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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|Bar}}). 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 $u\in L^1_{loc} (\Omega)$ 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\}\, .
\end{equation}
If $V(u, \Omega)< \infty$  then we say that $u$ has '''bounded variation'''. The space of functions  $u\in L^1 (\Omega)$ which have bounded variation are denoted by $BV  (\Omega)$.

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

'''Prposition 7'''
A    function $u\in L^1_{loc} (\Omega)$, then $V(u, \Omega)<\infty$ if  and only the [[Generalized derivative|distributional derivative]] of $u$  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, 
\[
V (u, [a,b]) = \inf \{TV (\tilde{u}): \tilde{u}= u \quad\mbox{a.e.}\}\, .
\]
(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 8'''
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==
An   important tool which allows often to reduce problems for $BV$   functions  to problems for Caccioppoli sets is the following   generalization of the  [[Coarea formula]], first proved by Fleming and   Rishel in {{Cite|FR}}.

'''Theorem 29'''
For  any   open set $\Omega\subset \mathbb R^n$ and any $u\in L^1 (\Omega)$,  the   map $t\mapsto {\rm Per}\, (\{u>t\}, \Omega)$ is Lebesgue  measurable   and one has
\[
V (u,\Omega) = \int_{-\infty}^\infty {\rm Per}\, (\{u>t\}, \Omega)\, dt\, 
\]
In     particular, if $u\in BV (\Omega)$, then $U_t:=\{u>t\}$ is a    Caccioppoli  set for a.e. $t$ and, for any Borel set  $B\subset \Omega$,
\[
|Du| (B) = \int_{-\infty}^\infty |D{\bf 1}_{U_t}| (B)\, dt \qquad\mbox{and}\qquad
Du (B) = \int_{-\infty}^\infty D{\bf 1}_{U_t} (B)\, dt\, 
\]
(where the maps $t\mapsto |D{\bf 1}_{U_t}| (B)$ and $t\mapsto D{\bf 1}_{U_t} (B)$ are both Lebesgue measurable).

Cp.  with Theorem 3.40 in {{Cite|AFP}}.
===Banach indicatrix===
Let $f:[a,b]\to \mathbb R$. The [[Banach indicatrix]] $N (y,f)$ is then the cardinality of the set $\{f=y\}$. A special
case of the coarea formula, first proved by Banach in {{Cite|Ba}}, is the identity
\[
TV (f) = \int_{-\infty}^\infty N (f,y)\, dy
\]
which holds for every '''continuous''' function $f:[a,b]\to\mathbb R$.

===Vitushkin variation===
In  {{Cite|Vi}} Vitushkin proposed a notion of variation for functions of  several variables based on the [[Banach indicatrix|Banach indicatrix]].  Let $f: [0,1]^n\to \mathbb R$ be a Lebesgue measurable function and  $k\in \{1, \ldots, n\}$. The variation $V_k (f)$ of order $k$ of $f$ on  $[0,1]^n$ is the number
\[
\int_{-\infty}^\infty v_{k-1} (\{f=t\})\, dt
\]
where $v_{k-1} (E)$ denotes the [[Variation  of a set|Variation of the set]] $E$ of order $k-1$. The Vitushkin variation
enjoys the following properties:

a) $V_n (f+g)\leq V_n (f) + V_n (g)$.

b) If a  sequence of functions $f_j$ converges  uniformly to $f$ in $[0,1]^n$, then
\[
V_k (f) \leq \liminf_{j\to\infty}\; V_k (f_j)\, .
\]

c) If the  function $f$ is continuous and all its  variations are finite, then $f$ has a total  differential almost-everywhere.

d) If the function  $f$ is absolutely  continuous then
\[
V_n (f) = \int_{[0,1]^n} |\nabla f (x)|\, dx
\]
and hence coincides with the variation $V (f, [0,1]^n)$. However, the two quantities differ in general.

e)  If the  function $f$ is continuous, it has bounded variations of all  orders and can be extended periodically with period $1$ in all  variables, then its Fourier series converges uniformly to it  (Pringsheim's  theorem).

If the function $f$ has  continuous  derivatives of all orders up to and including $n-k+1$, then  its  variation of order $k$ is finite. The smoothness conditions  cannot  be improved for any $k$.


==References==
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|-
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|-
|valign="top"|{{Ref|Bar}}|| N.K. Bary,   "A treatise on trigonometric series" , Pergamon  (1964).  {{MR|0171116}}  {{ZBL|0129.28002}} 
|-
|valign="top"|{{Ref|Ca}}||    R. Caccioppoli, "Misura e integrazione sugli insiemei  dimensionalmente   orientati I, II", Rend. Acc. Naz. Lincei (8), {\bf  12} (1952) pp. 3-11   and 137-146.
|-
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|-
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|-
|valign="top"|{{Ref|EG}}||    L.C.  Evans, R.F. Gariepy, "Measure theory and fine properties of     functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL,      1992. {{MR|1158660}} {{ZBL|0804.2800}} 
|-
|valign="top"|{{Ref|Fe}}||     H. Federer, "Geometric measure  theory". Volume 153 of Die   Grundlehren   der mathematischen  Wissenschaften. Springer-Verlag New   York Inc., New   York, 1969.  {{MR|0257325}} {{ZBL|0874.49001}}
|-
|valign="top"|{{Ref|Fi}}||    G. Fichera, "Lezioni sulle trasformazioni lineari", Istituto   matematico  dell'Università di Trieste, vol. I, 1954.
|-
|valign="top"|{{Ref|FR}}||    W. H. Fleming, R. Rishel, "An integral formula for total gradient    variation", Arch. Math., '''11''' (1960) pp. 218-222.  {{MR|0114892}}  {{ZBL|0094.26301}} 
|-
|valign="top"|{{Ref|Jo}}|| C. Jordan,   "Sur la série de Fourier"  ''C.R. Acad. Sci. Paris'' , '''92'''  (1881)  pp. 228–230   JFM {{ZBL|13.0184.01}} 
|-
|valign="top"|{{Ref|Le}}|| H. Lebesgue,   "Leçons sur l'intégration  et la récherche des fonctions primitives", Gauthier-Villars  (1928).  {{MR|2857993}}  JFM {{ZBL|54.0257.01}} 
|-
|valign="top"|{{Ref|Ro}}|| H.L. Royden,   "Real analysis" , Macmillan  (1969). {{MR|0151555}} {{ZBL|0197.03501}}   
|-
|valign="top"|{{Ref|To}}||    L. Tonelli, "Sulle funzioni di due variabili generalmente a  variazione   limitata", Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2),  '''5''' (1936)  pp.  315-320.   
|-
|valign="top"|{{Ref|Vi}}|| A.G. Vitushkin,   "On higher-dimensional variations", Moscow  (1955).   
|-
|valign="top"|{{Ref|Wi}}||  N. Wiener,   "The quadratic variation of a function and its Fourier  coefficients"  ''J. Math. and Phys.'' , '''3'''  (1924)  pp. 72–94.   JFM {{ZBL|50.0203.01}} 
|-
|valign="top"|{{Ref|Yo}}||  L.C. Young,   "Sur une généralisation de la notion de variation de   puissance $p$ borneé au sens de  M. Wiener, et sur la convergence des  series de Fourier"  ''C.R. Acad.  Sci. Paris Sér. I Math.'' , '''204'''   (1937)  pp. 470–472   {{ZBL|0016.10501}} JFM {{ZBL|63.0182.03}} 
|-
|}