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