Continued fractions built from convex sets and convex functions
Abstract
In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalisation of continued fractions. General sufficient conditions for convergence of continued fractions with deterministic terms are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform (where also necessary and sufficient conditions of convergence for continued fractions with constant terms are obtained) and the family of non-negative convex functions with the Legendre--Fenchel and Artstein-Avidan--Milman transforms.
Cite
@article{arxiv.1405.2779,
title = {Continued fractions built from convex sets and convex functions},
author = {Ilya Molchanov},
journal= {arXiv preprint arXiv:1405.2779},
year = {2014}
}
Comments
18 pages. This version deals with the deterministic case only and is due to appear in Communications in Contemporary Mathematics. The random case will be posted separately