On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals (Q1123332)

In this paper is proposed a modified Gaussian rule \(\phi^*_ n(wf;t)\) for computing of the integral \(\phi\) (wf;t) in the Cauchy principal value sense with the weight w and there is proved the convergence in closed sets contained in the integration interval. The main purpose of the present work is to prove uniform convergence of the sequence \(\{\phi^*_ n(wf;t)\}\) on the whole integration interval and to give estimates for the remainder term. The same results are shown for particular subsequences of the Gaussian rules \(\phi_ m(wf;t)\) for the evaluation of Cauchy principal value integrals. There is given an application to the numerical solutions of singular integral equations.