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Jordan decomposition (of a function)

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2020 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL]

A canonical decomposition theorem (due to Jordan) for functions of bounded variation of one real variable.

Theorem If is a function of bounded variation then there is a pair of nondecreasing functions f^+ and f^- such that f= f^+- f^- and TV (f) = f^+ (b)-f^+ (a) + f^- (b)- f^- (a) (where TV (f) denotes the total variation of f (see also Function of bounded variation). The pair is unique up to addition of a constant, i.e. if g^+ and g^- is a second pair with the same property, then g^+-f^+=g^--f^-\equiv {\rm const}.

For a proof see Section 5.2 of [Ro].

The decomposition is related to the Jordan decomposition (of a signed measure). More precisely, if we denote by \mu, \mu^+ and \mu^- the generalized derivatives of, respectively, f, f^+ and f^- we then have that

  • \mu is a signed measure
  • \mu^+ and \mu^- are (nonnegative) measures

on the Borel sets of \mathbb R and \mu = \mu^+-\mu^- is the Jordan decomposition of \mu. For more details we refer to Function of bounded variation.

References

[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Co] D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
[Jo] C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230
[Ro] H.L. Royden, "Real analysis" , Macmillan (1969) MR0151555 Zbl 0197.03501
How to Cite This Entry:
Jordan decomposition (of a function). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_decomposition_(of_a_function)&oldid=29165