第一类振荡积分的推广及其估计

Generalization of the First Kind of Oscillatory Integrals and Its Estimates

研究经典的第一类振荡积分的推广,即将经典的第一类振荡积分中的指数函数换成满足某个线性微分方程的实函数后所对应的振荡积分I(λ).首先讨论I(λ)的局部性质,即I(λ)随λ的变化情况(限定λ>0);其次给出当相位函数φ(x)满足|φ^(k)(x)|≥1时,I(λ)的一个大小估计.

This paper focus on the generalization of the first kind of oscillatory integrals I(λ),namely the one derived from replacing the exponential function of the integrand in the classical first kind of oscillatory integrals by a real function,satisfying some differential equation.First,we investigate the local properties of I(λ),that is,how I(λ)develops with λvarying(restricting that λ>0). Then we gave an estimate of I(λ),when the phase functionφ(x)satisfies |φ^(k)(x)|≥1.