View source for Negative variation of a function


{{MSC|26A45}}

[[Category:Analysis]]

{{TEX|done}}

Also called ''negative increment of a function''

One of the two terms whose sum is the complete increment or [[Variation of a function|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
\[
TV^- (f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .
\]
See also [[Positive variation of a function|Positive variation of a function]] and [[Variation of a function|Variation of a function]].

====References====
{|
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|valign="top"|{{Ref|AFP}}|| L. Ambrosio, N.   Fusco, D.  Pallara, "Functions of bounded    variations   and  free   discontinuity  problems". Oxford Mathematical    Monographs. The      Clarendon Press,  Oxford University Press, New York,    2000.       {{MR|1857292}}{{ZBL|0957.49001}} 
|-
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
|-

|valign="top"|{{Ref|Le}}|| H. Lebesgue,   "Leçons sur l'intégration  et la récherche des fonctions primitives", Gauthier-Villars  (1928).
|-
|valign="top"|{{Ref|Ro}}|| H.L. Royden,   "Real analysis" , Macmillan  (1969). {{MR|0151555}} {{ZBL|0197.03501}}
|-
|}