Namespaces
Variants
Actions

Difference between revisions of "Variation of a function"

From Encyclopedia of Mathematics
Jump to: navigation, search
m
(removing strange system messages at the top)
 
(5 intermediate revisions by 2 users not shown)
Line 1: Line 1:
A numerical characteristic of functions of one real variable which is connected with differentiability properties.
+
{{MSC|26A45}} (Functions of one variable)
  
1) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961101.png" /> be a complex-valued function defined on an interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961102.png" />; its variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961103.png" /> is the least upper bound of sums of the type
+
{{MSC|26B30|28A15,26B15,49Q15}} (Functions of severable variables)
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961104.png" /></td> </tr></table>
+
[[Category:Analysis]]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961105.png" /> is an arbitrary system of points on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961106.png" /> (cp. with [[Function of bounded variation#Total variation|Function of bounded variation]]. This definition was given by C. Jordan [[#References|[1]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961107.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961108.png" /> has (is of) bounded (finite) variation over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v0961109.png" />, and the class of all such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611010.png" />. A real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611011.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611012.png" /> if and only if it can be represented in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611013.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611015.png" /> are functions which increase on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611016.png" /> (the [[Jordan decomposition|Jordan decomposition]] of functions of bounded variation). The sum, the difference and the product of two functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611017.png" /> are also functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611018.png" />. This is also true of the quotient of two functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611019.png" /> if the modulus of the denominator is larger than a positive constant on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611020.png" />. Every function in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611021.png" /> is bounded and cannot have more than a countable set of discontinuity points, all of which are of the first kind. All these properties of functions in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611022.png" /> were established by Jordan [[#References|[1]]] (see also [[#References|[2]]]).
+
{{TEX|done}}
  
Functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611023.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611024.png" /> are almost-everywhere differentiable on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611025.png" /> and may be represented as
+
Also  called ''total variation''. A numerical characteristic of functions of  one or more real variables which is connected with differentiability  properties.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611026.png" /></td> </tr></table>
+
==Functions of one variable==
 +
===Classical definition===
 +
Let $I\subset \mathbb R$ be an interval. The total variation is defined in the following way.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611027.png" /> is an absolutely continuous function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611028.png" /> is a singular function and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611029.png" /> is a saltus function (the [[Lebesgue decomposition|Lebesgue decomposition]] of a function of bounded variation). Such a decomposition is unique if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611030.png" /> [[#References|[3]]], [[#References|[2]]].
+
'''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}}).
  
The class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611031.png" /> was originally introduced by Jordan in the context of the generalization of the Dirichlet criterion for the convergence of Fourier series of piecewise-monotone functions. It was shown by him that Fourier series of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611032.png" />-periodic functions in the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611033.png" /> converge at all points of the real axis. Functions of bounded variation subsequently found extensive application in various branches of mathematics, especially in the theory of the Stieltjes integral.
+
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]].
  
One sometimes also considers classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611034.png" />, defined as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611035.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611037.png" />) be a continuous function which increases monotonically if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611038.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611039.png" /> be the least upper bound of sums of the type
+
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}.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611040.png" /></td> </tr></table>
+
===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}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611041.png" /> is an arbitrary system of points in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611042.png" />. The quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611043.png" /> is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611045.png" />-variation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611046.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611047.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611048.png" />, one says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611049.png" /> has bounded <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611050.png" />-variation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611051.png" />, while the class of such functions is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611052.png" /> ([[#References|[4]]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611053.png" />, one obtains Jordan's class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611055.png" />, while if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611056.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611057.png" />, one obtains Wiener's classes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611059.png" /> [[#References|[5]]]. The definition of the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611060.png" /> was proposed by L.C. Young [[#References|[6]]].
+
'''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).
  
If
+
For  a proof  see Section 4 of  Chapter 4 in {{Cite|Co}}. Obvious  generalizations  hold in the case of  different domains of definition.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611061.png" /></td> </tr></table>
+
===Jordan decomposition===
 +
{{Anchor|Jordan decomposition}}
 +
A fundamental characterization of functions of bounded variation of one variable is due to Jordan.
  
then
+
'''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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611062.png" /></td> </tr></table>
+
(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.
  
