An extremum problem for distribution functions (Q1272300)

Let us consider the functions of the form \[ f( t)= \sum_{\nu \in Z} \widehat {f}_\nu e^{i\nu t},\quad \sum_{\nu \in Z} |\widehat {f}_\nu|< + \infty . \tag{1} \] The following result is proved. Theorem 1. For any \(n \in N\) and any nonnegative function \(f \not\equiv 0\) of the form \((1)\) whose Fourier coefficients satisfy the conditions \[ \widehat f_\nu = {1\over 2\pi} \int_{0}^{2\pi} f(t)e^{i\nu t} dt \leq 0,\quad |\nu|\geq n,\quad \nu \in Z, \] the following inequality is valid: \[ \text{meas}\{ t \in [0,2\pi):f(t) > 0\} \geq {2\pi\over n}. \] The assertion of this theorem is sharp for any \(n \in N\). This is shown by the example of the \(2\pi - \)periodic function: \[ f(t) \equiv f_n (t) = \begin{cases} |\sin nt |,& |t|\leq {\pi\over n},\\ 0,& {\pi\over n} \leq |t|\leq \pi. \end{cases} . \]