On the monotonicity of sequences of quadrature formulae
For a given quadrature formula \(\int^ 1_ 0 f(x)dx\approx\sum^ r_{k=0}\sum^{\nu_ k-1}_{\lambda=0} a_{k\lambda}f^{(\lambda)}(x_ k):=L(f)\), \(0\leq x_ 1<x_ 2<\dots<x_ r\leq 1\), and sufficiently smooth function \(f\), the sequence \(\{S_ n(f)\}^ \infty_{n=1}\) of successive approximations of the definite integral \(I(f)=:\int^ 1_ 0 f(x)dx\) is examined, where \(S_ n(f)\) is obtained by the composition of \(L\), i.e., \(S_ n(f)=L(\varphi_ n)\), where \(\varphi_ n(x)=1/n\sum^{n-1}_{k=0} f(k+x/n)\). Sufficient conditions for monotonicity of the sequence \(\{S_ n(f)\}^ \infty_{n=1}\) are obtained. As particular cases the monotonicity of well-known Newton-Cotes and Gauss quadratures is shown. Finally, a recovery theorem based on the monotonicity results is presented.
scientific article; zbMATH DE number 4216335