Error term improvements for van der Corput transforms

Vandehey, Joseph

Permalink

https://hdl.handle.net/2142/44398 Copy

Description

Title

Error term improvements for van der Corput transforms

Author(s)

Vandehey, Joseph

Issue Date

2013-05-24T22:10:16Z

Director of Research (if dissertation) or Advisor (if thesis)

• Ford, Kevin
• Boca, Florin

Doctoral Committee Chair(s)

• Athreya, Jayadev S.
• Hildebrand, A.J.

Committee Member(s)

• Ford, Kevin
• Zaharescu, Alexandru
• Boca, Florin

Department of Study

Mathematics

Discipline

Mathematics

Degree Granting Institution

University of Illinois at Urbana-Champaign

Degree Name

Ph.D.

Degree Level

Dissertation

Keyword(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).

Graduation Semester

2013-05

Permalink

http://hdl.handle.net/2142/44398

Copyright and License Information

Copyright 2013 Joseph Vandehey

Owning Collections