Technical Report CS-2021-02

Title: Exponential Convergence of Some Recent Numerical Quadrature Methods for Hadamard Finite Parts of Singular Integrals of Periodic Analytic Functions
Authors: Avram Sidi
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Abstract: Let $$I[f]=\intBar^b_a f(x)\,dx,\quad f(x)=\frac{g(x)}{(x-t)^m},\quad m=1,2,\ldots,\quad a<t<b,$$ assuming that $g\in C^\infty[a,b]$ such that $f(x)$ is $T$-periodic, $T=b-a$, and $f(x)\in C^\infty(\mathbb{R}_t)$ with $\mathbb{R}_t=\mathbb{R}\setminus\{t+ kT\}^\infty_{k=-\infty}$. Here $\intBar^b_a f(x)\,dx$ stands for the Hadamard Finite Part (HFP) of the singular integral $\int^b_af(x)\,dx$ that does not exist in the regular sense. In a recent work, we unified the treatment of these HFP integrals by using a generalization of the Euler--Maclaurin expansion due to the author and developed a number of numerical quadrature formulas $\widehat{T}^{(s)}_{m,n}[f]$ of trapezoidal type for $I[f]$ for all $m$. For example, three numerical quadrature formulas of trapezoidal type result from this approach for the case $m=3$, and these are \begin{align*} \widehat{T}^{(0)}_{3,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, \quad h=\frac{T}{n},\\ \widehat{T}^{(1)}_{3,n}[f]&=h\sum^n_{j=1}f(t+jh-h/2)-\pi^2\,g'(t)\,h^{-1},\quad h=\frac{T}{n},\\ \widehat{T}^{(2)}_{3,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),\quad h=\frac{T}{n}.\end{align*} For all $m$ and $s$, we showed that all of the numerical quadrature formulas $\widehat{T}^{(s)}_{m,n}[f]$ have spectral accuracy; that is, $$ \widehat{T}^{(s)}_{m,n}[f]-I[f]=o(n^{-\mu})\quad\text{as $n\to\infty$}\quad \forall \mu>0.$$ In this work, we continue our study of convergence and extend it to functions $f(x)$ that possess certain analyticity properties. Specifically, we assume that $f(z)$, as a function of the complex variable $z$, is also analytic in the infinite strip $|\Im z|<\sigma$ for some $\sigma>0$, excluding the poles of order $m$ at the points $t+kT$, $k=0,\pm1,\pm2,\ldots.$ For $m=1,2,3,4$ and relevant $s$, we prove that $$ \widehat{T}^{(s)}_{m,n}[f]-I[f]=O\big(\exp(-2\pi n\rho/T)\big)\quad\text{as $n\to\infty$}\quad \forall \rho<\sigma.$$
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