View source for Arzelà variation

{{MSC|26B30|26A45}} 

[[Category:Analysis]]

{{TEX|done}}

A generalization to functions of several variables of the [[Variation of a function]] of one variable, proposed by 
C. Arzelà in {{Cite|Ar}} (see also {{Cite|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 a function)|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====
{|
|-
|valign="top"|{{Ref|Ar}}|| C. Arzelà,   ''Rend. Accad. Sci. Bologna'' , '''9''' :  2  (1905)  pp. 100–107.   
|-
|valign="top"|{{Ref|Ha}}|| H. Hahn,   "Theorie der reellen Funktionen" , '''1''' , Springer  (1921).   JFM {{ZBL|48.0261.09}} 
|-
|}