TR#: | CS-2019-02 |

Class: | CS |

Title: | Exactness and Convergence Properties of Some Recent Numerical Quadrature Formulas for Supersingular Integrals of Periodic Functions |

Authors: | Avram Sidi |

Currently accessibly only within the Technion network | |

Abstract: | In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals $I[f]=\intBar^b_a f(x)\,dx$, where $f(x)=g(x)/(x-t)^3,$ assuming that $g\in C^\infty[a,b]$ and $f(x)$ is $T$-periodic, $T=b-a$. With $h=T/n$, these numerical quadrature formulas read \begin{align*} \widehat{T}{}^{(0)}_n[f]&=h\sum^{n-1}_{j=1}f(t+jh) -\frac{\pi^2}{3}\,g'(t)\,h^{-1}+\frac{1}{6}\,g'''(t)\,h,\\ \widehat{T}{}^{(1)}_n[f]&=h\sum^n_{j=1}f(t+jh-h/2) -\pi^2\,g'(t)\,h^{-1}, \\ \widehat{T}{}^{(2)}_n[f]&=2h\sum^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum^{2n}_{j=1}f(t+jh/2-h/4). \end{align*} We also showed that these formulas have spectral accuracy; that is, $$\widehat{T}{}^{(s)}_n[f]-I[f]=O(h^\mu)\quad\text{as $n\to\infty$}\quad \forall \mu>0.$$ In the present work, we continue our study of these formulas for the special case in which $f(x)=\frac{\cos\frac{\pi(x-t)}{T}}{\sin^3\frac{\pi(x-t)}{T}}\,u(x)$, where $u(x)$ is in $C^\infty(\mathbb{R})$ and is $T$-periodic. Actually, we prove that $\widehat{T}{}^{(s)}_n[f]$, $s=0,1,2,$ are exact for a class of singular integrals involving $T$-periodic trigonometric polynomials of degree at most $n-1$; that is, $$ \widehat{T}{}^{(s)}_n[f]=I[f]\quad\text{when\ \ $f(x)=\frac{\cos\frac{\pi(x-t)}{T}}{\sin^3\frac{\pi(x-t)}{T}}\,\sum^{n-1}_{m=-(n-1)} c_m\exp(\mrm{i}2m\pi x/T)$.}$$ We also prove that, when $u(z)$ is analytic in a strip $\big|\text{Im}\,z\big|<\sigma$ of the complex $z$-plane, the errors in all three $\widehat{T}{}^{(s)}_n[f]$ are $O(e^{-2n\pi\sigma/T})$ as $n\to\infty$, for all practical purposes. |

Copyright | The above paper is copyright by the Technion, Author(s), or others. Please contact the author(s) for more information |

Remark: Any link to this technical report should be to this page (http://www.cs.technion.ac.il/users/wwwb/cgi-bin/tr-info.cgi/2019/CS/CS-2019-02), rather than to the URL of the PDF files directly. The latter URLs may change without notice.

To the list of the CS technical reports of 2019

To the main CS technical reports page