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