On a generalisation of the Banach Indicatrix Theorem
Volume 148 / 2017
Colloquium Mathematicum 148 (2017), 301-313
MSC: Primary 26A45.
DOI: 10.4064/cm6583-3-2017
Published online: 24 March 2017
Abstract
We prove that for any regulated function $f:[a,b]\rightarrow \mathbb {R}$ and $c\geq 0,$ the infimum of the total variations of functions approximating $f$ with accuracy $c/2$ is equal to $\int _{\mathbb {R}} n_{c}^{y} \,dy,$ where $n_{c}^{y}$ is the number of times $f$ crosses the interval $[y,y+c].$
