Effective numerical evaluation of the double Hilbert transform. (20th January 2020)

• Record Type: Journal Article
• Authors: Sun, Xiaoyun; Dang, Pei; Leong, Ieng Tak; Ku, Min
• Is Part Of: Mathematical methods in the applied sciences. Volume 43:Number 7(2020)
• Page Start: 4086
• Page End: 4106
• Publication Date: 2020-01-20
• Subjects: 2D adaptive Fourier decomposition -- 2D mechanical quadrature method -- double Hilbert transform -- trigonometric interpolation; Mathematics -- Periodicals; Technology -- Mathematics -- Periodicals; 519
• DOI: 10.1002/mma.6176
• Languages: English
• ISSNs: 0170-4214

Abstract

Abstract : In this paper, we propose two methods to compute the double Hilbert transform of periodic functions. First, we establish the quadratic formula of trigonometric interpolation type for double Hilbert transform and obtain an estimation of the remainder. We call this method 2D mechanical quadrature method (2D‐MQM). Numerical experiments show that 2D‐MQM outperforms the library function "hilbert" in Matlab when the values of the functions being handled are very large or approach to infinity. Second, we propose a complex analytic method to calculate the double Hilbert transform, which is based on the 2D adaptive Fourier decomposition, and the method is called as 2D‐HAFD. In contrast to the pointwise approximation, 2D‐HAFD provides explicit rational functional approximations and is valid for all signals of finite energy.