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, Ilya Spitkovsky

    Research output: Contribution to journalArticle

    Abstract

    This paper indicates a method of calculating the defect numbers of the boundary value problem in terms of the -numbers of the Hankel operator constructed in a specified way with respect to the coefficients and. On the basis of this result the authors establish that the estimates, obtained in 1975 by A. M. Nikolaĭchuk and one of the authors (Ukrainian Math. J. 27 (1975), 629-639), 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, specified on a circle, by a function, 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.

    Original languageEnglish (US)
    Pages (from-to)205-224
    Number of pages20
    JournalMathematics of the USSR - Sbornik
    Volume45
    Issue number2
    DOIs
    StatePublished - Feb 28 1983

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    Riemann Boundary Value Problem
    Hermitian matrix
    Meromorphic Function
    Factorization
    Defects
    Approximation
    Estimate
    Pole
    Hankel Operator
    Laurent Series
    Coincidence
    Multiplier
    Solvability
    Determinant
    Circle
    Boundary Value Problem
    Partial
    Coefficient

    ASJC Scopus subject areas

    • Mathematics(all)

    Cite this

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    abstract = "This paper indicates a method of calculating the defect numbers of the boundary value problem in terms of the -numbers of the Hankel operator constructed in a specified way with respect to the coefficients and. On the basis of this result the authors establish that the estimates, obtained in 1975 by A. M. Nikolaĭchuk and one of the authors (Ukrainian Math. J. 27 (1975), 629-639), 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, specified on a circle, by a function, 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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