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 $\widehat f_0$ 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 $h\to0$ whose leading term is found by solving an $n$-dimensional extremum problem for entire functions of exponential type.