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{{MSC|26A45}} (Functions of one variable)

{{MSC|26B30|28A15,26B15,49Q15}} (Functions of severable variables)

[[Category:Analysis]]

{{TEX|done}}

Also called ''total variation''. A numerical characteristic of functions of one or more real variables which is connected with differentiability properties.

==Functions of one variable==
===Classical definition===
Let $I\subset \mathbb R$ be an interval. The total variation is defined in the following way.

'''Definition 1'''
Consider the collection $\Pi$ of ordered $(N+1)$-ples of points $a_1<a_2 < \ldots <   a_{N+1}\in I$,
where $N$ is an arbitrary natural number. The total variation of a function $f: I\to \mathbb R$ is given by
\begin{equation}\label{e:TV}
TV\, (f) := \sup \left\{ \sum_{i=1}^N |f(a_{i+1})-f(a_i)| : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, 
\end{equation}
(cp. with Section 4.4 of {{cite|Co}} or Section 10.2 of {{Cite|Ro}}).

If the total variation is finite, then $f$ is called a [[Function of bounded variation|function of bounded variation]]. For examples, properties and issues related to the space of functions of bounded variation we refer to [[Function of bounded variation]].

The  definition of total variation of a function of one real variable  can be  easily generalized when the target is a [[Metric space|metric  space]]  $(X,d)$: it suffices to substitute $|f(a_{i+1})-f(a_i)|$ with  $d (f(a_{i+1}),  f(a_i))$ in \ref{e:TV}.

===Modern definition and relation to measure theory===
Classically  right-continuous functions of bounded variations can be  mapped  one-to-one to [[Signed measure|signed measures]]. More  precisely,  consider a signed measure $\mu$ on (the [[Borel set|Borel  subsets ]] of)  $\mathbb R$ with finite total variation (see [[Signed  measure]] for the  definition). We then define the function
\begin{equation}\label{e:F_mu}
F_\mu (x) := \mu (]-\infty, x])\, .
\end{equation}

'''Theorem 2'''
*   For every signed measure $\mu$ with finite total variation, $F_\mu$ is  a  right-continuous function of bounded variation such that $\lim_{x\to   -\infty} F_\mu (x) = 0$ and $TV (f)$ equals the total variation of  $\mu$ (i.e. $|\mu| (\mathbb R))$.
* For every right-continuous  function $f:\mathbb R\to  \mathbb R$ of bounded variation with  $\lim_{x\to-\infty} f (x) =  0$ there is a unique signed measure $\mu$  such that $f=F_\mu$.
Moreover, the total variation of $f$ equals the total variation of the measure $\mu$ (cp. with [[Signed measure]] for the definition).

For  a proof see Section 4 of  Chapter 4 in {{Cite|Co}}. Obvious  generalizations hold in the case of  different domains of definition. 

===Jordan decomposition===
{{Anchor|Jordan decomposition}}
A fundamental characterization of functions of bounded variation of one variable is due to Jordan.

'''Theorem 3'''
Let   $I\subset \mathbb R$ be an interval. A function $f: I\to\mathbb R$ has   bounded variation if and only if it can be written as the difference  of  two bounded nondecreasing functions.

(Cp. with  Theorem 4  of Section 5.2 in {{Cite|Ro}}). Indeed it is possible to find  a  canonical representation of any function of bounded variation as   difference of nondecreasing functions.

'''Theorem 4'''
If   $f:[a,b] \to\mathbb R$ is a function of bounded variation then there  is  a pair of nondecreasing functions $f^+$ and $f^-$ such that $f= f^+-   f^-$ and $TV (f) = f^+ (b)-f^+ (a) + f^- (b)- f^- (a)$. The pair is   unique up to addition of a constant, i.e. if $g^+$ and $g^-$ is a second   pair with the same property, then $g^+-g^-=f^+-f^-\equiv {\rm const}$.

(Cp.   with Theorem 3 of Section 5.2 in {{Cite|Ro}}). The latter   representation of a function of bounded variation is also called   [[Jordan decomposition]].
====Negative and positive variations====
It is possible to define the negative and positive variations of $f$ in the following way.

'''Definition 5'''
Let $I\subset \mathbb R$ be an interval and $\Pi$ be as in '''Definition 1'''. The negative and positive variations of $f:I\to\mathbb R$ are then defined as
\[
TV^+ (f):= \sup \left\{ \sum_{i=1}^N \max \{(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\,
\]
\[
TV^- (f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .
\]

If $f$ is a function of bounded variation on $[a,b]$ we can define $f^+ (x) = TV^+ (f|_{[a,x]})$ and $f^- (x) = TV^- (f|_{[a,x]})$. Then it turns out that, up to constants, these two functions give the Jordan decomposition of '''Theorem 4''', cp. with Lemma 3 in Section 2, Chapter 5 of {{Cite|Ro}}.
===Historical remark===
The variation of a function of one real variable was considered for the first time by C.  Jordan in  {{Cite|Jo}} to study the  pointwise convergence of [[Fourier  series]], cp. with  [[Jordan  criterion]] and [[Function of bounded variation]].
==Wiener's and Young's generalizations==
One sometimes also considers classes $BV_\Phi ([a,b])$  defined as follows. Let $\Phi: [0, \infty[\to [0, \infty[$ be a continuous function with $\Phi (0)=0$ which increases monotonically. If $f:[a,b]\to \mathbb R$, we let $TV_{\phi} (f)$ be the least  upper bound of sums of the type
\[
\sum_{i=1}^N \Phi (|f (x_{i+1}- f(x_i)|)
\]
where  $a\leq x_1 < \ldots < x_{N+1}<b$ is an arbitrary family of points. The quantity $TV_\Phi (f)$ is called the $\Phi$-variation of $f$ on $[a,b]$. If $TV_\Phi (f)<\infty$ one says that $f$ has bounded $\Phi$-variation on  $[a,b]$, while the class  of such functions is denoted by $BV_\Phi ([a,b])$ (see {{Cite|Ba}}). If $\Phi (u)=u$, one obtains  Jordan's class $BV ([a,b])$, while if $\Phi (u)=u^p$, one obtains  Wiener's classes $BV_p ([a,b])$ (see {{Cite|Wi}}). The definition of the class $BV_\Phi ([a,b])$ was proposed by  L.C. Young in {{Cite|Yo}}.

