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A numerical characteristic of a function of several variables, which may be regarded as the multi-dimensional analogue of the variation of a function in one unknown. Let 
be a real-valued function given on an 
-dimensional parallelepipedon 
, 
and let 
be the class of all continuous vector functions 
, 
, such that each of the functions 
is non-decreasing on 
, and with 
, 
, 
. Then
 |
where 
is an arbitrary system of points in 
. This definition for the case 
was proposed by C. Arzelà [1] (see also [2], p. 543). If 
, one says that 
has bounded (finite) Arzelà variation on 
, and the class of all such functions is denoted by 
. For a function 
to belong to the class 
it is necessary and sufficient that there exists a decomposition 
, where 
and 
are finite non-decreasing functions on 
. A function 
is called non-decreasing on 
if
 |
for 
(
). The class 
contains the class of functions of bounded Hardy variation on 
.
References
| [1] | C. Arzelà, Rend. Accad. Sci. Bologna , 9 : 2 (1905) pp. 100–107 |
| [2] | H. Hahn, "Theorie der reellen Funktionen" , 1 , Springer (1921) |
Arzelà variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Arzel%C3%A0_variation&oldid=22035