Bounded variation

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In mathematics, bounded variation refers a to real-valued functions whose total variation is bounded i.e. is less than infinity: the graph of a function having this property is well behaved in a precise sense. For a continuous function of a single variable, being of bounded variation means that the distance along the direction of y-axis (i.e. the distance calculated neglecting the contribution of motion along x-axis) traveled by an ideal point moving along the graph of the given function (which, under given hypothesis, is also a continuous path) has a finite value. For a continuous function of several variables, the meaning of the definition is the same, except for the fact that the continuous path to be considered cannot be the whole graph of the given function (which is an hypersurface in this case), but can be every intersection of the graph itself with a plane parallel to a fixed x-axis and to the y-axis.

Functions of bounded variation are precisely those with respect to which one may find Riemann-Stieltjes integrals of all continuous functions.

Another characterization states that the functions of bounded variation on a closed interval are exactly those f which can be written as a difference g − h, where both g and h are monotone.

In the case of several variables, a function f defined on an open subset of \scriptstyle\mathbb{R}^n is said to have bounded variation if its distributional derivative is a finite vector Radon measure.

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[edit] History

According to Golubov, BV functions of a single variable were first introduced by Camille Jordan, in the paper (Jordan 1881) dealing with the convergence of Fourier series. The first step in the generalization of this concept to functions of several variables was due to Leonida Tonelli, who introduced a class of continuous BV functions in 1926 (Cesari 1986, pp. 47-48), to extend his direct method for finding solutions to problems in the calculus of variations in more than one variable. Ten years after, in 1936, Lamberto Cesari changed the continuity requirement in Tonelli's definition to a less restrictive integrability requirement, obtaining for the first time the class of functions of bounded variation of several variables in its full generality: as Jordan did before him, he applied the concept to resolve of a problem concerning the convergence of Fourier series, but for functions of two variables. After him, several authors applied BV functions to study Fourier series in several variables, geometric measure theory, calculus of variations, and mathematical physics: Renato Caccioppoli and Ennio de Giorgi used them to define measure of non smooth boundaries of sets (see voice "Caccioppoli set" for further informations), Edward D. Conway and Joel A. Smoller applied them to the study of a single nonlinear hyperbolic partial differential equation of first order in the paper (Conway & Smoller 1966), proving that the solution of the Cauchy problem for such equations is a function of bounded variation, provided the initial value belongs to the same class.

[edit] Formal definition

[edit] BV functions of one variable

Definition 1. The total variation of a real-valued function f, defined on a interval \scriptstyle [a , b] \subset \mathbb{R} is the quantity

V^a_b(f)=\sup_{P \in \mathcal{P}} \sum_{i=0}^{n_P-1} | f(x_{i+1})-f(x_i) |. \,

where the supremum is taken over the set \scriptstyle \mathcal{P} =\left\{P=\{ x_0, \dots , x_{n_p}\}|P\text{ is a partition of } [a,b] \right\} of all partitions of the interval considered.

If f is differentiable and its derivative is integrable, its total variation is the vertical component of the arc-length of its graph, that is to say,

V^a_b(f) = \int _a^b |f'(x)|\, dx.

Definition 2. A real-valued function f on the real line is said to be of bounded variation (BV function) on a chosen interval if its total variation is finite, i.e.

f \in BV([a,b]) \iff V^a_b(f) < +\infty

[edit] BV functions of several variables

Functions of bounded variation, BV functions, are functions whose distributional derivative is a finite Radon measure. More precisely:

Definition 1 Let Ω be an open subset of \scriptstyle\mathbb{R}^n. A function u in L1(Ω) is said of bounded variation (BV function), and write

u\in BV(\Omega)

if there exists a finite vector Radon measure \scriptstyle Du\in\mathcal M(\Omega,\mathbb{R}^n) such that the following equality holds

\int_\Omega u(x)\,\mathrm{div}\phi(x)\, dx = - \int_\Omega \langle \phi(x), Du(x)\rangle \qquad \forall \phi\in C_c^1(\Omega,\mathbb{R}^n)

that is, u defines a linear functional on the space \scriptstyle C_c^1(\Omega,\mathbb{R}^n) of continuously differentiable vector functions \scriptstyle\phi of compact support contained in Ω: the vector measure Du represents therefore the distributional or weak gradient of u.

An equivalent definition is the following.

Definition 2 Given a function u belonging to \scriptstyle L^1(\Omega), the total variation of u in Ω is defined as

V(u,\Omega):=\sup\left\{\int_\Omega u\mathrm{div}\phi\colon \phi\in C_c^1(\Omega,\mathbb{R}^n),\ \Vert \phi\Vert_{L^\infty(\Omega)}\le 1\right\}.

where \scriptstyle \Vert\;\Vert_{L^\infty(\Omega)} is the essential supremum norm.

The space of functions of bounded variation (BV functions) can then be defined as

BV(\Omega)=\{ u\in L^1(\Omega)\colon V(u,\Omega)<+\infty\}

Notice that the Sobolev space W1,1(Ω) is a proper subset of BV(Ω). In fact, for each u in W1,1(Ω) it is possible to choose a measure \scriptstyle \mu:=\nabla u \mathcal L (where \scriptstyle\mathcal L is the Lebesgue measure on Ω) such that the equality

\int u\mathrm{div}\phi = -\int \phi\, d\mu = -\int \phi \nabla u \qquad \forall \phi\in C_c^1

holds, since it is nothing more than the definition of weak derivative, and hence holds true. One can easily find an example of a BV function which is not W1,1.

