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

TL;DR

This paper improves Pólya's results on the distribution of zeros of Fourier transforms of positive, integrable functions supported on [0,1], providing new conditions and applying them to beta distributions to extend the known zero regions.

Contribution

It offers refined criteria for the zeros of Fourier transforms of certain functions and applies these to beta distributions, enlarging the applicable parameter regions.

Findings

Enhanced conditions for zeros to be simple and regularly distributed.

Extended the region in parameter space where zeros are well-understood.

Applied results specifically to beta probability density functions.

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.