Weighted stationary phase of higher orders
Weighted stationary phase of higher orders
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DOI：
10.1007/s11464-016-0615-y
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发表时间：
2016-01
影响因子：
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通讯作者：
M. McKee;Haiwei Sun;Y. Ye
中科院分区：
文献类型：
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作者：
M. McKee;Haiwei Sun;Y. Ye
本文的主题是有限区间[αβ]上Phasef(X)加权的BYG(X)具有指数振动性的积分：当Phasef(X)在(αβ)中只有一个驻点时，证明了该积分在≥~2上的一个n阶渐近展开式：这个渐近展开式加强了M.N.Huxley关于n=1的经典结果.V.Blmer，R.Khan和M.Young在f(X)和g(X)光滑且g(X)紧支撑于R：的假设下证明了类似的渐近展开式。然而，本文仅假设这两个函数在[αβ]2n+3和2n+1次上连续可微。由于对OfG(X)及其导数在端点α和β的消失没有要求，目前的渐近展开式在主项和误差项中包含显式的边界项。因此，本文的渐近展开式适用于分析、解析数论等领域中更广泛的一类问题。
The subject matter of this paper is an integral with exponential oscillation of phasef(x) weighted byg(x) on a finite interval [α β]: When the phasef(x) has a single stationary point in (α β), an nth-order asymptotic expansion of this integral is proved forn≥ 2: This asymptotic expansion sharpens the classical result forn= 1 by M. N. Huxley. A similar asymptotic expansion was proved by V. Blomer, R. Khan and M. Young under the assumptions thatf(x) andg(x) are smooth andg(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 ofg(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.