PlanetMath: total variation

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total variation

(Definition)

Let $\gamma:[a,b]\rightarrow X$ be a function mapping an interval to a metric space . We say that $\gamma$ is of bounded variation if there is a constant such that, for each partition $P=\{a=t_0<t_1<\cdots < t_n=b\}$ of ,

$\displaystyle v(\gamma, P)= \sum_{k=1}^n d(\gamma(t_k),\gamma(t_{k-1})) \leq M.$

The total variation $V_\gamma$ of $\gamma$ is defined by

$\displaystyle V_\gamma = \sup\{v(\gamma,P):\textnormal{$P$\ is a partition of $[a,b]$}\}.$

It can be shown that, if is either $\mathbb{R}$ or $\mathbb{C}$ , every continuously differentiable (or piecewise continuously differentiable) function $\gamma:[a,b]\rightarrow X$ is of bounded variation, and

$\displaystyle V_\gamma = \int_a^b \vert\gamma'(t)\vert dt.$

Also, if $\gamma$ is of bounded variation and $f:[a,b]\rightarrow X$ is continuous, then the Riemann-Stieltjes integral $\int_a^b fd\gamma$ is finite.

If $\gamma$ is also continuous, it is said to be a rectifiable path, and $V(\gamma)$ is the length of its trace.

If $X=\mathbb{R}$ , it can be shown that $\gamma$ is of bounded variation if and only if it is the difference of two monotonic functions.

"total variation" is owned by Koro.

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Also defines:

bounded variation, rectifiable path

Attachments:: function of not bounded variation (Example) by pahio
continuous derivative implies bounded variation (Theorem) by pahio
Fourier series of function of bounded variation (Theorem) by pahio

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Cross-references: monotonic functions, difference, trace, length, finite, Riemann-Stieltjes integral, continuous, piecewise, continuously differentiable, partition, metric space, interval, mapping, function
There are 10 references to this entry.

This is version 5 of total variation, born on 2003-02-08, modified 2008-03-27.
Object id is 3996, canonical name is TotalVariation.
Accessed 10436 times total.

Classification:

AMS MSC:	26B30 (Real functions :: Functions of several variables :: Absolutely continuous functions, functions of bounded variation)
	26A45 (Real functions :: Functions of one variable :: Functions of bounded variation, generalizations)

Pending Errata and Addenda

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