# Negative variation of a function

2010 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL]

Also called negative increment of a function

One of the two terms whose sum is the complete increment or variation of a function $f$ on a given interval.

Definition Consider an interval $I=[a,b]\subset \mathbb R$ and the collection $\Pi$ of ordered $(N+1)$-ples of points $a_1<a_2 < \ldots < a_{N+1}\in I$, where $N$ is an arbitrary natural number. The negative variation of a function $f: I\to \mathbb R$ is given by $TV^- (f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .$

The concept of negative variation of a function was introduced by C. Jordan in [Jo] and it is used to prove the Jordan decomposition of a function of bounded variation See also Positive variation of a function and Variation of a function.

How to Cite This Entry:
Negative variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_variation_of_a_function&oldid=27951
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article