Frontiers of Mathematics in China >
Weighted stationary phase of higher orders
Received date: 28 Jan 2016
Accepted date: 09 Nov 2016
Published date: 20 Apr 2017
Copyright
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.
Key words: First derivative test; weighted stationary phase
Mark MCKEE , Haiwei SUN , Yangbo YE . Weighted stationary phase of higher orders[J]. Frontiers of Mathematics in China, 2017 , 12(3) : 675 -702 . DOI: 10.1007/s11464-016-0615-y
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