Error term improvements for van der Corput transforms

Asymptotic analysis;exponential sum;trigonometric sum;van der Corput transform

Vandehey, Joseph

关键词: Asymptotic analysis;    exponential sum;    trigonometric sum;    van der Corput transform;   

Others  :  https://www.ideals.illinois.edu/bitstream/handle/2142/44398/Joseph_Vandehey.pdf?sequence=1&isAllowed=y

美国|英语

来源: The Illinois Digital Environment for Access to Learning and Scholarship

【 摘 要 】

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).