, Volume 69, Issue 3-4, pp 313-319

Extremum Problem for Periodic Functions Supported in a Ball

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Abstract

We consider the Turan n-dimensional extremum problem of finding the value of An(hB n ) which is equal to the maximum zero Fourier coefficient \(\widehat f_0\) of periodic functions f supported in the Euclidean ball hB n 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(hB n 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.