Mathematical Notes

, Volume 70, Issue 5–6, pp 860–865 | Cite as

Multidimensional Versions of Paley's Inequality

  • V. A. Yudin


For trigonometric polynomials of two variables whose spectrum lies both in the interior and on the boundary of a strictly convex domain, the lower bounds of their norms on % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY-Hhbbf9v8qqaq% Fr0xc9pk0xbba9q8WqFfea0-yr0RYxir-Jbba9q8aq0-yq-He9q8qq% Q8frFve9Fve9Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaaca% WGmbaaaa!3934!\[L\] are given. They are expressed in terms of the Fourier coefficients whose numbers are located on the boundary.

Paley's inequality trigonometric polynomial of two variables lower bounds for the norms of trigonometric polynomials \(n\)>n-dimensional torus lacunary sequence 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Zygmund, Trigonometric Series, vol. 2, Cambridge Univ. Press, Cambridge, 1960.Google Scholar
  2. 2.
    B. Smith, “Two trigonometric designs: one-sided Riesz products and Littlewood products. General inequalities 3.,” Intern. Ser. of Numer. Math., 64 (1983), 141–148.Google Scholar
  3. 3.
    V. A. Yudin, “Integral norms of trigonometric polynomials,” Mat. Zametki [Math. Notes], 70 (2001), no. 2, 308–315 [275-282].Google Scholar
  4. 4.
    R. Cooke, “A Cantor-Lebesgue theorem in two dimensions,” Proc. Amer. Math. Soc., 30 (1971), 547–550.Google Scholar
  5. 5.
    Sh. A. Alimov, V. A. Il′in, and E. M. Nikishin, “Questions of convergence of multiple trigonometric series and of spectral expansions. I,” Uspekhi Mat. Nauk [Russian Math. Surveys], 31 (1976), no. 6, 28–83, “II,” Uspekhi Mat. Nauk [Russian Math. Surveys], 32 (1977), no. 1, 107-130.Google Scholar
  6. 6.
    A. N. Podkorytov, “On the Lebesgue constants of double Fourier series,” Vestnik Leningrad State Univ. (1977), no. 7, 79–84.Google Scholar
  7. 7.
    A. A. Yudin and V. A. Yudin, “Discrete embedding theorems and Lebesgue constants,” Mat. Zametki [Math. Notes], 22 (1977), no. 3, 381–394.Google Scholar
  8. 8.
    J. F. Fournier, “On a theorem of Paley and the Littlewood conjecture,” Archiv für Mathematik, vol. 17 (1979), 199–216.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • V. A. Yudin
    • 1
  1. 1.Moscow Power Engineering InstituteRussia

Personalised recommendations