If
\[
\limsup_{u\to 0^+} \frac{\Phi_1 (u)}{\Phi_2 (u)} < \infty
\]
then
\[
BV_{\Phi_2} ([a,b])\subset BV_{\Phi_1} ([a,b)]\, .
\]
In particular, on any interval $[a,b]$,
\[
BV_p ([a,b])\subset BV_q ([a,b]) \subset BV_{\exp (-u^{-\alpha})} ([a,b]) \subset 
BV_{\exp (-u^{-\beta})} ([a,b])\, .
\]
for $1\leq p < q$ and $0<\alpha<\beta<\infty$, these being  proper inclusions.

==Functions of several variables==
===Historical remarks===
After  the introduction by Jordan of functions of bounded variations  of one  real variable, several authors attempted to generalize the  concept to  functions of more than one variable. The first attempt was  made by  Arzelà and Hardy in 1905, see [[Arzelà variation]] and [[Hardy   variation]], followed by Vitali, Fréchet, Tonelli and Pierpont, cp.  with  [[Vitali variation]], [[Fréchet variation]], [[Tonelli plane   variation]] and [[Pierpont variation]] (moreover, the definition of   Vitali variation was also considered independently by Lebesgue and De la   Vallée-Poussin). However, the point of view which became popular and  it  is nowadays accepted in the literature as most efficient  generalization  of the $1$-dimensional theory is due to De Giorgi and  Fichera (see  {{Cite|DG}} and {{Cite|Fi}}). Though with different  definitions, the  approaches by De Giorgi and Fichera are  equivalent  (and very close in spirit) to the ''distributional theory''  described  below. A promiment role in the further developing of the  theory was  also played by Fleming, Federer and Volpert. Moreover, Krickeberg and   Fleming showed, independently, that the current definition of functions   of bounded variation is indeed equivalent to a slight modification of   Tonelli's one {{Cite|To}}, proposed by Cesari {{Cite|Ce}}, cp. with the   section '''Tonelli-Cesari variation''' of [[Function of bounded variation]]. We refer to Section 3.12   of {{Cite|AFP}} for a thorough discussion of the topic. 
====Link to the theory of currents====
Functions   of bounded variation in $\mathbb R^n$ can be identified with   $n$-dimensional normal [[Current|currents]] in $\mathbb R^n$. This is  the point  of view of Federer, {{Cite|Fe}}, which thus derives most of  the  conclusions of the theory of $BV$ functions as special cases of  more  general theorems for normal currents,

===Definition===
Following {{Cite|EG}}:

'''Definition 6'''
Let $\Omega\subset \mathbb R^n$ be open. The total variation of $u\in L^1 (\Omega)$ is given by
\begin{equation}\label{e:diverg}
V   (u, \Omega) := \sup \left\{ \int_\Omega u\, {\rm div}\, \psi : \psi\in   C^\infty_c (\Omega, \mathbb R^n), \, \|\psi\|_{C^0}\leq 1\right\}\, .
If $V(u, \Omega)<\infty$ then we say that $u$ has '''bounded variation''' and the corresponding space of functions is denoted by $BV (\Omega)$.
\end{equation}

As a consequence of the [[Radon-Nikodym theorem]] we then have

'''Prposition 7'''
A   function $u\in L^1 (\Omega)$, then $V(u, \Omega)<\infty$ if and only the [[Generalized derivative|distributional derivative]] of $f$ is a Radon measure $Du$. Moreover $V (u,\Omega) = |Du|   (\Omega)$, where $|Du|$ is the total variation measure of $Du$.

When $n=1$ and $\Omega=[a,b]$, then $V (u, [a,b])<\infty$ if and only if there exists a function $\tilde{u}$ such that
$u=\tilde{u}$ a.e. and $TV (\tilde{u})<\infty$. Moreover, 
\[
V (u, [a,b]) = \inf \{TV (\tilde{u}): \tilde{u}= t \quad\mbox{a.e.}\}\, .
\]
(Cp. with [[Function of bounded variation]]).

===Caccioppoli sets===
A  special class of $BV_{loc}$ functions which play a fundamental role in  the  theory (and had also a pivotal role in its historical development)  is  the set of those $f\in BV$ which takes only the values $0$ and $1$  and  are, therefore, the indicator functions of a set.

'''Definition 23'''
Let   $\Omega\subset \mathbb R^n$ be an open set and $E\subset \Omega$ a   measurable set such that $ V({\bf 1}_E, \Omega)<\infty$. The $E$ is  called a  ''Caccioppoli set'' or a ''set of finite perimeter'' and its  perimeter  in $\Omega$ is defined to be
\[
{\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, .
\]

==Coarea formula==
===Banach indicatrix===
===Vitushkin variation===