Geodesic
Recovering a function from its trigonometric integral
Sbornik. Mathematics, Tome 201 (2010) no. 7, pp. 1053-1068 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

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. 
@article{SM_2010_201_7_a5,
     author = {T. A. Sworowska},
     title = {Recovering a~function from its trigonometric integral},
     journal = {Sbornik. Mathematics},
     pages = {1053--1068},
     year = {2010},
     volume = {201},
     number = {7},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2010_201_7_a5/}
}
TY  - JOUR
AU  - T. A. Sworowska
TI  - Recovering a function from its trigonometric integral
JO  - Sbornik. Mathematics
PY  - 2010
SP  - 1053
EP  - 1068
VL  - 201
IS  - 7
UR  - http://geodesic.mathdoc.fr/item/SM_2010_201_7_a5/
LA  - en
ID  - SM_2010_201_7_a5
ER  - 
%0 Journal Article
%A T. A. Sworowska
%T Recovering a function from its trigonometric integral
%J Sbornik. Mathematics
%D 2010
%P 1053-1068
%V 201
%N 7
%U http://geodesic.mathdoc.fr/item/SM_2010_201_7_a5/
%G en
%F SM_2010_201_7_a5
T. A. Sworowska. Recovering a function from its trigonometric integral. Sbornik. Mathematics, Tome 201 (2010) no. 7, pp. 1053-1068. http://geodesic.mathdoc.fr/item/SM_2010_201_7_a5/

[1] A. Zygmund, Trigonometric series, v. I, II, Cambridge Univ. Press, New York, 1959 | MR | MR | Zbl | Zbl

[2] A. Denjoy, “Une extension de l'intégrale de M. Lebesgue”, C. R. Acad. Sci. Paris, 154 (1912), 859–862 | Zbl

[3] D. Preiss, B. S. Thomson, “The approximate symmetric integral”, Canad. J. Math., 41:3 (1989), 508–555 | MR | Zbl

[4] K. M. Ostaszewski, Henstock integral in the plane, Mem. Amer. Math. Soc., 63, No 353, Amer. Math. Soc., Providence, RI, 1986 | MR

[5] B. S. Thomson, Symmetric properties of real functions, Monogr. Textbooks Pure Appl. Math., 183, Marcel Dekker, New York, 1994 | MR | Zbl

[6] G. G. Gevorkyan, “O edinstvennosti integralov Fure, skhodyaschikhsya v metrikakh $L_p$, $p>0$, na kazhdom konechnom intervale”, Dokl. AN Arm. SSR, 84:1 (1987), 21–25 | MR | Zbl

[7] N. Wiener, The Fourier integral and certain of its applications, Dover, New York, 1959 | MR | Zbl | Zbl

[8] N. Wiener, “Generalized harmonic analysis”, Acta Math., 55:1 (1930), 117–258 | DOI | MR | Zbl

[9] S. Bochner, Lectures on Fourier integrals, Princeton Univ. Press, Princeton, NJ, 1959 | MR | Zbl | Zbl

[10] W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1974 | MR | Zbl