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Fast Algorithms for Computing the Hilbert Transform of a Given Function with cubic splines

COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION 2024年 1-14页

作者： Yu, Bo Du, Jiaxin Qin, Xiaoxiao China Three Gorges Univ Coll Sci Three Gorges Math Res Ctr Yichang 443002 Hubei Peoples R China

It is important to compute the Hilbert transform of a given function defined on a finite interval. In 2013, Micchelli and his collaborators proposed a fast algorithm, which is called the Hilbert spline transform, to calculate the Hilbert transform of a given function on a finite interval with the computational complexity O(nlogn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(n\log n)$$\end{document}, where the spline knots were chosen to be the midpoints of sampling points. A natural question is that, whether or not the spline knots can be chosen to be the same as the sampling points. This paper gives a positive answer to this question. Besides, the analytic expression of the Hilbert transform of B-splines of any order is also established. Furthermore, the problem of how to choose spline coefficients, using the quasi-interpolation method or interpolation method, is also considered, although both make sure an optimal approximation order. Several interesting numerical examples are implemented and compared with most of the existing methods. Numerical results show that the proposed algorithm has a relatively high computational accuracy as well as a relatively low computational complexity.

关键词： The Hilbert transform The Hilbert spline transform quasi-interpolation cubic spline approximation interpolation cubic spline approximation Fast algorithm

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