In particular, on any interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611063.png" />,
+
'''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 $g^+-g^-=f^+-f^-\equiv {\rm  const}$.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611064.png" /></td> </tr></table>
+
(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.
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611066.png" />, these being proper inclusions.
+
'''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
 +
\[
 +
TV^+ (f):= \sup \left\{ \sum_{i=1}^N \max \{(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\,
 +
\]
 +
\[
 +
TV^- (f) := \sup \left\{ \sum_{i=1}^N \max \{-(f(a_{i+1})-f(a_i)), 0\} : (a_1, \ldots, a_{N+1})\in\Pi\right\}\, .
 +
\]
  
====References====
+
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}}.
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> C. Jordan,  "Sur la série de Fourier" ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''92''' : 5 (1881)  pp. 228–230</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> I.P. Natanson,  "Theory of functions of a real variable" , '''1–2''' , F. Ungar (1955–1961)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> H. Lebesgue,   "Leçons sur l'intégration et la récherche des fonctions primitives" , Gauthier-Villars (1928)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  N.K. [N.K. Bari] Bary,  "A treatise on trigonometric series" , Pergamon (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> N. Wiener,  "The quadratic variation of a function and its Fourier coefficients" ''J. Math. and Phys.'' , '''3''' (1924) pp. 72–94</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L.C. Young,   "Sur une généralisation de la notion de variation de puissance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611067.png" /> borneé au sens de M. Wiener, et sur la convergence des series de Fourier" ''C.R. Acad. Sci. Paris Sér. I Math.'' , '''204''' (1937pp. 470–472</TD></TR></table>
+
===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,
  
====Comments====
+
===Definition===
The variation of a function as defined above is often called the total variation. It is the sum of the negative and positive variations (cf. [[Negative variation of a function|Negative variation of a function]]; [[Positive variation of a function|Positive variation of a function]]). One has
+
Following {{Cite|EG}}:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611068.png" /></td> </tr></table>
+
'''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 $BV  (\Omega)$.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611069.png" /> is the [[Banach indicatrix|Banach indicatrix]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611070.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611071.png" />, then
+
As a consequence of the [[Radon-Nikodym theorem]] we then have
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611072.png" /></td> </tr></table>
+
'''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 $Du$.
  
====References====
+
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
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E. Hewitt,  K.R. Stromberg,  "Real and abstract analysis" , Springer  (1965) pp. 266; 270; 272</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Saks,   "Theory of the integral" , Hafner  (1937) (Translated from French)</TD></TR></table>
+
$u=\tilde{u}$ 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]]).
  
Several different definitions of variation exist for functions of several variables ([[Arzelà variation|Arzelà variation]]; [[Vitali variation|Vitali variation]]; [[Pierpont variation|Pierpont variation]]; [[Tonelli plane variation|Tonelli plane variation]]; [[Fréchet variation|Fréchet variation]]; [[Hardy variation|Hardy variation]]). The following definition, [[#References|[1]]], based on the use of the [[Banach indicatrix|Banach indicatrix]], also proved very fruitful. Let a real-valued function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611073.png" /> be given and be Lebesgue-measurable on an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611074.png" />-dimensional cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611075.png" />. The variation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611076.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611077.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611078.png" />, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611079.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611080.png" /> is the number
+
===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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611081.png" /></td> </tr></table>
+
'''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
 +
\[
 +
{\rm Per}\, (E, \Omega) = V ({\bf 1}_E, \Omega)\, .
 +
\]
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611082.png" /> denotes the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611083.png" />-th variation of the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611084.png" /> (cf. [[Variation of a set|Variation of a set]]), while the integral is understood in the sense of Lebesgue. This definition allows one to transfer many properties of functions of bounded variation in one variable to functions of several variables. For instance,
+
==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}}.
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611085.png" />;
+
'''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).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611086.png" /></td> </tr></table>
+
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 $f:[a,b]\to\mathbb R$.
  
b) If a sequence of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611087.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611088.png" /> converges uniformly to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611089.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611090.png" />, then
+
===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:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611091.png" /></td> </tr></table>
+
a) $V_n (f+g)\leq V_n (f) + V_n (g)$.
  
c) If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611092.png" /> is continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611093.png" /> and all its variations are finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611094.png" /> has a total differential almost-everywhere.
+
b) If a  sequence of functions $f_j$ converges  uniformly to $f$ in $[0,1]^n$, then
 +
\[
 +
V_k (f) \leq \liminf_{j\to\infty}\; V_k (f_j)\, .
 +
\]
  
d) If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611095.png" /> is absolutely continuous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611096.png" />, then
+
c) If the function $f$ is continuous and all its  variations are finite, then $f$ has a total  differential almost-everywhere.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611097.png" /></td> </tr></table>
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611098.png" /> is continuous in a cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v09611099.png" /> with side-length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110100.png" />, if it has bounded variations of all orders in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110101.png" /> and if it can be periodically extended with period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110102.png" /> for all arguments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110103.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110104.png" />, in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110105.png" />-dimensional space, then its Fourier series converges uniformly to it on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110106.png" /> (Pringsheim's theorem).
+
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).
  
