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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
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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=16607