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

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

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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