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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. 64,
N 4, 2009. 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.