Geodesic
An estimate of the rate of approximation of a~continuous function and its conjugate by Fourier sums on a~set of total measure
Izvestiya. Mathematics , Tome 8 (1974) no. 6, pp. 1372-1386

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In this paper, estimates of the rate of convergence almost everywhere of the Fourier series of a continuous function and its conjugate are obtained. These estimates, expressed in terms of its best approximation functions and its modulus of continuity, cannot be strengthened in a number of cases. 
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     author = {K. I. Oskolkov},
     title = {An estimate of the rate of approximation of a~continuous function and its conjugate by {Fourier} sums on a~set of total measure},
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K. I. Oskolkov. An estimate of the rate of approximation of a~continuous function and its conjugate by Fourier sums on a~set of total measure. Izvestiya. Mathematics , Tome 8 (1974) no. 6, pp. 1372-1386. http://geodesic.mathdoc.fr/item/IM2_1974_8_6_a11/