Difference between revisions of "Jordan decomposition (of a function)"
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'''Theorem''' | '''Theorem''' | ||
| − | If $f:[a,b] \to\mathbb R$ 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 [[Variation of a function|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^+-g^- | + | If $f:[a,b] \to\mathbb R$ 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 [[Variation of a function|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 {{Cite|Ro}}. | For a proof see Section 5.2 of {{Cite|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 derivative|generalized derivatives]] of, respectively, $f$, $f^+$ and $f^-$ we then have that | ||
| + | * $\mu$ is a [[Signed measure|signed measure]] | ||
| + | * $\mu^+$ and $\mu^-$ are (nonnegative) measures | ||
| + | on the [[Borel set|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==== | ====References==== | ||
Latest revision as of 14:05, 10 December 2012
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 $f:[a,b] \to\mathbb R$ 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 |
Jordan decomposition (of a function). Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Jordan_decomposition_(of_a_function)&oldid=27854