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01-09-2021 | Issue 3/2021

Calcolo 3/2021

Exactness and convergence properties of some recent numerical quadrature formulas for supersingular integrals of periodic functions

Calcolo > Issue 3/2021
Avram Sidi
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In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals \(I[f]={\mathop\int{\!\!\!\!\!\!=}}^{\,\,b}_{\!\!a} f(x)\,dx\), 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
$$\begin{aligned} {\widehat{T}}{}^{(0)}_n[f]&=h\sum ^{n-1}_{j=1}f(t+jh) -\frac{\pi ^2}{3}\,g'(t)\,h^{-1}+\frac{1}{6}\,g'''(t)\,h,\\ {\widehat{T}}{}^{(1)}_n[f]&=h\sum ^n_{j=1}f(t+jh-h/2) -\pi ^2\,g'(t)\,h^{-1}, \\ {\widehat{T}}{}^{(2)}_n[f]&=2h\sum ^n_{j=1}f(t+jh-h/2)- \frac{h}{2}\sum ^{2n}_{j=1}f(t+jh/2-h/4). \end{aligned}$$
We also showed that these formulas have spectral accuracy; that is,
$$\begin{aligned} {\widehat{T}}{}^{(s)}_n[f]-I[f]=o(n^{-\mu })\quad \text {as }n\rightarrow \infty \quad \forall \mu >0. \end{aligned}$$
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,
$$\begin{aligned} {\widehat{T}}{}^{(s)}_n[f]=I[f]\quad \text {when } f(x)=\frac{\cos \frac{\pi (x-t)}{T}}{\sin ^3\frac{\pi (x-t)}{T}}\,\sum ^{n-1}_{m=-(n-1)} c_m\exp (\mathrm {i}2m\pi x/T). \end{aligned}$$
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 \(n\rightarrow \infty \), for all practical purposes.

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