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A function whose variation is bounded (see [[Variation of a function|Variation of a function]]). The concept of a function of bounded variation was introduced by C. Jordan [[#References|[1]]] for functions of one real variable in connection with a generalization of the [[Dirichlet theorem|Dirichlet theorem]] on the convergence of Fourier series of piecewise-monotone functions (see [[Jordan criterion|Jordan criterion]] for the convergence of a Fourier series). A function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041970/f0419701.png" /> defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041970/f0419702.png" /> is a function of bounded variation if and only if it can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041970/f0419703.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041970/f0419704.png" /> (resp., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041970/f0419705.png" />) is an increasing (resp., decreasing) function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041970/f0419706.png" /> (see [[Jordan decomposition|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|Absolute continuity]]), a [[Singular function|singular function]] and a [[Jump function|jump function]] (see [[Lebesgue decomposition|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|Arzelà variation]]; [[Vitali variation|Vitali variation]]; [[Pierpont variation|Pierpont variation]]; [[Tonelli plane variation|Tonelli plane variation]]; [[Fréchet variation|Fréchet variation]]; [[Hardy variation|Hardy variation]]).
 
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  C. Jordan,  "Sur la série de Fourier"  ''C.R. Acad. Sci. Paris'' , '''92'''  (1881)  pp. 228–230</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.P. Natanson,  "Theorie der Funktionen einer reellen Veränderlichen" , H. Deutsch , Frankfurt a.M.  (1961)  (Translated from Russian)</TD></TR></table>
 

Revision as of 18:14, 27 August 2012

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