On the convergence of interpolatory-type quadrature rules for evaluating Cauchy integrals
The Cauchy principal value integral \(Q(f; t)\) with respect to the singularity \(t\) is transformed to \(H(h;x)\), where \(h(y)= f(\cos y)\) and \(t= \cos x\), and approximated by quadrature rules \(H_N(h;x)\) based on trigonometric interpolations. The \(N+1\) weights depend also from the singularity and the size of the stability factor \(\Lambda_N(x)= \sum''|\omega^N_k(x)|\) is of great importance for stability and convergence. The main results are the almost optimal estimate of the stability factor and the estimate of the remainder term \[ \Lambda_N(x)\leq C\Biggl({\log N+1\over \sin^2x}\Biggr)\quad\text{and}\quad |R_N(h; x)|\leq CM{N\log N\over 2^{N-1}}, \] respectively. Further, if the derivative \(f^{(p)}(t)\) satisfies the Hölder continuity of order \(\rho\), then the remainder term satisfies \(O((A+B\log N+ N^{2\nu})/N^{p+\rho})\).
scientific article; zbMATH DE number 1872924