A note on the optimal quadrature in \(H^ p\) (Q797763)

In the paper we study quadrature formulae of the form \(\int^{1}_{- 1}f(x)dx\approx\sum^{n}_{k=1}\sum^{\nu_ k-1}_{\lambda =0}a_{k\lambda}f^{(\lambda)}(x_ k)\) where \(-1<x_ 1<...<x_ n<1\). The existence of optimal nodes with preassigned multiplicities is proved for the Hardy spaces \(H^ p\) with \(1<p<\infty\). This result is then used to show that the exact order of convergence for the optimal quadrature formula with N nodes (including multiplicity) is \(N^{1/(2q)}\exp (- \pi\sqrt{N/q})\) where \(1/p+1/q=1\) and 1\(\leq p\leq\infty \).