Geodesic
On a generalisation of the Hahn–Jordan decomposition for real càdlàg functions
Rafał M. Łochowski1
1 Department of Mathematics and Mathematical Economics Warsaw School of Economics Madalińskiego 6/8 02-513 Warszawa, Poland and African Institute for Mathematical Sciences 6 Melrose Road Muizenberg 7945, South Africa
Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 121-138 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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.
DOI : 10.4064/cm132-1-10
Mots-clés : real function positive constant another function which has smallest total variation among functions uniformly approximating accuracy solution expressed terms truncated variation upward truncated variation downward truncated variation introduced earlier work author always finite even total variation infinite may viewed generalisation hahn jordan decomposition real functions present partial results general functions

Rafał M. Łochowski 1

1 Department of Mathematics and Mathematical Economics Warsaw School of Economics Madalińskiego 6/8 02-513 Warszawa, Poland and African Institute for Mathematical Sciences 6 Melrose Road Muizenberg 7945, South Africa 
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Rafał M. Łochowski. On a generalisation of the Hahn–Jordan decomposition for real càdlàg functions. Colloquium Mathematicum, Tome 132 (2013) no. 1, pp. 121-138. doi: 10.4064/cm132-1-10

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