 1982, no. 7, 17-21, (in Russian) Zbl0511.42018
  8. [7] Inin O.T., A truncated matrix trigonometric moment problem, Izv. Vyssh. Uchebn. Zaved. Mat., 1969, no. 5 (84), 49-57 (in Russian) 
  9. [8] Krein M.G., Nudelman A.A., The Markov moment problem and extremal problems. Ideas and problems of P. L. Cebysev and A. A. Markov and their further development, Translations of Mathematical Monographs. Vol. 50. Providence, R.I., American Mathematical Society (AMS), 1977 
  10. [9] Zagorodnyuk S.M., The truncated matrix trigonometric moment problem: the operator approach, Ukrainian Math. J., 2011, 63, no. 6, 786-797 [WoS] 
  11. [10] Zagorodnyuk S.M., Nevanlinna formula for the truncated matrix trigonometric moment problem, Ukrainian Math. J., 2013, 64, no. 8, 1199-1214 [11] Zagorodnyuk S.M., Generalized resolvents of symmetric and isometric operators: the Shtraus approach, Ann. Funct. Anal., 2013, http://www.emis.de/journals/AFA/ Zbl1273.42002

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