Convergence of Gauss-Lobatto type quadrature formulas for the evaluation of principal value singular integrals
Let w(x) be the generalized smooth Jacobi weight defined by \[ w(x)=\psi (x)(1-x)^{\alpha}\prod^{s}_{k=1}| x-t_ k|^{\gamma_ k}(1+x)^{\beta}, \] \(\alpha,\beta,\gamma_ k>-1\) \((k=1,2,...,s)\), \(-1<t_ 1<t_ 2<...<t_ s<1\), \(\psi\) (x) positive and continuous on [-1,1] and such that \(\int^{1}_{0}u^{-1}\omega (\psi;u)du<\infty,\) where \(\omega\) (\(\psi\) ;u) is the modulus of continuity of \(\psi\). The author considers the Gauss-Lobatto type quadrature rule \[ \int^{1}_{- 1}w(x)f(x)dx=\sum^{m}_{i=1}A_{m,i}f(x_{m,i})+\sum^{p- 1}_{j=0}[\bar H_{m,j}f^{(i)}(-1)+\bar K_{m,j}f^{(i)}(1)]+E[f] \] which can be applied for evaluating Cauchy singular integrals. Bounds for the weights \(\bar H{}_{m,j}\) and for the remainder E[f] are given.
scientific article; zbMATH DE number 4023943