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Title:Error term improvements for van der Corput transforms
Author(s):Vandehey, Joseph
Director of Research:Ford, Kevin; Boca, Florin
Doctoral Committee Chair(s):Athreya, Jayadev S.; Hildebrand, A.J.
Doctoral Committee Member(s):Ford, Kevin; Zaharescu, Alexandru; Boca, Florin
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Subject(s):Asymptotic analysis
exponential sum
trigonometric sum
van der Corput transform
Abstract:We improve the error term in the van der Corput transform for exponential sums, \sum g(n) exp(2 \pi i f(n)). For many smooth functions g and f, we can show that the largest factor of the error term is given by a simple explicit function, which can be used to show that previous results, such as those of Karatsuba and Korolev, are sharp. Of particular note, the methods of this paper avoid the use of the truncated Poisson formula, and thus can be applied to much longer intervals [a, b] with far better results. As an example of the strength of these results, we provide a detailed analysis of the error term in the case g(x) = 1 and f(x) = (x/3)^(3/2).
Issue Date:2013-05-24
Rights Information:Copyright 2013 Joseph Vandehey
Date Available in IDEALS:2013-05-24
Date Deposited:2013-05

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