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

Tom 132 / 2013

Rafał M. Łochowski 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
    Madalińskiego 6/8
    02-513 Warszawa, Poland
    and
    African Institute for Mathematical Sciences
    6 Melrose Road
    Muizenberg 7945, South Africa
    e-mail
    e-mail

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