Exactness and Convergence Properties of Some Recent Numerical Quadrature Formulas for Supersingular Integrals of Periodic Functions
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In a recent work, we developed three new compact numerical quadrature formulas for finite-range periodic supersingular integrals I[f]=^b_a f(x) dx, 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^(0)_n[f] =h∑^n-1_j=1f(t+jh) -π^2/3 g'(t) h^-1+1/6 g”'(t) h, T^(1)_n[f] =h∑^n_j=1f(t+jh-h/2) -π^2 g'(t) h^-1, T^(2)_n[f] =2h∑^n_j=1f(t+jh-h/2)- h/2∑^2n_j=1f(t+jh/2-h/4). We also showed that these formulas have spectral accuracy; that is, T^(s)_n[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)=cosπ(x-t)/T/sin^3π(x-t)/T u(x), where u(x) is in C^∞(ℝ) and is T-periodic. Actually, we prove that 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, T^(s)_n[f]=I[f] when f(x)=cosπ(x-t)/T/sin^3π(x-t)/T ∑^n-1_m=-(n-1) c_m(i2mπ 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^(s)_n[f] are O(e^-2nπσ/T) as n→∞, for all practical purposes.
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