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

In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals I [ f ] = ∫ = a b f (x) d x , where f (x) = g (x) / (x - t) 3 , assuming that g ∈ C ∞ [ a , b ] and f(x) is T-periodic, T = b - a . With h = T / n , these numerical quadrature formulas read T ^ n (0) [ f ] = h ∑ j = 1 n - 1 f (t + j h) - π 2 3 g ′ (t) h - 1 + 1 6 g ′ ′ ′ (t) h , T ^ n (1) [ f ] = h ∑ j = 1 n f (t + j h - h / 2) - π 2 g ′ (t) h - 1 , T ^ n (2) [ f ] = 2 h ∑ j = 1 n f (t + j h - h / 2) - h 2 ∑ j = 1 2 n f (t + j h / 2 - h / 4). We also showed that these formulas have spectral accuracy; that is, T ^ n (s) [ f ] - I [ f ] = o (n - μ) as n → ∞ ∀ μ > 0. In the present work, we continue our study of these formulas for the special case in which f (x) = cos π (x - t) T sin 3 π (x - t) T u (x) , where u(x) is in C ∞ (R) and is T-periodic. Actually, we prove that T ^ n (s) [ 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, T ^ n (s) [ f ] = I [ f ] when f (x) = cos π (x - t) T sin 3 π (x - t) T ∑ m = - (n - 1) n - 1 c m exp (i 2 m π x / T). We also prove that, when u(z) is analytic in a strip | Im z | < σ of the complex z-plane, the errors in all three T ^ n (s) [ f ] are O (e - 2 n π σ / T) as n → ∞ , for all practical purposes. [ABSTRACT FROM AUTHOR]