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Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika, 2010, Number 1, Pages 12–18
(Mi vmumm746)
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Mathematics
Estimation of Dirichlet kernel difference in the norm of $\mathrm{L}$
V. O. Tonkov Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
Abstract:
This work is related to the problem of estimation of the norm of a trigonometrical polynomials through their coefficient in $\mathrm{L}$. It is proved that the norm of the difference of Dirichlet's kernels in $\mathrm{L}$ has the precise order $\ln(n-m)$ and the lower estimate is also valid with the coefficient $4/\pi^{2}$. A theorem and two lemmas are presented showing that the coefficients $c$ at $\ln(n-m)$ in an asymptotc estimate uniform with resepect to $m$ and $n$ may be greater than $4/\pi^{2}$ and its value in examples depends on arithmetic properties of $n$ and $m$.
Key words:
norm of a trigonometrical polynomial in $\mathrm{L}$, asymptotic estimate.
Received: 09.06.2008
Citation:
V. O. Tonkov, “Estimation of Dirichlet kernel difference in the norm of $\mathrm{L}$”, Vestnik Moskov. Univ. Ser. 1. Mat. Mekh., 2010, no. 1, 12–18
Linking options:
https://www.mathnet.ru/eng/vmumm746 https://www.mathnet.ru/eng/vmumm/y2010/i1/p12
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