UNIFORM-CONVERGENCE OF OPTIMAL ORDER QUADRATURE-RULES FOR CAUCHY PRINCIPAL VALUE INTEGRALS

[摘要] For the numerical evaluation of Cauchy principal value integrals of the form f1(-1) f(x)/(x - lambda)dx, lambda is-an-element-of (-1, 1), f is-an-element-of C(s)[ - 1, 1], we consider a quandrature method based on spline interpolation of odd degree 2k + 1, k is-an-element-of N0. We show that these rules converge uniformly for lambda is-an-element-of ( - 1, 1). In particular, we calculate the exact order of magnitude of the error and show that it is equal to the order of the optimal remainder in the class of functions with bounded sth derivative if s is-an-element-of {2k + 1, 2k + 2}. Finally, we compare the rule to the well-known quandrature rule of Elliott and Paget which only converges pointwise.

[发布日期] 1994-12-30

[关键词] CAUCHY PRINCIPAL VALUE INTEGRALS;QUADRATURE FORMULA;SPLINE INTERPOLATION;OPTIMAL ORDER OF CONVERGENCE;PEANO CONSTANTS;UNIFORM CONVERGENCE