Pierpont variation
One of the numerical characteristics of a function of several variables which can be considered as a multi-dimensional analogue of the variation of a function of one variable. Let a function ,
be given on an
-dimensional parallelopipedon
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and let ,
, be a subdivision of the segment
into
,
equal segments by points
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These subdivisions generate a subdivision
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of the parallelopipedon into
parallelopipeda
with edges parallel to the coordinate axes.
Let
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where is the oscillation of the function
on
(cf. Oscillation of a function). Then
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If , then the function
is said to be of bounded (finite) Pierpont variation on
, and the class of all such functions is denoted by
. This definition was suggested by J. Pierpont [1]. The class
contains as a subset the class
of all functions of bounded Arzelà variation on
.
References
[1] | J. Pierpont, "Lectures on the theory of functions of real variables" , 1 , Dover, reprint (1959) |
[2] | H. Hahn, "Reellen Funktionen" , 1 , Chelsea, reprint (1948) |
Pierpont variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pierpont_variation&oldid=27978