Mathematics Prelims

October 22, 2008

Total and Bounded Variation

Filed under: Analysis,Real Analysis — cjohnson @ 5:57 pm

The total variation of a function $f$ over an interval $\left[a, b\right]$, which we’ll denote $V_{[a, b]}(f)$, describes how a function varies over that interval.  That is, suppose $P = \{a = x_0 < x_1 < ... < x_n = b\}$.  With respect to this particular $P$, the variation is simply

$\displaystyle \sum_{k=1}^n |f(x_k) - f(x_{k-1})|$.

The total variation of $f$ is then the supremum of sums such as the above, but over all partitions of $\left[a, b \right]$:

$\displaystyle V_{[a, b]}(f) = \sup_{P \vdash [a, b]} \sum_{k=1}^n |f(x_k) - f(x_{k-1})|$.

In the case of $f : \mathbb{R} \to \mathbb{C}$, with $f$ continuous, $V_{[a, b]}(f)$ gives the length of the curve in $\mathbb{C}$ defined by $f$.  We take the interval, and break it into lots of little pieces, then look at the corresponding pieces of that function in the complex plane.  As the absolute value in the complex plane gives the distance between two points, we’re basically connected the dots on our curve, and measure the lengths of the connecting line segments.  As we make our partition finer and finer, the pieces of the curve are getting smaller and smaller, giving a better approximation to the length of the curve.  When we take the supremum, we’re getting the “best” approximation (the true value).

Notice that the total variation doesn’t give us the length of a curve for $f : \mathbb{R} \to \mathbb{R}$, though.  For instance, consider the interval [0, 1] and the function f(x) = x.  Obviously the length of the curve in $\mathbb{R}^2$ is $\sqrt{2}$.  However, if we partition the interval, the terms $|f(x_k) - f(x_{k-1})|$ only give the vertical component of the length.  Summing up all these terms, however, we will only have one, not $\sqrt{2}$.

We say that $f$ is of bounded variation if its total variation is finite.  That is, if there exists an $M \in \mathbb{R}$ such that for any partition $P$ of $\left[a, b\right]$ we choose,

$\displaystyle \sum_{k=1}^n |f(x_k) - f(x_{k-1})| \leq M$.

Update: We have to be a little bit careful with the discussion of the length of the curve in the above.  A better way to state the above is to talk about the trace of the curve (range of the function).  If the function is continuous then total variation gives the length of the trace, which in the case $f : \mathbb{R} \to \mathbb{R}$ is the the length “travelled” along the path if projected to the y axis.  There’s a nice animation of this idea on Wikipedia’s page on total variation.