论文标题
带有正力和渐近扩展的相位函数的振荡积分
Oscillatory integrals with phase functions of positive real powers and asymptotic expansions
论文作者
论文摘要
关于将振荡积分扩展到参数的渐近系列的方法，非分类相的固定相的方法以及使用奇异性分辨率分辨率的方法的方法是已知的。本文的目的是将前者扩展到具有积极实际力量的退化相，而无需使用奇异性。为此，我们首先将菲涅尔积分概括为具有积极实力的相位函数的振荡积分。接下来，通过使用此结果，我们将具有渐近积分的渐近膨胀，用于退化相位，具有正真实力量，包括中度振荡，以及一个变量中更广泛的振幅类。此外，我们为退化相位获得振荡积分的渐近扩展，包括每个变量中的单个单元总和，包括$ a_ {k} $，$ e_6 $，$ e_6 $，$ e_8 $ in multivariable。
As to methods for expanding an oscillatory integral into an asymptotic series with respect to the parameter, the method of stationary phase for the non-degenerate phases and the method of using resolution of singularities for degenerate phases are known. The aim of this paper is to extend the former for degenerate phases with positive real powers without using resolution of singularities. For this aim, we first generalize Fresnel integrals as oscillatory integrals with phase functions of positive real powers. Next, by using this result, we have asymptotic expansions of oscillatory integrals for degenerate phases with positive real powers including moderate oscillations and for a wider amplitude class in one variable. Moreover, we obtain asymptotic expansions of oscillatory integrals for degenerate phases consisting of sums of monomials in each variable including the types $A_{k}$, $E_6$, $E_8$ in multivariable.
