Geodesic
A generalized integral and conjugate functions
Sbornik. Mathematics, Tome 28 (1976) no. 1, pp. 73-106 
I. A. Vinogradova. A generalized integral and conjugate functions. Sbornik. Mathematics, Tome 28 (1976) no. 1, pp. 73-106. http://geodesic.mathdoc.fr/item/SM_1976_28_1_a4/
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     author = {I. A. Vinogradova},
     title = {A~generalized integral and conjugate functions},
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     url = {http://geodesic.mathdoc.fr/item/SM_1976_28_1_a4/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

The author gives a descriptive definition of the $LG^*$-integral. The $LG^*$-integral extends the Lebesgue integral and coincides with it for nonnegative functions. For a function $f(x)$, $LG^*$-integrable on $[0,2\pi]$, the $LG^*$-Fourier series is defined and is almost everywhere $(C,1)$ summable to $f(x)$; the conjugate series is $(C,1)$ summable to $\widetilde f(x)$, which is also $LG^*$-integrable on $[0,2\pi]$, and is its $LG^*$-Fourier series. Bibliography: 12 titles.

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[9] P. L. Ulyanov, “Primenenie $A$-integrirovaniya k odnomu klassu trigonometricheskikh ryadov”, Matem. sb., 35 (77) (1954), 469–490 | MR

[10] P. L. Ulyanov, “$A$-integral i ego primenenie k teorii trigonometricheskikh ryadov”, Uspekhi matem. nauk, X:1 (66) (1955), 189–191

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[12] P. L. Ulyanov, “$A$-integral i sopryazhennye funktsii”, Uchenye zapiski MGU, 181 (1956), 139–157 | MR