View source for Positive variation of a function




{{MSC|26A45}}

[[Category:Analysis]]

{{TEX|done}}

Also called ''positive 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
\[
TV^+ (f) := \sup \left\{ \sum_{i=1}^N \max \{(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .
\]

The concept of positive variation of a function was introduced by C. Jordan in {{Cite|Jo}} and it is used to prove the [[Jordan decomposition (of a function)|Jordan decomposition]] of a [[Function of bounded variation|function of bounded variation]]. See also [[Negative variation of a function|Negative 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|Jo}}|| C. Jordan,   "Sur la série de Fourier"  ''C.R. Acad. Sci. Paris'' , '''92'''  (1881)  pp. 228–230
|-
|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}}
|-
|}