Unbounded divergence of simple quadrature formulas (Q1180592)

Quadrature formulas of the following type are considered: \[ \int_ a^ b x(t)du(t)=Q_ n(x)+R_ n(x), \qquad x\in C[a,b],\tag{QF} \] where \(Q_ n(x)=\sum_{i=1}^{m_ n} c_ n^ i x(t_ n^ i)\), \(c_ n^ i\in\mathbb{R}\), for \(n\in\mathbb{N}\), \((m_ n)_{n\in\mathbb{N}}\) being an increasing sequence of positive integers, \(t_ n^ i\in[a,b]\), \(i=1,\dots,m_ n\), and \(u: [a,b]\to\mathbb{R}\) is a non-constant continuous function of bounded variation. Based on a result about unbounded sequences in the dual \(X^*\) of a Banach space \(X\) and a classical result abut the convergence of quadrature formulas by Pólya it is shown that the set \(S=\{x\in C[a,b]\): \(\sup\{| Q_ n(x)|\): \(n\in\mathbb{N}\}=\infty\}\) is superdense in \(C[a,b]\), i.e., \(S\) is an uncountably infinite dense \(G_ \delta\)-set in \(C[a,b]\), if \(\sup\bigl\{ \sum_{i=1}^{m_ n} | c_ n^ i|\): \(n\in\mathbb{N}\bigr\}=\infty\). Then it is shown that (QF) is divergent, if \(c_ n^ i=c_ i\), \(t_ n^ i=t_ i\) for \(i\in\{1,\dots,m_ n\}\), \(n\in\mathbb{N}\), and all the nodes \(t_ i\), \(i\in\mathbb{N}\), are distinct. Finally, it is shown that under these assumptions (QF) is unboundedly divergent at some \(x\in C[a,b]\), if and only if \(\sum_{i=1}^ \infty| c_ i|=\infty\). Moreover, this condition implies the unbounded divergence of \((QF)\) on a superdense subset of \(C[a,b]\).