total variation

Let $\gamma :[a,b]\to X$ 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 of $[a,b]$,

$$v(\gamma ,P)=\sum _{k=1}^{n}d(\gamma (t_k),\gamma (t_{k-1}))\le M.$$

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

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

It can be shown that, if $X$ is either $ℝ$ or $ℂ$, every continuously differentiable (or piecewise continuously differentiable) function $\gamma :[a,b]\to X$ 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 $f:[a,b]\to X$ 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 $X=ℝ$, 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