Bounded variation functions

Note

Let $I\subseteq \mathbb{R}$ be an interval and $P=\{x_1,\dots ,x_n\}\subset I$ a partition. We define the variation of a funtion $f:I\to \mathbb{R}$ w.r.t. partition $P$ as

$$V(f,P):=\sum _{i=1}^{n-1}|f(x_{i+1})-f(x_i)|$$

and the total variation of $f$ as the supremum on all partitions

$$Var(f,I):=\underset{P\text{ part. of}I}{\sup }V(f,P)$$

If this quantity is finite, we say that $f$ has bounded variation and we denote the set of all bounded variation functions on $I$ as $BV(I)$. If $I=[a,b]$ then we write $V_a^b[f]$ (notation from Kolmogorov-Fomin).

Example

1. The function $f(x)=x\sin \frac{1}{x}$ is continuous on $[0,1]$ but $V_0^1[f]=+\infty$.
2. The function $f(x)=\sqrt{x}$ has $V_0^1[f]=1$.

Properties §

The set $BV(I)$ forms a linear space (with sum and scalar multiplication defined in the obbious way), since

1. $Var(\alpha f,I)=|\alpha |Var(f,I)$
2. $Var(f+g,I)\le Var(f,I)+Var(g,I)$ since $\sup (A+B)\le \sup (A)+\sup (B)$.

Since $Var(f,I)=0$ iff $f$ is a constant function, $Var(\cdot ,I)$ is a seminorm on this linear space. To get a norm one could define $\Vert \cdot \Vert :=|f(a)|+Var(f,I)$ where $a\in I$.

Let $f:[a,b]\to \mathbb{R}$.

1. If $f$ is monotone, then $f\in BV([a,b])$ and $V_a^b[f]=|f(b)-f(a)|$, since if w.l.o.g. $f$ is non-decreasing we can drop the absolute values and get a telescopic sum

$$V(f,P)=\sum _{i=1}^{n-1}f(x_{i+1})-f(x_i)=f(x_n)-f(x_i)\le f(b)-f(a)$$

2. If $f$ is Lipschitz with constant $L$ then $V_a^b[f]\le L(b-a)$, since

$$V(f,P)\le L\sum _{i=1}^{n-1}|x_{i+1}-x_i|\le L(b-a)$$

3. If $f\in C^1[a,b]$ then

$$V_a^b[f]={\int }_a^b|f^{\prime }(x)|dx$$

if we use the mean value theorem

$$Var(f,P)=\sum _{i=1}^{n-1}|f^{\prime }({\xi }_i)|(x_{i+1}-x_i)$$

but the supremum on all partitions of $[a,b]$ is the definition of the Riemann-Darboux intergral of $|f^{\prime }|$.

4. Let $a<b<c$, then

$$V_a^b[f]+V_b^c[f]=V_a^c[f]$$

since

$$Var(f,P)=\sum _{i=1}^r|f(x_{i+1})-f(x_i)|+\sum _{i=r+1}^{n-1}|f(x_{i+1})-f(x_i)|\le V_a^b[f]+V_b^c[f]$$

where we have split the sum such that $\{x_1,\dots x_{r+1}\}\subset [a,b]$ and $\{x_{r+1},\dots x_n\}\subset [b,c]$.

Let $P_1$ and $P_2$ be two partitions of the intervals $[a,b]$ and $[b,c]$ such that $\forall ϵ>0$ $Var(f,P_1)>V_a^b[f]-ϵ$ and $Var(f,P_2)>V_b^c[f]-ϵ$. Since $P_1\cup P_2$ is a partition for $[a,c]$

$$V_a^c[f]\ge Var(f,P_1\cup P_2)>V_a^b[f]+V_b^c[f]-2ϵ$$

since $ϵ$ was arbitrary the claim follows.

This implies, since the total variation is a non-negative quantity, that the function $v:[a,b]\to [0,\infty ]$ defined as

$$x\mapsto V_a^x[f]$$

is monotonic non-decreasing. From this follows a cool decomposition theorem.

Note

Jordan’s decomposition Every bounded variation function in an interval $f\in BV([a,b])$ can be written as the difference of two monotonic non-decreasing functions.

Note \phi = v-f, where v(x) = V_a^x[f]. Let x \leq y, then

Consider the function

$$\varphi (y)-\varphi (x)=V_x^y[f]-[f(y)-f(x)]\ge 0$$

we see that $\varphi$ is also monotonic non-decreasing, so $f$ can be written as

$$f=v-\varphi$$

which is the decomposition we were looking for. $\square$

Note

Clearly the decomposition is not unique, since we can add a constant. To obtain a unique decomposition one can fix the value of the first function to zero at point $a$ for example.

Tip

We can extend the theorem to unbounded intervals taking the $\inf$ or $\sup$, of course we need to restrict to bounded monotonic functions.

A consequence of this decomposition is that any bounded variation function has a well defined derivative a.e.

Decomposition of BV functions §

Let’s recap what we know about bounded variation functions. Let $g\in BV$, since in general $g\notin AC$

$$g\ne {\int }_a^x{g}^{\prime }dt-g(a)$$

define a new function as this difference

$$\chi :=g-{\int }_a^x{g}^{\prime }dt+g(a)$$

Note that ${\chi }^{\prime }=0$ almost everywhere. Then we can decompose $g$ as

$$g=\chi +\varphi ,\varphi :={\int }_a^x{g}^{\prime }dt-g(a)$$

where $\varphi$ is an absolutely continuous function. Then $\chi$ is still a bounded variation function that we can represent using the Jordan decomposition:

$$\chi =u-v$$

where $u,v$ are monotonic non-decreasing functions. We can further decompose $u$ and $v$ using Jump functions:

$$u=H_u+{\phi }_u,v=H_v+{\phi }_v$$

where ${\phi }_u,{\phi }_v$ are continuous functions. Putting all piecies togheter we get:

$$g=H+C+\varphi$$

where $H:=H_u+H_v$ is called the Jump part of $g$; $C:={\phi }_u+{\phi }_v$ is a continuous function known as the cantorian part of $g$, and $\varphi$ is the absolutely continuous part of $g$.

For example the Cantor-Vitali function coincides with its cantorian part (this is the reason for its name).

Moral of the story

Since $(C+H{)}^{\prime }=0$ a.e., the only part we can get back from integrating the derivative $g^{\prime }$ is the absolutely continuous one.