论文标题
对某些振荡函数的傅立叶宽段变换的渐近分析
An asymptotic analysis of the Fourier-Laplace transforms of certain oscillatory functions
论文作者
论文摘要
我们研究傅立叶宽段的家族转换$$ f_ {α，β}（z）= \ permatorName*{f.p。} \ int_ {0}^{\ infty} t^ββ\ exp（\ exp） \ operatotorname*{im} z <0，$ a> 1 $和$β\ in \ mathbb {c} $，其中hadamard有限零件用于在$ \ operatatorName*{re} re}β\ leq -1 $时正规化积分。我们证明，每个$ f_ {α，β} $都对整个复杂平面都具有分析性延续，并通过原点确定其沿任何线的渐近学。我们还运用了我们的想法来表明其中一些功能为Wiener-ikehara定理提供了具体的极端示例，以及Ingham-Karamata定理的量化版本，为这些复杂的Tauberian定理提供了新的简单和建设性的优化结果。
We study the family of Fourier-Laplace transforms $$ F_{α,β}(z)= \operatorname*{F.p.} \int_{0}^{\infty} t^β\exp(\mathrm{i} t^α-\mathrm{i} z t)\:\mathrm{d} t, \quad \operatorname*{Im} z<0, $$ for $α>1$ and $β\in\mathbb{C}$, where Hadamard finite part is used to regularize the integral when $\operatorname*{Re} β\leq -1$. We prove that each $F_{α,β}$ has analytic continuation to the whole complex plane and determine its asymptotics along any line through the origin. We also apply our ideas to show that some of these functions provide concrete extremal examples for the Wiener-Ikehara theorem and a quantified version of the Ingham-Karamata theorem, supplying new simple and constructive proofs of optimality results for these complex Tauberian theorems.
