A generalization of a theorem of Brass and Schmeisser

TL;DR

This paper extends a known inequality involving quadrature rules and convex functions to more general measures and all positive integers, broadening the applicability of the original theorem.

Contribution

It generalizes Brass and Schmeisser's theorem by replacing the quadrature with a measure-based integral and extending the result to all positive integers.

Findings

The inequality holds for any positive integer n.

The result applies to integrals with respect to arbitrary measures.

It covers cases with even n using Radau quadratures.

Abstract

Let $n$ be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature $$\mathcal{Q}=\alpha_1f(x_1)+\dots+\alpha_mf(x_m),$$ (with positive weights) of order at least $n+1$ and for every $n-$convex function $f,$ the value of $Q$ on $f$ lies between the values of Gauss and Lobatto quadratures of order $n+1$ calculated for the same function $f$. We generalize this result in two directions, replacing $Q$ by an integral with respect to a given measure and allowing the number $n$ to any positive integer (for even $n$ Radau quadratures replace Gauss and Lobatto ones