A method and a program for the numerical evaluation of the Hilbert transform of a real function

Abstract In the first part of this paper a method for the evaluation of the Hilbert transform of a function, approximated by piecewise polynomials, is presented. In the second part a program is presented to be used when the function is approximated by cubic splines. We show that the Hilbert transform displays the same strong convergence properties as the cubic splines. The asymptotic properties of the Hilbert transform are shown to be very well reproduced. Title of program: FHT Catalogue Id: ABVD_v1_0 Nature of problem This program calculates the Hilbert transform of a continuous function. The evaluation of the matrix elements of the Green's function in quantum mechanics involves the evaluation of the Hilbert transform of the matrix elements of the Hamiltonian operator. The same type of problem appears in many other fields of applied physics. The program assumes that a cubic spline interpolation has been performed over the integration region. As the Hilbert transform function is evaluated at a point of the com ... Versions of this program held in the CPC repository in Mendeley Data ABVD_v1_0; FHT; 10.1016/0010-4655(80)90008-9 This program has been imported from the CPC Program Library held at Queen's University Belfast (1969-2018)**摘要** 本文第一部分提出了一种针对以分段多项式逼近的函数的希尔伯特变换（Hilbert transform）评估方法。第二部分给出了适用于以三次样条（cubic splines）逼近的函数的配套程序。研究表明，希尔伯特变换具备与三次样条一致的强收敛性质，其渐近性质可得到极佳的重现。 程序标题：FHT 目录编号：ABVD_v1_0 问题本质：本程序用于计算连续函数的希尔伯特变换。在量子力学中，格林函数（Green's function）矩阵元的求解涉及哈密顿算符（Hamiltonian operator）矩阵元的希尔伯特变换计算，此类问题同样广泛存在于诸多应用物理学领域。本程序假定已在积分区域上完成三次样条插值（cubic spline interpolation）。当希尔伯特变换函数于某一……点处被求值时…… 本程序收录于Mendeley数据中的CPC程序库版本：ABVD_v1_0; FHT; 10.1016/0010-4655(80)90008-9 本程序源自贝尔法斯特女王大学维护的CPC程序库（1969-2018）。