Geodesic
On a~relationship in the theory of Fourier series
Matematičeskie zametki, Tome 15 (1974) no. 5, pp. 679-682

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In this paper we prove the validity of the inequality $$ \sup\limits_n\int_{-\pi}^\pi\Bigl|\frac{f(0)}2+\sum_{k=1}^nf\bigl(\frac{k\pi}n\bigr)e^{ikt}\Bigr|\,dt\le C\sum_{m=0}^\infty\Bigl|\int_0^\pi f(t)e^{imt}\,dt\Bigr| $$ for an arbitrary continuous function ($C$ is an absolute constant). An inequality in the opposite sense was obtained by one of us earlier. 
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     author = {\`E. S. Belinskii and R. M. Trigub},
     title = {On a~relationship in the theory of {Fourier} series},
     journal = {Matemati\v{c}eskie zametki},
     pages = {679--682},
     publisher = {mathdoc},
     volume = {15},
     number = {5},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_15_5_a2/}
}
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È. S. Belinskii; R. M. Trigub. On a~relationship in the theory of Fourier series. Matematičeskie zametki, Tome 15 (1974) no. 5, pp. 679-682. http://geodesic.mathdoc.fr/item/MZM_1974_15_5_a2/