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{π^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) -π^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, $$\hat{T}{}_n^{(s)}[f]-I[f]=O(n^{-\mu })\text{as n→∞}\forall \mu >0.$$ 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 $\hat{T}{}_n^{(s)}[f]$, $s=0,1,2,$ are exact for a class of singular integrals involving $T$-periodic trigonometric polynomials of degree at most $n-1$; that is, $$ \widehat{T}{}^{(s)}_n[f]=I[f]\quad\text{when\ \ $f(x)=\frac{\cos\frac{π(x-t)}{T}}{\sin^3\frac{π(x-t)}{T}}\,\sum^{n-1}_{m=-(n-1)} c_m\exp(\mrm{i}2mπx/T)$.}$$ We also prove that, when $u(z)$ is analytic in a strip $\left|\text{Im}z\right|<\sigma$ of the complex $z$-plane, the errors in all three $\hat{T}{}_n^{(s)}[f]$ are $O(e^{-2n\pi \sigma /T})$ as $n\to \infty$, for all practical purposes.

Number of papges: 13