Epsilon-complexity of continuous functions

TL;DR

This paper introduces a formal definition of epsilon-complexity for continuous functions and function classes, linking it to Kolmogorov's ideas, and provides explicit formulas and conjectures for functions within Holder classes.

Contribution

It proposes a new formal definition of epsilon-complexity for individual functions and classes, with explicit formulas for Holder classes and related conjectures.

Findings

Epsilon-complexity for Holder class characterized by two real numbers

Explicit formula for epsilon-complexity of functional classes

Conjectures on epsilon-complexity of functions from Holder class

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.