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On a generalisation of the Banach Indicatrix Theorem

Volume 148 / 2017

Rafał M. Łochowski 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 $[y,y+c].$

Authors

  • Rafał M. ŁochowskiDepartment of Mathematics and Mathematical Economics
    Warsaw School of Economics
    Madalińskiego 6/8
    02-513 Warszawa, Poland
    and
    African Institute for Mathematical Sciences
    6-8 Melrose Road
    Muizenberg 7945, South Africa
    e-mail

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