Localization of small zeros of sine and cosine Fourier transforms of a finite positive nondecreasing function

A. M. Sedletskii

Vestnik Moskovskogo universiteta. Matematika, mehanika, no. 4 (2009), pp. 35-41

Résumé

Let a function $f$ be integrable, positive, and nondecreasing in the interval $(0,1)$. Then by Polya's theorem all zeros of the corresponding cosine- and sine-Fourier transforms are real and simple; in this case positive zeros lie in the intervals $(\pi(n-1/2),\pi(n+1/2)),\;(\pi n,\pi(n+1)),\;n\in\mathbb{N},$ respectively. In the case of the sine-transforms it is required that $f$ cannot be a stepped function with retional discontinuity points. In this paper, zeros of the function with small numbers are included into intervals being proper subsets of the corresponding Polya intervals. A localization of small zeros of the Mittag-Leffler function $E_{1/2}(-z^2;\mu),\,\mu\in(1,2)\cup(2,3)$ is obtained as a corollary.