On the weights of certain quadratures for the numerical evaluation of Cauchy principal value integrals and their derivatives

Summary

We consider interpolatory quadrature rules of the type

$$\begin{gathered} \frac{{d^k }}{{d\lambda ^k }}\int\limits_{ - 1}^1 {(1 - x)^\alpha (1 + x)^\beta } \frac{{f(x)}}{{x - \lambda }}dx = \sum\limits_{i = 1}^n {w_{ni}^{(k)} (\lambda )f(x_{ni} ) + R_n^{(k)} (f),} \hfill \\ k = 0,1,..., \alpha ,\beta > 0,--- \hfill \\ \end{gathered}$$

where the nodes {x _ni } are the zeros of the Jacobi polynomialP ^(α,β) _n (x). In particular we prove that

$$\sum\limits_{i = 1}^n {\left| {w_{ni}^{(k)} (\lambda )} \right| = O(n^k \log n)}$$

uniformly in −1+ε≦λ≦1−ε. Furthermore we show that whenk=1,f(x) ∈C ^p[−1, 1],p≧1, andf ^(p)(x)∈H _μ[−1, 1], 0<μ≦1, we have

$$R_n^{(1)} (f) = O(n^{ - p - \mu + 1 + \delta } ),$$

δ>0 small as we like, uniformly in −1+ε≦λ≦1−ε.