Nonlinear programming and an optimization problem on infinitely differentiable functions

Abstract

A nonnegative, infinitely differentiable function ø defined on the real line is called a Friedrichs mollifier function if it has support in [0, 1] and ∫ ¹₀ ø(t)dt=1. In this article the following problem is considered. Determine Δ _k =inf∫ ¹₀ ∣vø^(k)(t)∣dt, k=1,..., where ø^(k) denotes thekth derivative of ø and the infimum is taken over the set of all mollifier functions. This problem has applications to monotone polynomial approximation as shown by this author elsewhere. In this article, the structure of the problem of determining Δ _k is analyzed, and it is shown that the problem is reducible to a nonlinear programming problem involving the minimization of a strictly convex function of [(k−1)/2] variables, subject to a simple ordering restriction on the variables. An optimization problem on the functions of bounded variation, which is equivalent to the nonlinear programming problem, is also developed. The results of this article and those from approximation of functions theory are applied elsewhere to derive numerical values of various mathematical quantities involved in this article, e.g., Δ _k =k~2^2k−1 for allk=1, 2, ..., and to establish certain inequalities of independent interest. This article concentrates on problem reduction and equivalence, and not numerical value.