Epsilon-complexity of continuous functions
Abstract
A formal definition of epsilon-complexity of an individual continuous function defined on a unit cube is proposed. This definition is consistent with the Kolmogorov's idea of the complexity of an object. A definition of epsilon-complexity for a class of continuous functions with a given modulus of continuity is also proposed. Additionally, an explicit formula for the epsilon-complexity of a functional class is obtained. As a consequence, the paper finds that the epsilon-complexity for the Holder class of functions can be characterized by a pair of real numbers. Based on these results the papers formulates a conjecture concerning the epsilon-complexity of an individual function from the Holder class. We also propose a conjecture about characterization of epsilon-complexity of a function from the Holder class given on a discrete grid.
Cite
@article{arxiv.1303.1777,
title = {Epsilon-complexity of continuous functions},
author = {Boris Darkhovsky and Alexandra Pyriatinska},
journal= {arXiv preprint arXiv:1303.1777},
year = {2013}
}