On a generalisation of the Banach indicatrix theorem

TL;DR

This paper generalizes the Banach indicatrix theorem by relating the minimal total variation of functions approximating a regulated function within a certain accuracy to an integral involving the crossing counts over intervals.

Contribution

It extends the classical Banach indicatrix theorem to regulated functions, establishing a new integral formula for minimal total variation approximations.

Findings

Established a new integral relation for total variation approximations.

Generalized the Banach indicatrix theorem to broader class of functions.

Provided a theoretical foundation for analyzing function crossings and variations.

Abstract

We prove that for any regulated function $f:\left[a,b\right]\rightarrow\mathbb{R}$ and $c\geq 0$ the infimum of the total variations of functions approximating $f$ with accuracy $c/2$ is equal $\int_{\mathbb{R}} n_{c}^{y} \mathrm{d} y,$ where $n_{c}^{y}$ is the number of times that $f$ crosses the interval $[y,y+c].$