# Wydawnictwa / Czasopisma IMPAN / Colloquium Mathematicum / Wszystkie zeszyty

## On a generalisation of the Hahn–Jordan decomposition for real càdlàg functions

### Tom 132 / 2013

Colloquium Mathematicum 132 (2013), 121-138 MSC: Primary 26A45. DOI: 10.4064/cm132-1-10

#### Streszczenie

For a real càdlàg function $f$ and a positive constant $c$ we find another càdlàg function which has the smallest total variation among all functions uniformly approximating $f$ with accuracy $c/2.$ The solution is expressed in terms of truncated variation, upward truncated variation and downward truncated variation introduced in earlier work of the author. They are always finite even if the total variation of $f$ is infinite, and they may be viewed as a generalisation of the Hahn–Jordan decomposition for real càdlàg functions. We also present partial results for more general functions.

#### Autorzy

• Rafał M. ŁochowskiDepartment of Mathematics and Mathematical Economics
Warsaw School of Economics
02-513 Warszawa, Poland
and
African Institute for Mathematical Sciences