Variation of a function

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 $f$ 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, \[ V (u, [a,b]) = \inf \{TV (\tilde{u}): \tilde{u}= u \quad\mbox{a.e.}\}\, . \] (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

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