Localization of small zeros of sine and cosine Fourier transforms of a finite positive nondecreasing function |
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Authors: | A M Sedletskii |
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Institution: | 1.Faculty of Mechanics and Mathematics,Moscow State University,Leninskie Gory, Moscow,Russia |
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Abstract: | 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 E
1/2(−z
2; μ), μ ∈ (1, 2) ∪ (2, 3) is obtained as a corollary. |
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Keywords: | |
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