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

We also showed that these formulas have spectral accuracy; that is,

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,

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.

在最近的一项工作中，我们为有限范围周期超奇异积分开发了三个新的紧凑数值求积公式\(I[f]={\mathop\int{\!\!\!\!\!\!=}}^{ \,\,b}_{\!\!a} f(x)\,dx\)，其中\(f(x)=g(x)/(xt)^3,\)假设\(g \in C^\infty [a,b]\)和f ( x ) 是T周期的，\(T=ba\)。随着\(h=T/n\)，这些数值求积公式读取

我们还表明这些公式具有光谱精度；那是，

在目前的工作中，我们继续研究这些特殊情况的公式，其中\(f(x)=\frac{\cos \frac{\pi (xt)}{T}}{\sin ^3\frac {\pi (xt)}{T}}\,u(x)\)，其中u ( x ) 在\(C^\infty ({\mathbb {R}})\) 中并且是T周期的。实际上，我们证明了\({\widehat{T}}{}^{(s)}_n[f]\) , \(s=0,1,2,\)对于一类涉及T -阶次至多\(n-1\) 的周期三角多项式；那是，

我们还证明，当u ( z )在复z平面的条带\(\big |\text {Im}\,z\big |<\sigma \)中解析时，所有三个\( {\widehat{T}}{}^{(s)}_n[f]\)是\(O(e^{-2n\pi \sigma /T})\)作为\(n\rightarrow \infty \ )，用于所有实际目的。