Zeros of the Fourier transform of a distribution (Q1173805)

Two theorems are proved in this paper. Theorem 1: The trigonometric polynomial \(f(\theta)=\sum_{n=-N}^ N c_ ne^{in\theta}\), \(c_ n\in\mathbb{C}\) is nonzero on at least one open interval of the length \(d>\pi/N\) except for the exceptional case where \(f(\theta)=c \sin N(\theta-\alpha)\), \(0\leq\alpha<2\pi\), \(c\in\mathbb{C}\). The exceptional case arises as the Fourier transform of the distribution \(T(u)=e^{-iN\alpha}\delta(u-N)-e^{iN\alpha}\delta(u+N)\), where \(\delta\) is the Dirac distribution. Theorem 2. Suppose \(f\not\equiv0\) is the Fourier transform of \(\phi\in L^ 2[-\sigma,\sigma]\). Then \(f\) is nonzero on at least one open interval of the real axis of length \(d>\pi/\sigma\). The author concludes the paper with two remarks.