Optimal quadrature formulae for differentiable functions
The existence of an optimal quadrature formula of the form \(\int^{b}_{a}f| x| dx\approx \sum^{n}_{k=1}\sum^{\nu_ k-1}_{\lambda =0}a_{k\lambda}f^{(\lambda)}(xk)\) with preassigned multiplicities \((\nu_ k)_ 1^ n\) in the classes \(LW_ q^ r<a,b>:=\{f\in C^{(r-1)}<a,b>:f^{(r-1)}\)- abs. cont., \(\| Lf\|_ q\leq 1\}\) for \(1<q\leq \infty\), where \(Ly=y^{(r)}+a_{r- 1}(x)y^{(r-1)}+...+a_ 0(x)y\) is a given linear differential operator which has the property W of Polya in \(<a,b>\) is proved and its characterization is given.
scientific article; zbMATH DE number 4013237