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 \(\widehat f_0\) 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 A_{1}([−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.

extremum problemperiodic functionFourier coefficientasymptotic expansionentire function of exponential type