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 Mat. Sb. (N.S.), 1982, Volume 117(159), Number 1, Pages 32–43 (Mi msb2179)

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

Order estimates of derivatives of the multidimensional periodic Dirichlet $\alpha$-kernel in a mixed norm

È. M. Galeev

Abstract: In this paper the author establishes a sharp order estimate, in the mixed norm of $L_p(\mathbf T^n)$ for $1<p<\infty$ and in $L_\infty(\mathbf T^n)$ ($\mathbf T^n =[-\pi,\pi]^n$ is the $n$-dimensional torus), of the derivatives of order $\beta \in \mathbf R^n$ of the multidimensional Dirichlet $\alpha$-kernel $D_{\alpha,\mu}$ and the function $F_{\alpha,\mu}$, $\alpha>0$, $\mu>0$, which are sums of exponentials $e^{i(k,t)}$ lying respectively inside and outside a “graduated hyperbolic cross”, i.e., the set $\{k\in\square_s\mid(\alpha,s)\leqslant \mu\}$, where $\square_s=\{k\in\mathbf Z^n\mid2^{s_{j-1}} \leqslant|k_j|<2^{s_j}, j=1,\ldots,n\}$, $s>0$.
Bibliography: 11 titles.

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English version:
Mathematics of the USSR-Sbornik, 1983, 45:1, 31–43

Bibliographic databases:

UDC: 517.5
MSC: Primary 42B99; Secondary 26A33, 46E30
Received: 12.12.1980

Citation: È. M. Galeev, “Order estimates of derivatives of the multidimensional periodic Dirichlet $\alpha$-kernel in a mixed norm”, Mat. Sb. (N.S.), 117(159):1 (1982), 32–43; Math. USSR-Sb., 45:1 (1983), 31–43

Citation in format AMSBIB
\Bibitem{Gal82} \by \E.~M.~Galeev \paper Order estimates of derivatives of the multidimensional periodic Dirichlet $\alpha$-kernel in a mixed norm \jour Mat. Sb. (N.S.) \yr 1982 \vol 117(159) \issue 1 \pages 32--43 \mathnet{http://mi.mathnet.ru/msb2179} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=642487} \zmath{https://zbmath.org/?q=an:0514.42022|0499.42008} \transl \jour Math. USSR-Sb. \yr 1983 \vol 45 \issue 1 \pages 31--43 \crossref{https://doi.org/10.1070/SM1983v045n01ABEH002584} `

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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. Ðinh Dung, “The approximation of classes of periodic functions of many variables”, Russian Math. Surveys, 38:6 (1983), 117–118
2. Belinskii E., “The Approximation of Periodic-Functions of Several-Variables by Floating System of Exponents and the Trigonometric Widths”, 284, no. 6, 1985, 1294–1297
3. È. M. Galeev, “Order estimates of smallest norms, with respect to the choice of $N$ harmonics, of derivatives of the Dirichlet and Favard kernels”, Math. USSR-Sb., 72:2 (1992), 567–578
4. Belinskii E., Galeev E., “On the Smallest Norm Value of Mixed Derivatives of Trigonometric Polynomials with Fixed Number of Harmonics”, Vestn. Mosk. Univ. Seriya 1 Mat. Mekhanika, 1991, no. 2, 3–7
5. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171
6. A. I. Kozko, “Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms”, Izv. Math., 62:6 (1998), 1189–1206
7. A. S. Romanyuk, “Approximation of Classes of Periodic Functions in Several Variables”, Math. Notes, 71:1 (2002), 98–109
8. M. B. Sikhov, “Inequalities of Bernstein and Jackson–Nikol'skii Type and Estimates of the Norms of Derivatives of Dirichlet Kernels”, Math. Notes, 80:1 (2006), 91–100
9. G. A. Akishev, “O tochnosti otsenok nailuchshego $M$-chlennogo priblizheniya klassa Besova”, Sib. elektron. matem. izv., 7 (2010), 255–274
10. S. N. Kudryavtsev, “A Littlewood–Paley type theorem and a corollary”, Izv. Math., 77:6 (2013), 1155–1194
11. Thomas Trogdon, “Rational Approximation, Oscillatory Cauchy Integrals, and Fourier Transforms”, Constr Approx, 2015
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