Arzelà variation

2020 Mathematics Subject Classification: Primary: 26B30 Secondary: 26A45 [MSN][ZBL]

A generalization to functions of several variables of the Variation of a function of one variable, proposed by C. Arzelà in [Ar] (see also [Ha], p. 543). However the modern theory of functions of bounded variation uses a different generalization (see Function of bounded variation and Variation of a function). Therefore the Arzelà variation is seldomly used nowadays.

Consider a rectangle $R:= [a_1, b_1]\times \ldots \times [a_n, b_n]\subset \mathbb R^n$ and denote by

• $\Gamma$ the class of continuous $\gamma= (\gamma_1, \ldots, \gamma_n):[0,1]\to R$ such that each component $\gamma_j$ is nondecreasing and maps $[0,1]$ onto $[a_j, b_j]$.
• $\Pi$ the family of $N+1$-tuples of points $0\leq t_1 < \ldots < t_{N+1}\leq 1$.

Definition The Arzelà variation of a function $f:R\to\mathbb R$ is then defined as \[ V_A (f):= \sup_{\gamma\in \Gamma}\; TV (f\circ \gamma) = \sup_{\gamma\in \Gamma}\; \left(\sup \left\{ \sum_{i=1}^N |f(\gamma (t_{i+1})) - f (\gamma (t_i))| : (t_1, \ldots , t_{N+1})\in \Pi\right\}\right) \] ($TV (f\circ \gamma)$ is then the classical total variation of the real variable function $f\circ \gamma$, see Variation of a function).

A function $f$ has finite Arzelà variation if and only if it can be written as the difference of two functions $f^+-f^-$ with the property that \[ f^{\pm} (x_1, \ldots, x_n) \leq f^{\pm} (y_1, \ldots, y_n) \qquad \mbox{if } x_i\leq y_i \, \forall i\, . \] This statement generalizes the Jordan decomposition of functions of bounded variation of one variable.

The class of functions with finite Arzelà variation contains the class of functions with finite Hardy variation.

References

[Ar]	C. Arzelà, Rend. Accad. Sci. Bologna , 9 : 2 (1905) pp. 100–107.
[Ha]	H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921). JFM Zbl 48.0261.09