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

Sbornik. Mathematics, Tome 45 (1983) no. 2, pp. 205-224

Résumé

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), No 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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