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Mat. Sb., 1993, Volume 184, Number 3, Pages 3–20 (Mi msb969)  

This article is cited in 11 scientific papers (total in 11 papers)

Norms of Dirichlet kernels and some other trigonometric polynomials in $L_p$-spaces

M. I. Dyachenko

M. V. Lomonosov Moscow State University

Abstract: The following problem is considered. Let $\mathbf{a}=\{a_{\mathbf{n}}\}_{\mathbf{n}=1}^{\mathbf{M}}=\{a_{n_1,…,n_m}\}_{n_1,…,n_m=1}^{M_1,…,M_m}$ be a finite $m$-fold sequence of nonnegative numbers such that if $\mathbf{n}\ge\mathbf{k}$ then $a_{\mathbf{n}}\le a_{\mathbf{k}}$, and $Q(\mathbf{x})=\sum_{\mathbf{n}=1}^{\mathbf{M}}a_{\mathbf{n}}e^{i\mathbf{nx}}$. The purpose of the work is to give best possible upper estimates of the norms $\|Q(\mathbf x)\|_p$ and $\|Q(\mathbf x)\|_{\mathbf{\delta},p}$ with $\boldsymbol\delta>0$ in terms of the coefficients $\{a_{\mathbf{n}}\}$. The Dirichlet kernels $D_U(\mathbf{x})=\sum_{\mathbf{n}\in U}e^{i\mathbf{nx}}$ with $U\in A_1$ present a particular case of $Q(\mathbf x)$.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1994, 78:2, 267–282

Bibliographic databases:

UDC: 517.51
MSC: 42A05
Received: 23.01.1992

Citation: M. I. Dyachenko, “Norms of Dirichlet kernels and some other trigonometric polynomials in $L_p$-spaces”, Mat. Sb., 184:3 (1993), 3–20; Russian Acad. Sci. Sb. Math., 78:2 (1994), 267–282

Citation in format AMSBIB
\Bibitem{Dya93}
\by M.~I.~Dyachenko
\paper Norms of Dirichlet kernels and some other trigonometric polynomials in $L_p$-spaces
\jour Mat. Sb.
\yr 1993
\vol 184
\issue 3
\pages 3--20
\mathnet{http://mi.mathnet.ru/msb969}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=1220616}
\zmath{https://zbmath.org/?q=an:0815.42001}
\transl
\jour Russian Acad. Sci. Sb. Math.
\yr 1994
\vol 78
\issue 2
\pages 267--282
\crossref{https://doi.org/10.1070/SM1994v078n02ABEH003469}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1994PD76700001}


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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. E. D. Nursultanov, “Net spaces and inequalities of Hardy–Littlewood type”, Sb. Math., 189:3 (1998), 399–419  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. E. D. Nursultanov, “Concerning the multiplicators of Fourier series in the trigonometric system”, Math. Notes, 63:2 (1998), 205–214  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. M. I. Dyachenko, “Fourier coefficients of piecewise-monotone functions of several variables”, Izv. Math., 62:2 (1998), 247–259  mathnet  crossref  crossref  mathscinet  zmath  isi
    4. D'yachenko M., “Fourier Transform of Piecewise Monotone Functions of Many Variables”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1998, no. 3, 25–38  mathscinet  zmath  isi
    5. O. S. Dragoshanskii, “Anisotropic norms of Dirichlet kernels and some other trigonometric polynomials”, Math. Notes, 67:5 (2000), 582–595  mathnet  crossref  crossref  mathscinet  zmath  isi
    6. M. I. Dyachenko, “$U$-Convergence of Fourier Series with Monotone and with Positive Coefficients”, Math. Notes, 70:3 (2001), 320–328  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. A. P. Antonov, “Smoothness of sums of trigonometric series with monotone coefficients”, Russian Math. (Iz. VUZ), 51:4 (2007), 18–26  mathnet  crossref  mathscinet  zmath
    8. Mikhail Dyachenko, Sergey Tikhonov, “Convergence of trigonometric series with general monotone coefficients”, Comptes Rendus Mathematique, 345:3 (2007), 123  crossref  mathscinet  zmath
    9. Mikhail Dyachenko, Sergey Tikhonov, “A Hardy–Littlewood theorem for multiple series”, Journal of Mathematical Analysis and Applications, 339:1 (2008), 503  crossref  mathscinet  zmath
    10. M. I. Dyachenko, E. D. Nursultanov, M. E. Nursultanov, “The Hardy–Littlewood Theorem for Multiple Fourier Series with Monotone Coefficients”, Math. Notes, 99:4 (2016), 503–510  mathnet  crossref  crossref  mathscinet  isi  elib
    11. D. G. Dzhumabaeva, M. I. Dyachenko, E. D. Nursultanov, “On convergence of multiple trigonometric series with monotone coefficients”, Siberian Math. J., 58:2 (2017), 205–214  mathnet  crossref  crossref  isi  elib  elib
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