On approximation of linear functionals over convex functions: Construction techniques and new directions

Recently there has been renewed interest in the problem of finding under and over estimations on the set of convex functions to a given non-negative linear functional; that is, approximations which estimate always below (or above) the functional over a family of convex functions. The most important example of such an approximation problem is given by the multidimensional versions of the midpoint (rectangle) rule and the trapezoidal rule, which provide under and over estimations to the true value of the integral on the set of convex functions (also known as the Hermite-Hadamard inequality). In this paper, we introduce a general method of constructing new families of under/over-estimators on the set of convex functions for a general class of linear functionals. In particular, under the regularity condition, namely the functions belonging to C2(Ω) (not necessarily convex), we will show that the error estimations based on such estimators are always controlled by the error associated with using the quadratic function. The result is also extended to the class of Lipschitz functions. We also propose a modified approximation technique to derive a general class of under/over estimators with better error estimates.