Negative variation of a function
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 \[ TV^- (f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, . \]
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.
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. |
[Jo] | C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230 |
[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 |
Negative variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Negative_variation_of_a_function&oldid=27951