Geodesic
Conjugate functions and the restricted Denjoy integral
Sbornik. Mathematics, Tome 33 (1977) no. 1, pp. 81-124 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the functions conjugate to Denjoy integrable functions. In particular, it is shown that if $f$ and its conjugate $\overline f$ are integrable in the restricted Denjoy sense then the conjugate series coincides with the Fourier–Denjoy series of the conjugate function, $(D^*)\sigma[\overline f]=(D^*)\overline\sigma [f]$, and the Riesz equation $(D^*)\int_0^{2\pi}\varphi\overline f\,dx=-(D^*)\int_0^{2\pi}f\overline\varphi\,dx$ holds provided that $\varphi$ and its conjugate function $\overline\varphi$ are of bounded variation. Bibliography: 20 titles. 
@article{SM_1977_33_1_a5,
     author = {T. P. Lukashenko},
     title = {Conjugate functions and the restricted {Denjoy} integral},
     journal = {Sbornik. Mathematics},
     pages = {81--124},
     year = {1977},
     volume = {33},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1977_33_1_a5/}
}
TY  - JOUR
AU  - T. P. Lukashenko
TI  - Conjugate functions and the restricted Denjoy integral
JO  - Sbornik. Mathematics
PY  - 1977
SP  - 81
EP  - 124
VL  - 33
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1977_33_1_a5/
LA  - en
ID  - SM_1977_33_1_a5
ER  - 
%0 Journal Article
%A T. P. Lukashenko
%T Conjugate functions and the restricted Denjoy integral
%J Sbornik. Mathematics
%D 1977
%P 81-124
%V 33
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1977_33_1_a5/
%G en
%F SM_1977_33_1_a5
T. P. Lukashenko. Conjugate functions and the restricted Denjoy integral. Sbornik. Mathematics, Tome 33 (1977) no. 1, pp. 81-124. http://geodesic.mathdoc.fr/item/SM_1977_33_1_a5/

[1] A. S. Bezikovich, “On a general metric property of summable functions”, J. London Math. Soc., 1 (1926), 120–128 | DOI

[2] E. C. Titchmarsh, “On conjugate functions”, Proc. London Math. Soc., 29 (1929), 49–80 | DOI

[3] I. I. Privalov, Integral Cauchy, Saratov, 1919

[4] A. N. Kolmogorov, “Sur un procédé d'integration de M. Denjoy”, Fundam. Math., 11 (1928), 27–28

[5] V. I. Smirnov, “Sur les valeurs limites des fonctions analytiques”, C. r. Acad. Sci., Paris, 188 (1929), 131–133

[6] N. K. Bari, Trigonometricheskie ryady, Fizmatgiz, Moskva, 1961 | MR

[7] T. P. Lukashenko, “O $D$-integriruemykh sopryazhennykh funktsiyakh”, Izv. AN SSSR, seriya matem., 37 (1973), 946–958 | Zbl

[8] A. I. Plessner, “Zur Theorie der konjugierten trigonometrischen Reihen”, Mitt. Math. Semin. Univ. Giessen, 10 (1923), 1–36

[9] J. Marcinkiewics, “Sur les séries de Fourier”, Fundam. Math., 27 (1936), 38–69

[10] A. Zigmund, Trigonometricheskie ryady, t. 1, izd-vo «Mir», Moskva, 1965 | MR

[11] A. Zigmund, Trigonometricheskie ryady, t. 2, izd-vo «Mir», Moskva, 1965 | MR

[12] S. Saks, Teoriya integrala, IL, Moskva, 1949

[13] I. P. Natanson, Teoriya funktsii veschestvennoi peremennoi, izd-vo «Nauka», Moskva, 1974

[14] I. A. Vinogradova, V. A. Skvortsov, “Obobschennye integraly i ryady Fure”, Itogi nauki. Matematicheskii analiz. 1970, VINITI AN SSSR, Moskva, 1971, 65–107 | MR

[15] T. P. Lukashenko, “Sopryazhennye funktsii i integraly”, Izv. AN SSSR, seriya matem., 36 (1972), 435–449 | Zbl

[16] G. A. Khuskivadze, “O sopryazhennykh funktsiyakh i integralakh tipa Koshi”, Trudy Tbil. matem. in-ta, 31 (1966), 5–54 | Zbl

[17] G. E. Shilov, Matematicheskii analiz, funktsii neskolkikh veschestvennykh peremennykh, chasti 1 i 2, izd-vo «Nauka», Moskva, 1972

[18] G. E. Shilov, Matematicheskii analiz, funktsii odnogo peremennogo, chasti 1–2, izd-vo «Nauka», Moskva, 1969 | MR

[19] A. G. Dzhvarsheishvili, “Integral Danzhua i nekotorye voprosy iz analiza”, Trudy Tbil. matem. in-ta, 25 (1958), 273–372

[20] P. L. Ulyanov, “$A$-integral i sopryazhennye funktsii”, Matematika, t. VIII, Uchenye zapiski MGU, 181, 1956, 139–157 | MR