Extremum Problem for Periodic Functions Supported in a Ball

We consider the Turan n-dimensional extremum problem of finding the value of A_n(hB ⁿ ) which is equal to the maximum zero Fourier coefficient 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 A_n(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.