Research Reports from the Department of Operations
Document Type
Report
Publication Date
4-1-1976
Abstract
A nonnegative, infinitely differentiable function φ defined on the real line is called a Friedrichs mollifier function if it has support in [0,1] and ∫₀¹ φ(t)dt = 1. In this article the following problem is considered: Determine Ak = inf ∫₀¹ φ(k)(t)dt, k=1,2,..., where φ(k) denotes the kth derivative of φ and the infimum is taken over the set of all mollifier functions, which is a convex set. This problem has applications to monotone polynomial approximation as shown by this author in Ref. 2. The problem is reducible to three equivalent problems--a nonlinear programming problem, a problem on the functions of bounded variation and an approximation problem involving Tchebycheff polynomials. The principle result of this article shows that Ak = k! 2^(k-1), k=1,2,.... The numerical values of the optimal solutions of the three problems are obtained as a function of k. Some inequalities of independent interest are also derived.
Keywords
Operations research, Nonlinear programming, Mathematical optimization, Approximation theory, Differentiable functions, Chebyshev polynomials
Publication Title
Technical Memorandums from the Department of Operations, School of Management, Case Western Reserve University
Issue
Technical memorandum no. 408
Rights
This work is in the public domain and may be freely downloaded for personal or academic use
Recommended Citation
Ubhaya, Vasant A., "Nonlinear Programming, Approximation, and Optimization on Infinitely Differentiable Functions" (1976). Research Reports from the Department of Operations. 341.
https://commons.case.edu/wsom-ops-reports/341
