|
|
Weighted stationary phase of higher orders |
Mark MCKEE1, Haiwei SUN2(), Yangbo YE1 |
1. Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, USA 2. School of Mathematics and Statistics, Shandong University, Weihai 264209, China |
|
|
Abstract The subject matter of this paper is an integral with exponential oscillation of phase f(x) weighted by g(x) on a finite interval [α, β]. When the phase f(x) has a single stationary point in (α, β), an nth-order asymptotic expansion of this integral is proved for . This asymptotic expansion sharpens the classical result for n = 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions that f(x) and g(x) are smooth and g(x) is compactly supported on . In the present paper, however, these functions are only assumed to be continuously differentiable on [α, β] 2n + 3 and 2n + 1 times, respectively. Because there are no requirements on the vanishing of g(x) and its derivatives at the endpoints α and β, the present asymptotic expansion contains explicit boundary terms in the main and error terms. The asymptotic expansion in this paper is thus applicable to a wider class of problems in analysis, analytic number theory, and other fields.
|
Keywords
First derivative test
weighted stationary phase
|
Corresponding Author(s):
Haiwei SUN
|
Issue Date: 20 April 2017
|
|
1 |
BlomerV, KhanR, YoungM. Distribution of mass of holomorphic cusp forms.Duke Math J, 2013, 162(14): 2609–2644
https://doi.org/10.1215/00127094-2380967
|
2 |
GradshteynI S, RyzhikI M. Table of Integrals, Series, and Products. 6th ed.San Diego: Academic Press, 2000; online errata:
|
3 |
HuxleyM N. Area, Lattice Points, and Exponential Sums.London Math Soc Monogr New Ser, Vol 13. Oxford: Clarendon Press, 1996
|
4 |
JutilaM, MotohashiY. Uniform bound for Hecke L-functions.Acta Math, 2005, 195(1): 61–115
https://doi.org/10.1007/BF02588051
|
5 |
McKeeM, HaiweiSun, YangboYe. Improved subconvexity bounds for GL(2)×GL(3) and GL(3) L-functions.Preprint
|
6 |
SalazarN, YangboYe, Spectral square moments of a resonance sum for Maass forms.Front Math China, 2017 (to appear)
https://doi.org/10.1007/s11464-016-0621-0
|
7 |
WolffT H. Lectures on Harmonic Analysis. Edited by Laba I, Shubin C.University Lecture Series, Vol 29. Providence: Amer Math Soc, 2003
https://doi.org/10.1090/ulect/029
|
|
Viewed |
|
|
|
Full text
|
|
|
|
|
Abstract
|
|
|
|
|
Cited |
|
|
|
|
|
Shared |
|
|
|
|
|
Discussed |
|
|
|
|