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 Mat. Sb. (N.S.), 1982, Volume 117(159), Number 2, Pages 196–215 (Mi msb2199)

Sharp estimates of defect numbers of a generalized Riemann boundary value problem, factorization of hermitian matrix-valued functions and some problems of approximation by meromorphic functions

G. S. Litvinchuk, I. M. Spitkovsky

Abstract: This paper indicates a method of calculating the defect numbers of the boundary value problem
$$\varphi^+(t)=A(t)\varphi^-(t)+B(t)\overline{\varphi^-(t)}+C(t),\qquad|t|=1,$$
in terms of the $s$-numbers of the Hankel operator constructed in a specified way with respect to the coefficients $A$ and $B$. On the basis of this result the authors establish that the estimates, obtained in 1975 by A. M. Nikolaichuk and one of the authors (Ukr. Mat. Zh., 27 (1975), № 6, p. 767–779), of the defect numbers in terms of the number of coincidences in a disk of the solutions of certain approximating problems are sharp. This paper also establishes, in passing, criteria for the solvability of the problem of approximating a function $f$, specified on a circle, by a function $R$, meromorphic in a disk, for which a portion of the poles (along with the principal parts of the Laurent series at these poles) is assumed to be given.
As auxiliary results expressions for partial indices are obtained, and properties of factorizing multipliers of Hermitian matrices of the second order with a negative determinant and a sign-preserving diagonal element are established.
Bibliography: 27 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 45:2, 205–224

Bibliographic databases:

UDC: 517.544.8+517.984.5+517.518.84
MSC: 30E25, 45E05, 30E10

Citation: G. S. Litvinchuk, I. M. Spitkovsky, “Sharp estimates of defect numbers of a generalized Riemann boundary value problem, factorization of hermitian matrix-valued functions and some problems of approximation by meromorphic functions”, Mat. Sb. (N.S.), 117(159):2 (1982), 196–215; Math. USSR-Sb., 45:2 (1983), 205–224

Citation in format AMSBIB
\Bibitem{LitSpi82} \by G.~S.~Litvinchuk, I.~M.~Spitkovsky \paper Sharp estimates of defect numbers of a generalized Riemann boundary value problem, factorization of hermitian matrix-valued functions and some problems of approximation by meromorphic functions \jour Mat. Sb. (N.S.) \yr 1982 \vol 117(159) \issue 2 \pages 196--215 \mathnet{http://mi.mathnet.ru/msb2199} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=644769} \zmath{https://zbmath.org/?q=an:0509.30033|0495.30032} \transl \jour Math. USSR-Sb. \yr 1983 \vol 45 \issue 2 \pages 205--224 \crossref{https://doi.org/10.1070/SM1983v045n02ABEH002595} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Latushkin Y., Litvinchuk G., “How to Calculate the Defect Numbers of the Generalized Riemann Boundary-Value Problem”, 1043, 1984, 303–305
2. Spitkovskii I., Tashbaev A., “Factorization of Piecewise-Constant Matrix Functions with 3 Points of Discontinuity in Lp,P Classes and Some its Applications”, 307, no. 2, 1989, 291–296
3. Ramirez J., Spitkovsky I., “On the Algebra Generated by a Poly-Bergman Projection and a Composition Operator”, Factorization, Singular Operators and Related Problems, Proceedings, eds. Samko S., Lebre A., DosSantos A., Springer, 2003, 273–289
4. S. N. Kiyasov, “Otsenki chastnykh indeksov gelderovskikh matrits-funktsii vtorogo poryadka”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 156, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2014, 44–50
5. A. F. Voronin, “O svyazi obobschennoi kraevoi zadachi Rimana i usechennogo uravneniya Vinera—Khopfa”, Sib. elektron. matem. izv., 15 (2018), 412–421
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