Sharp error bounds of a quadrature rule with one multiple node for the finite Hilbert transform in some classes of continuous differentiable functions (Q558403)

Using classical techniques, the author derives error bounds for a quadrature formula for Cauchy principal value integrals \(\int_a^b (\tau-t)^{-1} f(\tau) d \tau\) where \(a<t<b\). The quadrature formula under consideration is based on an approximation of \(f\) using an \(n\)-fold node at the point \(t\). The error bounds are given in terms of the \(L_p(a,b)\)-norm of \(f^{(n)}\), where \(1 \leq p \leq \infty\). The results are extended in the usual way to a compounded variant of the quadrature formula.