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 Trudy Inst. Mat. i Mekh. UrO RAN, 2010, Volume 16, Number 4, Pages 166–179 (Mi timm651)

Estimates for sums of moduli of blocks from trigonometric Fourier series

V. P. Zastavnyi

Donetsk National University, Ukraine

Abstract: We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets $A_k\subset\mathbb Z$ and a sequence of functions $\lambda_k\colon A_k\to\mathbb C$ provide the existence of a number $C$ such that any function $f\in L_1$ satisfies the inequality $\|U_{\mathcal A,\Lambda}(f)\|_p\le C\|f\|_1,$and what is the exact constant in this inequality? Here, $U_{\mathcal A,\Lambda}(f)(x)=\sum_{k=1}^\infty|\sum_{m\in A_k}\lambda_k(m)c_m(f)e^{imx}|$, and $c_m(f)$ are Fourier coefficients of the function $f\in L_1$. Problem 2: what conditions on a sequence of finite subsets $A_k\subset\mathbb Z$ guarantee that the a function $\sum_{k=1}^\infty|\sum_{m\in A_k}c_m(h)e^{imx}|$ belongs to $L_p$ for every function $h$ of bounded variation?

Keywords: trigonometric series; Hardy-Littlewood theorems.

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2011, 273, suppl. 1, S190–S204

UDC: 517.518

Citation: V. P. Zastavnyi, “Estimates for sums of moduli of blocks from trigonometric Fourier series”, Trudy Inst. Mat. i Mekh. UrO RAN, 16, no. 4, 2010, 166–179; Proc. Steklov Inst. Math. (Suppl.), 273, suppl. 1 (2011), S190–S204

Citation in format AMSBIB
\Bibitem{Zas10} \by V.~P.~Zastavnyi \paper Estimates for sums of moduli of blocks from trigonometric Fourier series \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2010 \vol 16 \issue 4 \pages 166--179 \mathnet{http://mi.mathnet.ru/timm651} \elib{http://elibrary.ru/item.asp?id=15318498} \transl \jour Proc. Steklov Inst. Math. (Suppl.) \yr 2011 \vol 273 \issue , suppl. 1 \pages S190--S204 \crossref{https://doi.org/10.1134/S0081543811050208} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-79959274964} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. S. A. Telyakovskii, “Series formed by the moduli of blocks of terms of trigonometric series. A survey”, J. Math. Sci., 209:1 (2015), 152–158
2. S. A. Telyakovskii, “Addition to V. P. Zastavnyi's paper “Estimates for sums of moduli of blocks in trigonometric Fourier series””, Proc. Steklov Inst. Math. (Suppl.), 297, suppl. 1 (2017), 186–190
3. V. P. Zastavnyi, A. S. Levadnaya, “Integriruemost so stepennym vesom summ iz modulei blokov trigonometricheskikh ryadov”, Tr. IMM UrO RAN, 23, no. 3, 2017, 125–133
4. Krasniqi X.Z., “On l-P-Integrability of a Special Double Sine Series Formed By Its Blocks”, J. Contemp. Math. Anal.-Armen. Aca., 52:1 (2017), 48–53
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