论文标题
希尔伯特变换的有限零件集成
Finite-Part Integration of the Hilbert Transform
论文作者
论文摘要
单方面和完整的希尔伯特变换通过有限零件集成的方法进行精确评估[E.A. Galapon，\ textit {proc。罗伊。 Soc。 a} \ textbf {473}，20160567（2017）]。通常，结果由两个术语组成 - 第一个是无限的不同积分的有限零件，第二个是由转换核的奇异性产生的贡献。第一个项恰恰是当转换内核以转换参数的正幂进行二元扩展时获得的结果，其次是按学期集成，而所得的发散积分分配的值等于其有限的零件。在所有情况下，都存在有限的部分贡献，而奇异贡献的存在或不存在取决于整合的间隔和函数在围绕原点转换下的均衡。从希尔伯特变换的确切评估中，获得了任意小参数的主要渐近行为。
The one-sided and full Hilbert transforms are evaluated exactly by means of the method of finite-part integration [E.A. Galapon, \textit{Proc. Roy. Soc. A} \textbf{473}, 20160567 (2017)]. In general, the result consists of two terms -- the first is an infinite series of finite-part of divergent integrals, and the second is a contribution arising from the singularity of the kernel of transformation. The first term is precisely the result obtained when the kernel of transformation is binomially expanded in positive powers of the parameter of transformation, followed by term-by-term integration, and the resulting divergent integrals assigned values equal to their finite-parts. In all cases, the finite-part contribution is present while the presence or absence of the singular contribution depends on the interval of integration and on the parity of the function under transformation about the origin. From the exact evaluation of the Hilbert transform, the dominant asymptotic behavior for arbitrarily small parameter is obtained.
