We improve the error term in the van der Corput transform for exponential
sums
  \sum_{a \le n \le b} g(n) exp(2\pi i f(n)).
  For many functions g and f, we can extract the next term in the asymptotic,
showing 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. We provide a detailed analysis of the error term
in the case g(x)=1 and f(x)=(x/3)^{3/2}.Comment: 72 pages, 10 figure

Corput Transform

J. Vandehey

English

arXiv

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)

Vandehey, Joseph

IDEALS @ Illinois

Error term improvements for van der Corput transforms

We improve the error term in the van der Corput transform for exponential sums, a≤n≤b g(n) exp(2π if (n)). For many functions g and f, we can extract a simple explicit main term for the error. In addition, 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 better estimates than previous results. As a result of these refinements, we are able to provide an improvement to the classical Kusmin–Landau inequality. 1

Joseph Vandehey

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ERROR TERM IMPROVEMENTS FOR VAN DER CORPUT TRANSFORMS

We improve the error term in the van der Corput transform for expo-nential sums ∑∗ a≤n≤b g(n)exp(2piif(n)). For many functions g and f, we can extract the next term in the asymptotic, showing 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. We provide a detailed analysis of the error term in the case g(x)  = 1 and f(x)  = (x/3)3/2

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license.txt: 4065 bytes, checksum: 629f12a987f803dfc9a1edc8c6d1c34c (MD5)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).Item withdrawn by Mark Zulauf (zulauf@illinois.edu) on 2013-04-16T13:54:28Z
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Error term improvements for van der Corput transforms

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