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−  A function whose variation is bounded (see [[Variation of a functionVariation 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 theoremDirichlet theorem]] on the convergence of Fourier series of piecewisemonotone functions (see [[Jordan criterionJordan 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 decompositionJordan 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 absolutelycontinuous function (see [[Absolute continuityAbsolute continuity]]), a [[Singular functionsingular function]] and a [[Jump functionjump function]] (see [[Lebesgue decompositionLebesgue 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 variationArzelà variation]]; [[Vitali variationVitali variation]]; [[Pierpont variationPierpont variation]]; [[Tonelli plane variationTonelli plane variation]]; [[Fréchet variationFréchet variation]]; [[Hardy variationHardy variation]]).
 
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−  ====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