[edit] Generalizations

[edit] Weighted BV functions

It is possible to generalize the above notion of total variation so that different variations are weighted differently. More precisely, let \scriptstyle \varphi : [0, +\infty)\longrightarrow [0, +\infty) be any increasing function such that \scriptstyle \varphi(0) = \varphi(0+) =\lim_{x\rightarrow 0_+}\varphi(x) = 0 (the weight function) and let \scriptstyle f : [0, T]\longrightarrow X be a function from the interval \scriptstyle [0 , T] \subset \mathbb{R} taking values in a normed vector space X. Then the \scriptstyle \boldsymbol\varphi-variation of f over [0,T] is defined as

\mathop{\varphi\mbox{-Var}}_{[0, T]} (f) := \sup \sum_{j = 0}^{k} \varphi \left( | f(t_{j + 1}) - t_{j} |_{X} \right),

where, as usual, the supremum is taken over all finite partitions of the interval [0,T], i.e. all the finite sets of real numbers ti such that

0 = t_{0} < t_{1} < \ldots < t_{k} = T.

The original notion of variation considered above is the special case of \scriptstyle \varphi-variation for which the weight function is the identity function: therefore a integrable function f is said to be a weighted BV function (of weight \scriptstyle \boldsymbol\varphi) if and only if its \scriptstyle \varphi-variation is finite.

f\in BV_\varphi([0, T];X)\iff \mathop{\varphi\mbox{-Var}}_{[0, T]} (f) <+\infty

The space \scriptstyle BV_\varphi([0, T];X) is a topological vector space with respect to the norm

\| f \|_{BV_\varphi} := \| f \|_{\infty} + \mathop{\varphi \mbox{-Var}}_{[0, T]} (f),

where \scriptstyle\| f \|_{\infty} denotes the usual supremum norm of f.

[edit] SBV functions

SBV functions i.e. Special functions of Bounded Variation where introduced by Luigi Ambrosio and Ennio de Giorgi in the paper (Ambrosio & De Giorgi 1988), dealing with free discontinuity variational problems: given a open subset Ω of \scriptstyle\mathbb{R}^n, the space \scriptstyle {SBV}(\Omega) is a proper subspace of \scriptstyle BV(\Omega), since the weak gradient of each function belonging to it const exatcly of the sum of a n-dimensional support and a n − 1-dimensional support measure and no lower-dimensional terms, as seen in the following definition.

Definition. Given a function u belonging to \scriptstyle L^1(\Omega), then \scriptstyle u\in {SBV}(\Omega) if and only if

1. There exist two Borel functions f and g of domain Ω and Codomain \scriptstyle \mathbb{R}^n such that

\int_\Omega\vert f\vert dH^n+ \int_\Omega\vert g\vert dH^{n-1}<+\infty.

2. For all of continuously differentiable vector functions \scriptstyle\phi of compact support contained in Ω, i.e. for all \scriptstyle \phi \in C_c^1(\Omega,\mathbb{R}^n) the following formula is true:

\int_\Omega u\mbox{div} \phi dH^n = \int_\Omega \langle \phi, f\rangle dH^n +\int_\Omega \langle \phi, g\rangle dH^{n-1}.

where Hα is the α-dimensional Hausdorff measure.

Details on the properties of SBV functions can be found in works cited in the bibliography section: particularly the paper (De Giorgi 1992) contains a useful bibliography.

[edit] Examples

The function

f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ \sin(1/x), & \mbox{if } x \neq 0 \end{cases}

is not of bounded variation on the interval [0,2 / π]

The function f(x)=sin(1/x) is not of bounded variation.
The function f(x)=sin(1/x) is not of bounded variation.

While it is harder to see, the function

f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x \sin(1/x), & \mbox{if } x \neq 0 \end{cases}

is not of bounded variation on the interval [0,2 / π] either.

The function f(x)=x sin(1/x) is not of bounded variation.
The function f(x)=x sin(1/x) is not of bounded variation.

At the same time, the function

f(x) = \begin{cases} 0, & \mbox{if }x =0 \\ x^2 \sin(1/x), & \mbox{if } x \neq 0 \end{cases}

is of bounded variation on the interval [0,2 / π].

The function f(x)=x2 sin(1/x) is of bounded variation.
The function f(x)=x2 sin(1/x) is of bounded variation.

[edit] Applications

[edit] Mathematics

Functions of bounded variation have been studied in connection with the set of discontinuities of functions and differentiability of real functions, and the following results are well-known. If f is a real function of bounded variation on an interval [a, b] then

[edit] Physics and engineering

The ability of BV functions to deal with discontinuities has made their use widespread in the applied sciences: solutions of problems in mechanics, physics, chemical kinetics are very often representable by functions of bounded variation.

[edit] See also

[edit] References

[edit] Bibliography

[edit] External links

[edit] Theory

  • Boris I. Golubov (and comments of Anatolii Georgievich Vitushkin) "Variation of a function", Springer-Verlag Online Encyclopaedia of Mathematics.
  • BV function on PlanetMath.
  • Jordan, Camille (1881), "Sur la série de Fourier", Comptes rendus des Académie des sciences de Paris 92: 228-230 (at Gallica). This is, according to Boris Golubov, the first paper on functions of bounded variation.
  • Rowland, Todd and Weisstein, Eric W. "Bounded Variation". From MathWorld--A Wolfram Web Resource.

[edit] Varia

This article incorporates material from BV function on PlanetMath, which is licensed under the GFDL.

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