Geodesic
On upper estimates of the partial sums of a trigonometric series in terms of lower estimates
Sbornik. Mathematics, Tome 77 (1994) no. 2, pp. 313-330 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\{a_k\}_{k=0}^\infty$ and $\{b_k\}_{k=0}^\infty$ be sequences of real numbers and let $ S_n(x)$ be defined by $$ S_n(x)=\sum^n_{k=0}\bigl(a_k\cos(kx)+b_k\sin(kx)\bigr),\qquad n=0,1,\dotsc\,. $$ It is proved that the estimate $$ \max_x S_n(x)\leqslant 4a_0 n^{1-\alpha}, $$ holds for each natural number $n$ such that $S_m(x)\geqslant0$ for all $x$ and $m=1,\,\dots,\,n$. Here $\alpha\in(0,\,1)$ is the unique root of the equation $$ \int^{3\pi /2}_0 t^{-\alpha}\cos t\,dt=0. $$ It is proved that the order $n^{1-\alpha}$ in this estimate cannot be improved. Various generalizations of this result are also obtained. 
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     title = {On upper estimates of the~partial sums of a~trigonometric series in terms of lower estimates},
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     volume = {77},
     number = {2},
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     url = {http://geodesic.mathdoc.fr/item/SM_1994_77_2_a4/}
}
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A. S. Belov. On upper estimates of the partial sums of a trigonometric series in terms of lower estimates. Sbornik. Mathematics, Tome 77 (1994) no. 2, pp. 313-330. http://geodesic.mathdoc.fr/item/SM_1994_77_2_a4/

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