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Mat. Zametki, 1977, Volume 22, Issue 3, Pages 381–394 (Mi mz8059)  

This article is cited in 10 scientific papers (total in 10 papers)

Discrete imbedding theorems and Lebesgue constants

A. A. Yudina, V. A. Yudinb

a Vladimir Pedagogical Institute
b Moscow Power Engineering Institute

Abstract: The order of growth of the Lebesgue constant for a “hyperbolic cross” is found:
$$ L_R=\int_{T^2}|\sum_{0<|\nu_1\nu_2|\le R_2}e^{2\pi i\nu x}| dx\asymp R^{1.2},\quad R\to\infty. $$
Estimates are obtained by applying a discrete imbedding theorem. It is proved that among all convex domains in $E^2$, the square gives rise to a Lebesgue constant with the slowest growth $\ln^2R$.

Full text: PDF file (692 kB)

English version:
Mathematical Notes, 1977, 22:3, 702–711

Bibliographic databases:

UDC: 517.5
Received: 05.07.1976

Citation: A. A. Yudin, V. A. Yudin, “Discrete imbedding theorems and Lebesgue constants”, Mat. Zametki, 22:3 (1977), 381–394; Math. Notes, 22:3 (1977), 702–711

Citation in format AMSBIB
\Bibitem{YudYud77}
\by A.~A.~Yudin, V.~A.~Yudin
\paper Discrete imbedding theorems and Lebesgue constants
\jour Mat. Zametki
\yr 1977
\vol 22
\issue 3
\pages 381--394
\mathnet{http://mi.mathnet.ru/mz8059}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=467281}
\zmath{https://zbmath.org/?q=an:0358.42010}
\transl
\jour Math. Notes
\yr 1977
\vol 22
\issue 3
\pages 702--711
\crossref{https://doi.org/10.1007/BF02412499}


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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. È. M. Galeev, “Order estimates of derivatives of the multidimensional periodic Dirichlet $\alpha$-kernel in a mixed norm”, Math. USSR-Sb., 45:1 (1983), 31–43  mathnet  crossref  mathscinet  zmath
    2. Yu. N. Subbotin, “The Lebesgue constants of certain $m$-dimensional interpolation polynomials”, Math. USSR-Sb., 46:4 (1983), 561–570  mathnet  crossref  mathscinet  zmath
    3. 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
    4. M. I. Dyachenko, “Norms of Dirichlet kernels and some other trigonometric polynomials in $L_p$-spaces”, Russian Acad. Sci. Sb. Math., 78:2 (1994), 267–282  mathnet  crossref  mathscinet  zmath  isi
    5. M. I. Dyachenko, “$u$-convergence of multiple Fourier series”, Izv. Math., 59:2 (1995), 353–366  mathnet  crossref  mathscinet  zmath  isi
    6. M. I. Dyachenko, “Uniform convergence of double Fourier series for classes of functions with anisotropic smoothness”, Math. Notes, 59:6 (1996), 680–686  mathnet  crossref  crossref  mathscinet  zmath  isi
    7. 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
    8. 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
    9. V. A. Yudin, “Multidimensional Versions of Paley's Inequality”, Math. Notes, 70:6 (2001), 860–865  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    10. M. I. Dyachenko, “Uniform Convergence of Hyperbolic Partial Sums of Multiple Fourier Series”, Math. Notes, 76:5 (2004), 673–681  mathnet  crossref  crossref  mathscinet  zmath  isi
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