Fourier transforms having only real zeros

Abstract

Let G ( z ) G(z) be a real entire function of order less than 2 2 with only real zeros. Then we classify certain distribution functions F F such that the Fourier transform H ( z ) = ∫ − ∞ ∞ G ( i t ) e i z t d F ( t ) H(z)=\int _{-\infty }^{\infty }G(it)e^{izt}\,dF(t) has only real zeros.