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Difference between revisions of "Negative variation of a function"

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''negative increment of a function''
 
  
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One of the two terms whose sum is the complete increment or [[Variation of a function|variation of a function]] on a given interval. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662401.png" /> be a function of a real variable, defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662402.png" /> and taking finite real values.
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{{MSC|26A45}}
  
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662403.png" /> be an arbitrary partition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662404.png" /> and let
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[[Category:Analysis]]
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662405.png" /></td> </tr></table>
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{{TEX|done}}
  
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where the summation is over those numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662406.png" /> for which the difference <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662407.png" /> is non-positive. The quantity
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Also called ''negative increment of a function''
  
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662408.png" /></td> </tr></table>
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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.  
  
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is called the negative variation (negative increment) of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n0662409.png" /> on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n06624010.png" />. It is always true that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/n/n066/n066240/n06624011.png" />. See also [[Positive variation of a function|Positive variation of a function]]; [[Variation of a function|Variation of a function]].
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'''Definition'''
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Consider an interval $I=[a,b]\subset \mathbb R$ and the collection $\Pi$ of ordered $(N+1)$-ples of points $a_1<a_2 < \ldots <   a_{N+1}\in I$,
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where $N$ is an arbitrary natural number. The negative variation of a function $f: I\to \mathbb R$ is given by
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\[
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TV^- (f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .
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\]
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See also [[Positive variation of a function|Positive variation of a function]] and [[Variation of a function|Variation of a function]].
  
 
====References====
 
====References====
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<table><TR><TD valign="top">[1]</TD> <TD valign="top">  H. Lebesgue,  "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars  (1928)</TD></TR></table>
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{|
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|-
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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}}
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|-
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|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
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|-
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|valign="top"|{{Ref|Le}}|| H. Lebesgue,  "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars  (1928).
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|-
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|valign="top"|{{Ref|Ro}}|| H.L. Royden,  "Real analysis" , Macmillan  (1969). {{MR|0151555}} {{ZBL|0197.03501}}
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|-
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|}

Revision as of 08:41, 16 September 2012

2020 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 \[ TV^- (f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, . \] See also Positive variation of a function and Variation of a function.

References

[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. MR1857292Zbl 0957.49001
[Co] D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
[Le] H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars (1928).
[Ro] H.L. Royden, "Real analysis" , Macmillan (1969). MR0151555 Zbl 0197.03501
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=27949
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article
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