周期函数超奇异积分的某些最新数值求积公式的精确性和收敛性
Exactness and Convergence Properties of Some Recent Numerical Quadrature Formulas for Supersingular Integrals of Periodic Functions
摘要：
在最近的工作中，我们开发了三个新的紧凑的数值求积公式有限范围周期超奇异积分$I[f]=\intBar^b_a f(x)\,dx$，其中$f(x)=g(x)/(x-t)^3,$假设$g\in C^\infty[a,b]$和$f(x)$是$T$ -周期，$T=b-a$。与$h=T/n$，这些数值求积公式读\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*}我们还表明，这些公式具有频谱精度，即$$\widehat{T}{}^{(s)}_n[f]-I[f]=O(n^{-μ})\quad\text{as $n\to\infty$}\quad \forall μ>0.$$在目前的工作中，我们继续我们的研究这些公式的特殊情况下，其中$f(x)=\frac{\cos\frac{π(x-t)}{T}}{\sin^3\frac{π(x-t)}{T}}\,u(x)$，其中$u(x)$是在$C^\infty(\mathbb{R})$和$T$ -周期。实际上，我们证明了$\widehat{T}{}^{(s)}_n[f]$，$s=0,1,2,$是准确的一类奇异积分涉及$T$ -周期三角多项式的次数最多$n-1$，即$$ \widehat{T}{}^{(s)}_n[f]=I[f]\quad\text{when\ \ $f(x)=\frac{\cos\frac{π(x-t)}{T}}{\sin^3\frac{π(x-t)}{T}}\,\sum^{n-1}_{m=-(n-1)} c_m\exp(\mrm{i}2mπx/T)$.}$$我们还证明，当$u(z)$是分析在一个带$\big|\text{Im}\,z\big|<σ$的复杂$z$ -平面，在所有三个错误$\widehat{T}{}^{(s)}_n[f]$ $O(e^{-2nπσ/T})$作为$n\to\infty$，为所有实际目的。
英文摘要：
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, $$\widehat{T}{}^{(s)}_n[f]-I[f]=O(n^{-μ})\quad\text{as $n\to\infty$}\quad \forall μ>0.$$ In the present work, we continue our study of these formulas for the special case in which $f(x)=\frac{\cos\frac{π(x-t)}{T}}{\sin^3\frac{π(x-t)}{T}}\,u(x)$, 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, $$ \widehat{T}{}^{(s)}_n[f]=I[f]\quad\text{when\ \ $f(x)=\frac{\cos\frac{π(x-t)}{T}}{\sin^3\frac{π(x-t)}{T}}\,\sum^{n-1}_{m=-(n-1)} c_m\exp(\mrm{i}2mπx/T)$.}$$ We also prove that, when $u(z)$ is analytic in a strip $\big|\text{Im}\,z\big|<σ$ of the complex $z$-plane, the errors in all three $\widehat{T}{}^{(s)}_n[f]$ are $O(e^{-2nπσ/T})$ as $n\to\infty$, for all practical purposes.
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