Localization of small zeros of sine and cosine Fourier transforms of a finite positive nondecreasing function / A. M. Sedletskii. // Vestnik Moskovskogo Universiteta. Seriya 1. Matematika. Mekhanika. 2009. № 4. P. 35-41 [Moscow Univ. Math. Bulletin. Vol. 72, N 2, 2017. P. 172-177].

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.

Key words: sine- and cosine-Fourier transform, zeros of entire function, Mittag-Leffler's function.