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

TL;DR

This paper analyzes the exactness and convergence of three recent numerical quadrature formulas for periodic supersingular integrals, demonstrating their spectral accuracy and exactness for specific classes of functions, including trigonometric polynomials.

Contribution

The paper extends previous work by proving the exactness of these quadrature formulas for certain singular integrals involving periodic trigonometric polynomials and establishes exponential error decay for analytic functions.

Findings

Formulas are exact for specific classes of singular integrals with trigonometric polynomial structure.

Errors decay exponentially for functions analytic in a strip around the real axis.

Spectral accuracy of the quadrature formulas is confirmed for a broad class of periodic functions.

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, $$\widehat{T}{}^{(s)}_n[f]-I[f]=O(n^{-\mu})\quad\text{as $n\to\infty$}\quad \forall \mu>0.$$ 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.