Geodesic
On functions conjugate to Denjoy–Khinchin integrable functions
Izvestiya. Mathematics, Tome 5 (1971) no. 2, pp. 393-420 
T. P. Lukashenko. On functions conjugate to Denjoy–Khinchin integrable functions. Izvestiya. Mathematics, Tome 5 (1971) no. 2, pp. 393-420. http://geodesic.mathdoc.fr/item/IM2_1971_5_2_a6/
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     author = {T. P. Lukashenko},
     title = {On functions conjugate to {Denjoy{\textendash}Khinchin} integrable functions},
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     pages = {393--420},
     year = {1971},
     volume = {5},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1971_5_2_a6/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper an example is constructed of a function $f(x)$ that is integrable in the sense of Denjoy–Khinchin for which the conjugate function $\bar f(x)$ is integrable in the same sense and has a positive zeroth Fourier coefficient.

[1] Khinchin A. Ya., “O protsesse integrirovaniya Denjoy”, Matem. sb., 30:4 (1918), 543–557 | Zbl

[2] Marcinkiewics J., “Sur les series de Fourier”, Fund. Math., 27 (1936), 36–69

[3] Zigmund A., Trigonometricheskie ryady, t. I, Mir, M., 1965 | MR

[4] Bari N. K., Trigonometricheskie ryady, GIFML, M., 1961 | MR