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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  mathnet  crossref  mathscinet  zmath  isi
    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  mathscinet  isi
    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  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    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  mathscinet  isi
    5. M. I. Dyachenko, “Some problems in the theory of multiple trigonometric series”, Russian Math. Surveys, 47:5 (1992), 103–171  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    6. A. I. Kozko, “Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms”, Izv. Math., 62:6 (1998), 1189–1206  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. A. S. Romanyuk, “Approximation of Classes of Periodic Functions in Several Variables”, Math. Notes, 71:1 (2002), 98–109  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    9. G. A. Akishev, “O tochnosti otsenok nailuchshego $M$-chlennogo priblizheniya klassa Besova”, Sib. elektron. matem. izv., 7 (2010), 255–274  mathnet  elib
    10. S. N. Kudryavtsev, “A Littlewood–Paley type theorem and a corollary”, Izv. Math., 77:6 (2013), 1155–1194  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    11. Thomas Trogdon, “Rational Approximation, Oscillatory Cauchy Integrals, and Fourier Transforms”, Constr Approx, 2015  crossref  mathscinet
  • Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics (from 1967)
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