Exponential convergence of some recent numerical quadrature methods for Hadamard finite parts of singular integrals of periodic analytic functions

Let I[f] = (sic)(a)(b) f(x) dx, f(x) = g(x)/(x - t)(m), m = 1, 2, . . ., a < t< b, assuming that g is an element of C-infinity[a, b] such that f (x) is T -periodic, T = b - a, and f (x) is an element of C-infinity (R-t) with R-t = R\{t + kT}(k=-infinity)(infinity). Here (sic)(a)(b) f (x) dx stands for the Hadamard Finite Part (HFP) of the singular integral integral(b)(a) f ( 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 (T) over cap ((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 (T) over cap ((0))(3,n)[f] = h Sigma(n=1)(j=1) f(t + jh) - pi(2)/3 g'(t) h(-1) + 1/6 g''' (t) h, h = T/n, (T) over cap ((1))(3,n)[f] = h Sigma(n)(j=1) f(t + jh - h/2) - pi(2) g'(t) h(-1), h = T/n, (T) over cap ((2))(3,n)[f] = 2h Sigma(n)(j=1) f(t + jh - h/2) - h/2 Sigma(2n)(j=1) f(t + jh/2 - h/4), h = T/n. For all m and s, we showed that all of the numerical quadrature formulas (T) over cap ((s))(m,n)[f] have spectral accuracy; that is, (T) over cap ((s))(m,n)[f] - I[f] = o(n(-mu)) as n -> infinity for all 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 vertical bar Im z vertical bar < sigma for some sigma > 0, excluding the poles of order m at the points t + kT, k = 0, +/- 1, +/- 2, .... For m = 1, 2, 3, 4 and relevant s, we prove that (T) over cap ((s))(m,n)[f] - I[f] = Oexp(-2 pi n rho/T)) as n -> infinity for all rho < sigma.

Automated summary

In this study, we unified the treatment of HFP integrals and developed numerical quadrature formulas to improve the accuracy of computations. We also extended the convergence analysis to functions with certain analytic properties and proved error bounds for the numerical quadrature formulas.