Extremum Problem for Periodic Functions Supported in a Ball
- 36 Downloads
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.
Unable to display preview. Download preview PDF.
- 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.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.E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, Princeton, 1971.Google Scholar
- 4.S. M. Nikol′skii, Approximation of Functions of Several Variables and Embedding Theorems [in Russian], Nauka, Moscow, 1977.Google Scholar
- 5.H. Bateman and A. Erdélyi, Higher Transcendental Functions, vol. 1, McGraw-Hill, New York- Toronto-London, 1953.Google Scholar
- 6.N. I. Akhiezer, Lectures on Approximation Theory [in Russian], Nauka, Moscow, 1965.Google Scholar
- 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