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Effective numerical evaluation of the double Hilbert transform
Author(s) -
Sun Xiaoyun,
Dang Pei,
Leong Ieng Tak,
Ku Min
Publication year - 2020
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.6176
Subject(s) - mathematics , hilbert transform , pointwise , trigonometric interpolation , interpolation (computer graphics) , hilbert–huang transform , mathematical analysis , energy (signal processing) , polynomial interpolation , polynomial , linear interpolation , animation , statistics , spectral density , computer graphics (images) , computer science
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.