total variation

Let $\gamma:[a,b]\rightarrow X$ be a function mapping an interval $[a,b]$ to a metric space $(X,d)$. We say that $\gamma$ is of bounded variation  if there is a constant $M$ such that, for each partition $P=\{a=t_{0} of $[a,b]$,

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

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

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

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

 $V_{\gamma}=\int_{a}^{b}|\gamma^{\prime}(t)|dt.$

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.

 Title total variation Canonical name TotalVariation Date of creation 2013-03-22 13:26:09 Last modified on 2013-03-22 13:26:09 Owner Koro (127) Last modified by Koro (127) Numerical id 8 Author Koro (127) Entry type Definition Classification msc 26A45 Classification msc 26B30 Related topic BVFunction Related topic IntegralRepresentationOfLengthOfSmoothCurve Related topic OscillationOfAFunction Defines bounded variation Defines rectifiable path