Optimal Quadrature for Convex Functions and Generalizations

Abstract

Let C _uv denote the class of functions f, convex on [a, b], and satisfying ${f'_ + }\left( a \right) \geqslant u,f' - (b) \leqslant v$ . We display an affine formula that yields an optimal estimate of $\int_b^a {f(x)dx}$ for f ϵ C _uv , among all methods based solely on function evaluation at N points of [a, b]. A similar approach achieves analogous results for monotone functions and for n-convex functions.