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
and let , , be a subdivision of the segment into , equal segments by points
These subdivisions generate a subdivision
of the parallelopipedon into parallelopipeda with edges parallel to the coordinate axes.
where is the oscillation of the function on (cf. Oscillation of a function). Then
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 . The class contains as a subset the class of all functions of bounded Arzelà variation on .
|||J. Pierpont, "Lectures on the theory of functions of real variables" , 1 , Dover, reprint (1959)|
|||H. Hahn, "Reellen Funktionen" , 1 , Chelsea, reprint (1948)|
Pierpont variation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pierpont_variation&oldid=13447