Optimal cubature formulas in a reflexive Banach space (Q1856355)

Let \(\Omega \subset {\mathbb R}^n\) be a bounded domain with a sufficiently smooth boundary and let \(X\) be a Banach space whose elements form a subset of \(C({\overline \Omega})\). A sequence of cubature formulas is considered, each formula using a finite set \( \text{Nod}^{(k)} \subset C({\overline \Omega})\), \( \text{Nod}^{(k)} \subset \text{Nod}^{(k-1)}\). Given an integral over \(\Omega\) or, more generally, a linear functional \({\l}\) on \(X\), one finds a best approximation \(l_k\) in \(X^*\) to \(l\) by a linear combination of delta-functions \(\delta(x-x_j)\), \(x_j \in \text{Nod}^{(k)}\). The paper under review studies, under some conditions on \(X\), the existence, uniqueness, characterization, and convergence of the error, \(\|l-l_k\|\), to zero as \(k \to \infty\).