In this paper, we are concerned with the numerical approximations of one-sided Hilbert transforms with oscillatory kernel by means of the multiple integrals. This type of Hilbert transform has two computing difficulties: singularity and oscillation. To avoid the singularity, we transfer the Hilbert transform to an individual oscillatory integral which can be analytically calculated and a non-singular integral which can be well evaluated by the multiple integrals. Numerical examples are provided to illustrate the advantages of the proposed methods.

中文翻译：

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高振荡希尔伯特变换的数值逼近

在本文中，我们关注带有振荡核的单边希尔伯特变换的数值积分，该数值逼近是多个积分。这种类型的希尔伯特变换具有两个计算难题：奇点和振荡。为了避免奇异性，我们将希尔伯特变换转移到可以分析计算的单个振荡积分和可以通过多个积分很好地评估的非奇异积分。数值例子说明了所提出方法的优点。