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 Mat. Sb., 1992, Volume 183, Number 11, Pages 55–74 (Mi msb1089)

On upper estimates of the partial sums of a trigonometric series in terms of lower estimates

A. S. Belov

Abstract: 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
$$S_n(x)=\sum^n_{k=0}(a_k\cos(kx)+b_k\sin(kx)),\qquad n=0,1,\dotsc .$$
It is proved that the estimate
$$\max_x S_n(x)\leq 4a_0 n^{1-\alpha},$$
holds for each natural number $n$ such that $S_m(x)\geq0$ for all $x$ and $m=1, …, n$. 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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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1994, 77:2, 313–330

Bibliographic databases:

UDC: 517.5
MSC: Primary 42A05; Secondary 42A32, 42B05

Citation: A. S. Belov, “On upper estimates of the partial sums of a trigonometric series in terms of lower estimates”, Mat. Sb., 183:11 (1992), 55–74; Russian Acad. Sci. Sb. Math., 77:2 (1994), 313–330

Citation in format AMSBIB
\Bibitem{Bel92} \by A.~S.~Belov \paper On upper estimates of the~partial sums of a~trigonometric series in terms of lower estimates \jour Mat. Sb. \yr 1992 \vol 183 \issue 11 \pages 55--74 \mathnet{http://mi.mathnet.ru/msb1089} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1208214} \zmath{https://zbmath.org/?q=an:0796.42001} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1994SbMat..77..313B} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1994 \vol 77 \issue 2 \pages 313--330 \crossref{https://doi.org/10.1070/SM1994v077n02ABEH003443} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994NF83500005} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. S. Belov, “Some upper bounds of partial sums of a trigonometric series cannot be sharpened via lower bounds”, Math. Notes, 58:3 (1995), 984–989
2. A. S. Belov, “Non-Fourier-Lebesgue trigonometric series with nonnegative partial sums”, Math. Notes, 59:1 (1996), 18–30
3. A. S. Belov, “Some estimates for non-negative trigonometric polynomials and properties of these polynomials”, Izv. Math., 67:4 (2003), 637–653
4. D. G. Vasilchenkova, V. I. Danchenko, “Vydelenie garmonik iz trigonometricheskikh mnogochlenov amplitudno-fazovymi operatorami”, Algebra i analiz, 32:2 (2020), 21–44
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