A generalization of a theorem of Brass and Schmeisser

Abstract

Let n be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature $$\mathcal {Q}=\alpha _1f(x_1)+\dots +\alpha _mf(x_m)$$ Q = α 1 f ( x 1 ) + ⋯ + α m f ( x m ) (with positive weights) of order at least $$n+1$$ n + 1 and for every $$n-$$ n - convex function f, the value of Q on f lies between the values of Gauss-Legendre and Gauss-Lobatto quadratures of order $$n+1$$ 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 Gauss-Radau quadratures replace Gauss-Legendre and Gauss-Lobatto ones).