摘要
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 n ≥ 2. 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 R. 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.
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 n ≥ 2. 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 R. 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.
基金
The authors would like to thank anonymous referees for critical readings and thoughtful comments. Gratitude is also due to Xiumin Ren who gave the authors many helpful suggestions. The second author was partially supported by the National Natural Science Foundation of China （Grant No. 11601271）.
