Geodesic
A~generalized integral and conjugate functions
Sbornik. Mathematics, Tome 28 (1976) no. 1, pp. 73-106

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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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     author = {I. A. Vinogradova},
     title = {A~generalized integral and conjugate functions},
     journal = {Sbornik. Mathematics},
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     year = {1976},
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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/