The zeros of certain Fourier transforms:Improvements of P\'olya's results

Abstract

As for the Fourier transforms of positive and integrable functions supported in the unit interval, we make a list of improvements for P\'olya's results on the distribution of their positive zeros and give new sufficient conditions under which those zeros are simple and regularly distributed. As an application, we take the two-parameter family of beta probability density functions defined by \begin{equation*} f(t)= \frac{1}{B(\alpha, \beta)}\, (1-t)^{\alpha-1} t^{\beta-1},\quad 0<t<1, \end{equation*} where $\,\alpha>0,\,\beta>0,$ and specify the distribution of zeros of the associated Fourier transforms for some region of $(\alpha, \beta)$ in the first quadrant which turns out to be much larger than the region where P\'olya's results are applicable.