Jump function

2020 Mathematics Subject Classification: Primary: 26A45 Secondary: 28A15 [MSN][ZBL]

One of the three components in the Lebesgue decomposition of a function of bounded variation depending on one real variable, rediscovered and extended to functions of several variables by De Giorgi and its school (see [AFP]). According to Lebesgue, if $I\subset\mathbb R$ is an interval, a right-continuous function of bounded variation $f: I\to\mathbb R$ can be decomposed in a canonical way into three functions $f_a+f_j+f_c$. The function $f_j$ is the jump part of $f$ (or jump function of $f$, using the terminology of Lebesgue [Le]) and it is defined by \begin{equation}\label{e:jump} f_j (x)= \sum_{y\leq x}\, f (y^+) - f(y^-)\, . \end{equation} Therefore its distributional derivative is the atomic part of the distributional derivative of $f$. The jump part can also be characterized as the (right-continuous) function $g$ with smallest variation such that $f-g$ is continuous (cp. with Remark 1 at page 163 of [Le]). Observe therefore that the total variation of $f$ is the sum of the total variation of $f_j$ and the total variation of $f-f_j$.

The term jump function is used also for those functions of bounded variation $f$ such that $f=f_j$, i.e. so that their distributional derivative is a purely atomic measure.

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
[DS]	N. Dunford, J.T. Schwartz, "Linear operators. General theory", 1, Interscience (1958) MR0117523 Zbl 0635.47001
[Ha]	P.R. Halmos, "Measure theory", v. Nostrand (1950) MR0033869 Zbl 0040.16802
[Le]	H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars (1928).
[Na]	I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)
[Sa]	S. Saks, "Theory of the integral" , Hafner (1952) MR0167578 Zbl 63.0183.05
[Sz]	B. Szökefalvi-Nagy, "Introduction to real functions and orthogonal expansions" , Oxford Univ. Press (1965)