Lower bounds for the error of quadrature formulas for Hilbert spaces

We prove lower bounds for the worst case error of quadrature formulas that use given sample points X-n = {x(1), ..., x(n)}. We are mainly interested in optimal point sets X-n, but also prove lower bounds that hold with high probability for sets of independently and uniformly distributed points. As a tool, we use a recent result (and extensions thereof) of Vybiral on the positive semi-definiteness of certain matrices related to the product theorem of Schur. The new technique also works for spaces of analytic functions where known methods based on decomposable kernels cannot be applied. (C) 2020 Elsevier Inc. All rights reserved.

Automated summary

We investigate the lower bounds for the worst case error of quadrature formulas using given sample points, focusing on optimal point sets and independently and uniformly distributed points. By utilizing recent results on the positive semi-definiteness of certain matrices related to the product theorem of Schur by Vybiral, our new technique extends to spaces of analytic functions where traditional methods are not applicable.