Optimal Quadrature of Convex Functions

Abstract

At the instigation of H. Brass we consider the following question: Let us assume that for given fixed nodes a=x₁<x₂ ...<x_n =b the values f(x_i)=:f_i , i = 1,...,n, of a convex function f:[a,b]→ IR are known. How can we compute the endpoints I, S (I<S) of the interval

$$ J\;: = \left\{ {\int\limits_a^b {g(x)dx\;:\;g\;convex\;and\;g({x_i}) = {f_i}} ,\;i = 1,...,n} \right\}? $$