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Function of bounded variation

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A function whose variation is bounded (see Variation of a function). The concept of a function of bounded variation was introduced by C. Jordan [1] for functions of one real variable in connection with a generalization of the Dirichlet theorem on the convergence of Fourier series of piecewise-monotone functions (see Jordan criterion for the convergence of a Fourier series). A function defined on an interval is a function of bounded variation if and only if it can be represented in the form , where (resp., ) is an increasing (resp., decreasing) function on (see Jordan decomposition of a function of bounded variation). Every function of bounded variation is bounded and can have at most countably many points of discontinuity, all of the first kind. A function of bounded variation can be represented as the sum of an absolutely-continuous function (see Absolute continuity), a singular function and a jump function (see Lebesgue decomposition of a function of bounded variation).

In the case of several variables there is no unique concept of a function of bounded variation; there are several definitions of the variation of a function in this case (see Arzelà variation; Vitali variation; Pierpont variation; Tonelli plane variation; Fréchet variation; Hardy variation).

References

[1] C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230
[2] I.P. Natanson, "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M. (1961) (Translated from Russian)
How to Cite This Entry:
Function of bounded variation. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Function_of_bounded_variation&oldid=11751
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article