Recovering a function from its trigonometric integral

T. A. Sworowska

M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: The approximate symmetric Henstock-Kurzweil integral is shown as solving the problem of the recovery of a function from its trigonometric integral. This being so, we generalize Offord's theorem, which is an analogue of de la Vallée Poussin's theorem for trigonometric series. A new condition for a function to be representable by a singular Fourier integral is also obtained.
Bibliography: 10 titles.

Keywords: trigonometric integral, approximate symmetric integral, Preiss-Thomson theorem, Offord's theorem, singular Fourier integral.

UDC: 517.52

MSC: 26A36, 26A39

Received: 10.06.2009 and 03.12.2009

DOI: 10.4213/sm7588

 English version: Sbornik: Mathematics, 2010, 201:7, 1053–1068

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