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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-10,
author = {Rafa\l\ M. \L ochowski},
title = {On a generalisation of the Hahn-Jordan decomposition for real c\`adl\`ag functions},
journal = {Colloquium Mathematicae},
volume = {131},
year = {2013},
pages = {121-138},
zbl = {1279.26024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-10}
}
Rafał M. Łochowski. On a generalisation of the Hahn-Jordan decomposition for real càdlàg functions. Colloquium Mathematicae, Tome 131 (2013) pp. 121-138. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm132-1-10/