Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals

We show that, if f epsilon C-k[-1,1] (k greater than or equal to 2), the error term of every modified positive interpolatory quadrature rule for Cauchy principal value integrals of the type f(-1)(1) w(x)x-lambda\f(x) dx, lambda epsilon (-1,1), fulfills R(n) [f;lambda] = O(n(-k) ln n) uniformly for all lambda epsilon (-1,1), and hence it is of optimal order of magnitude in the classes C-k[-1,1] (k = 2, 3, 4,...). Here, w is a weight function with the property 0 less than or equal to w(x)root 1-x(2) less than or equal to C. We give explicit upper bounds for the Peanotype error constants of such rules. This improves and completes earlier results by Criscuolo and Mastroianni (Calcolo 22 (1985), 391-441 and Numer. Math. 54 (1989), 445-461) and Ioakimidis (Math. Comp. 44 (1985), 191-198). For the special case of the Gaussian rule, we show that the restriction k greater than or equal to 2 can be dropped. The results are based on a new representation of the Peano kernels of these formulae via the Peano kernels of the underlying classical quadrature formulae. This representation may also be useful in connection with some different problems.