The zeros of certain Fourier transforms:Improvements of Pólya's results

2021/10/05 by Yong-Kum Cho, Cho, Yong-Kum, Young Woong Park +1 · 1 citation
Computer Science · Mathematics · #30C15 #33C20 #42A38 #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.2110.01885

openalex publication_date 2021/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

As for the Fourier transforms of positive and integrable functions supported in the unit interval, we make a list of improvements for Pólya'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 \beginequation* f(t)= (1)/(B(α, β)) (1-t)^α-1 t^β-1, 00, β>0, and specify the distribution of zeros of the associated Fourier transforms for some region of (α, β) in the first quadrant which turns out to be much larger than the region where Pólya's results are applicable.

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