Localization of Small Zeros of Sine and Cosine Fourier Transforms of a Finite Positive Nondecreasing Function / A.M. Sedletskii // Vestn. Mosk. Univ., Matem. Mekhan. 2009. № 4. P. 35-41 [Moscow Univ. Math. Bulletin. Vol. 64, No 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 (π(n-1/2), π(n+1/2)), (πn, π(n+1)), n∈N, respectively. In the case of sine transforms it is required that f cannot be a stepped function with rational 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 E1/2(-z2;μ), μ∈(1,2)∪(2,3) is obtained as a corollary.
Key words: sine- and cosine-Fourier transform, zeros of entire function, Mittag-Leffler's function