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The Hilbert problem: The case of infinitely many discontinuity points of coefficients

Authors
  • Salimov, R. B.1
  • Shabalin, P. L.1
  • 1 Kazan State University of Architecture and Building, Kazan, Russia , Kazan (Russia)
Type
Published Article
Journal
Siberian Mathematical Journal
Publisher
Springer US
Publication Date
Jul 01, 2008
Volume
49
Issue
4
Pages
718–733
Identifiers
DOI: 10.1007/s11202-008-0069-x
Source
Springer Nature
Keywords
License
Yellow

Abstract

We obtain a solution to the Hilbert boundary value problem in the theory of analytic functions on the half-plane in the case that the coefficients of the boundary condition have countably many discontinuity points of the first kind. We elaborate the two substantially different situations: the series consisting of the jumps of the argument of the coefficient function and the increments of its continuous part converges and this series diverges. Accordingly, Hilbert problems with finite and infinite indices result. We derive formulas for the general solution and investigate the pictures of solvability of these problems.

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