A sufficient condition for being of bounded variation is: If the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110107.png" /> has continuous derivatives of all orders up to and including <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110108.png" /> in the cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110109.png" />, then its variation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110110.png" /> is finite. This theorem is a final theorem in the sense that the smoothness conditions cannot be improved for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096110/v096110111.png" />.
+
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====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Vitushkin,  "On higher-dimensional variations" , Moscow  (1955)  (In Russian)</TD></TR></table>
 
  
''A.G. Vitushkin''
+
==References==
 +
{|
 +
|-
 +
|valign="top"|{{Ref|AFP}}||      L. Ambrosio, N.  Fusco, D.  Pallara, "Functions of bounded    variations  and  free  discontinuity  problems". Oxford Mathematical    Monographs. The    Clarendon Press,  Oxford University Press, New York,    2000.      {{MR|1857292}}{{ZBL|0957.49001}}
 +
|-
 +
|valign="top"|{{Ref|Ba}}||  S. Banach,  "Sur les lignes rectifiables et les surfaces dont l'aire  est finie"  ''Fund. Math.'' , '''7'''  (1925)  pp. 225–236.  JFM {{ZBL|51.0199.03}}
 +
|-
 +
|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.
 +
|-
 +
|valign="top"|{{Ref|Co}}|| D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
 +
|-
 +
|valign="top"|{{Ref|DG}}||    E. De Giorgi, "Su una teoria generale della misura $n-1$-dimensionale    in uno spazio a $r$ dimensioni", Ann. Mat. Pura Appl. (4), '''36'''    (1954) pp. 191-213.  {{ZBL|0055.28504}}
 +
|-
 +
|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}}
 +
|-
 +
|}

Latest revision as of 16:07, 12 October 2012

2020 Mathematics Subject Classification: Primary: 26A45 [MSN][ZBL] (Functions of one variable)

2020 Mathematics Subject Classification: Primary: 26B30 Secondary: 28A1526B1549Q15 [MSN][ZBL] (Functions of severable variables)

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 [Co] or Section 10.2 of [Ro]).

If the total variation is finite, then $f$ is called a 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 $(X,d)$: 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 measures. More precisely, consider a signed measure $\mu$ on (the 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 [Co]. Obvious generalizations hold in the case of different domains of definition.

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 [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 [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 [Ro].

Historical remark

The variation of a function of one real variable was considered for the first time by C. Jordan in [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 [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 [Wi]). The definition of the class $BV_\Phi ([a,b])$ was proposed by L.C. Young in [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 [DG] and [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 [To], proposed by Cesari [Ce], cp. with the section Tonelli-Cesari variation of Function of bounded variation. We refer to Section 3.12 of [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 currents in $\mathbb R^n$. This is the point of view of Federer, [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 [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 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 [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 [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 [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 [Vi] Vitushkin proposed a notion of variation for functions of several variables based on the 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 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

[AFP] L. Ambrosio, N. Fusco, D. Pallara, "Functions of bounded variations and free discontinuity problems". Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. MR1857292Zbl 0957.49001
[Ba] S. Banach, "Sur les lignes rectifiables et les surfaces dont l'aire est finie" Fund. Math. , 7 (1925) pp. 225–236. JFM Zbl 51.0199.03
[Bar] N.K. Bary, "A treatise on trigonometric series" , Pergamon (1964). MR0171116 Zbl 0129.28002
[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.
[Co] D. L. Cohn, "Measure theory". Birkhäuser, Boston 1993.
[DG] E. De Giorgi, "Su una teoria generale della misura $n-1$-dimensionale in uno spazio a $r$ dimensioni", Ann. Mat. Pura Appl. (4), 36 (1954) pp. 191-213. Zbl 0055.28504
[EG] L.C. Evans, R.F. Gariepy, "Measure theory and fine properties of functions" Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992. MR1158660 Zbl 0804.2800
[Fe] H. Federer, "Geometric measure theory". Volume 153 of Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag New York Inc., New York, 1969. MR0257325 Zbl 0874.49001
[Fi] G. Fichera, "Lezioni sulle trasformazioni lineari", Istituto matematico dell'Università di Trieste, vol. I, 1954.
[FR] W. H. Fleming, R. Rishel, "An integral formula for total gradient variation", Arch. Math., 11 (1960) pp. 218-222. MR0114892 Zbl 0094.26301
[Jo] C. Jordan, "Sur la série de Fourier" C.R. Acad. Sci. Paris , 92 (1881) pp. 228–230 JFM Zbl 13.0184.01
[Le] H. Lebesgue, "Leçons sur l'intégration et la récherche des fonctions primitives", Gauthier-Villars (1928). MR2857993 JFM Zbl 54.0257.01
[Ro] H.L. Royden, "Real analysis" , Macmillan (1969). MR0151555 Zbl 0197.03501
[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.
[Vi] A.G. Vitushkin, "On higher-dimensional variations", Moscow (1955).
[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
[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
How to Cite This Entry:
Variation of a function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variation_of_a_function&oldid=27856
This article was adapted from an original article by B.I. Golubov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article