Mathematical Notes

, Volume 69, Issue 3–4, pp 313–319

Extremum Problem for Periodic Functions Supported in a Ball

  • D. V. Gorbachev


We consider the Turan n-dimensional extremum problem of finding the value of An(hBn) which is equal to the maximum zero Fourier coefficient \(\widehat f_0\) of periodic functions f supported in the Euclidean ball hBn of radius h, having nonnegative Fourier coefficients, and satisfying the condition f(0)= 1. This problem originates from applications to number theory. The case of A1([−h,h]) was studied by S. B. Stechkin. For An(hBn we obtain an asymptotic series as h → 0 whose leading term is found by solving an n-dimensional extremum problem for entire functions of exponential type.

extremum problem periodic function Fourier coefficient asymptotic expansion entire function of exponential type 


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  1. 1.
    S. B. Stechkin, “An extremum problem for trigonometric series with nonnegative coefficients,” in: Selected Works. Mathematics [in Russian], Nauka, Moscow, 1998, pp. 244–245.Google Scholar
  2. 2.
    N. N. Andreev, “Extremum problems for periodic functions with a small support,” Vestnik Moskov. Univ. Ser. I Mat. Mekh. [Moscow Univ. Math. Bull.] (1997), no. 1, 29–32.Google Scholar
  3. 3.
    E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.Google Scholar
  4. 4.
    S. M. Nikol′skii, Approximation of Functions of Several Variables and Embedding Theorems [in Russian], Nauka, Moscow, 1977.Google Scholar
  5. 5.
    H. Bateman and A. Erdélyi, Higher Transcendental Functions, vol. 1, McGraw-Hill, New York- Toronto-London, 1953.Google Scholar
  6. 6.
    N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow, 1965.Google Scholar
  7. 7.
    R. B. Ghanem and C. Frappier, “Explicit quadrature formulae for entire functions of exponential type,” J. Approx. Th., 92 (1998), 267–279.Google Scholar

Copyright information

© Plenum Publishing Corporation 2001

Authors and Affiliations

  • D. V. Gorbachev
    • 1
  1. 1.Tula State UniversityRussia

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