In mathematical analysis, a function of bounded variation, also known as function, is a real-valued function whose total variation is bounded (finite): 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 the -axis, neglecting the contribution of motion along -axis, traveled by a point moving along the graph 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 a hypersurface in this case), but can be every intersection of the graph itself with a hyperplane (in the case of functions of two variables, a plane) parallel to a fixed -axis and to the -axis.
A function has bounded variation if and only if the Riemann–Stieltjes integral exists for every continuous function .
Another characterization states that the functions of bounded variation on a compact interval are exactly those which can be written as a difference , where both and are bounded monotone. In particular, a BV function may have discontinuities, but at most countably many.
In the case of several variables, a function defined on an open subset of is said to have bounded variation if its distributional derivative is a vector-valued finite Radon measure.
One of the most important aspects of functions of bounded variation is that they form an algebra of discontinuous functions whose first derivative exists almost everywhere: due to this fact, they can and frequently are used to define generalized solutions of nonlinear problems involving functionals and ordinary and partial differential equations in mathematics, physics and engineering.
We have the following chains of inclusions for continuous functions over a closed, bounded interval of the real line:
Continuously differentiable ⊆ Lipschitz continuous ⊆ absolutely continuous ⊆ continuous and bounded variation ⊆ differentiable almost everywhereHistory According to Boris Golubov, BV functions of a single variable were first introduced by Camille Jordan, in the paper dealing with the convergence of Fourier series. "The properties of functions of bounded variation became widely known because they were discussed by Jordan in a note appended to the third volume of his Course d’analyse (1887).Thomas W. Hawkins Jr. (1970) Lebesgue’s Theory of Integration: Its Origins and Development, page 85 University of Wisconsin Press
The first successful step in the generalization of this concept to functions of several variables was due to Leonida Tonelli,Tonelli introduced what is now called after him Tonelli plane variation: for an analysis of this concept and its relations to other generalizations, see the entry "Total variation". who introduced a class of continuous BV functions in 1926 , to extend his direct method for finding solutions to problems in the calculus of variations in more than one variable. Ten years after, in , 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 nonsmooth boundaries of sets (see the entry "Caccioppoli set" for further information). Olga Arsenievna Oleinik introduced her view of generalized solutions for nonlinear partial differential equations as functions from the space BV in the paper , and was able to construct a generalized solution of bounded variation of a first order partial differential equation in the paper : few years later, Edward D. Conway and Joel A. Smoller applied BV-functions to the study of a single nonlinear hyperbolic partial differential equation of first order in the paper , 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. Aizik Isaakovich Vol'pert developed extensively a calculus for BV functions: in the paper he proved the chain rule for BV functions and in the book he, jointly with his pupil Sergei Ivanovich Hudjaev, explored extensively the properties of BV functions and their application. His chain rule formula was later extended by Luigi Ambrosio and Gianni Dal Maso in the paper .
Formal definition
BV functions of one variable The total variation of a real-valued (or more generally complex-valued) function f, defined on an interval is the quantity
where the supremum is taken over the set of all partitions of the interval considered.
If f is differentiable and its derivative is Riemann-integrable, its total variation is the vertical component of the arc-length of its graph, that is to say,
A real-valued function 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.
It can be proved that a real function is of bounded variation in if and only if it can be written as the difference of two non-decreasing functions and on : this result is known as the Jordan decomposition of a function and it is related to the Jordan decomposition of a measure.
Through the Stieltjes integral, any function of bounded variation on a closed interval defines a bounded linear functional on . In this special case,See for example . the Riesz–Markov–Kakutani representation theorem states that every bounded linear functional arises uniquely in this way. The normalized positive functionals or probability measures correspond to positive non-decreasing lower semicontinuous functions. This point of view has been important in spectral theory,For a general reference on this topic, see in particular in its application to ordinary differential equations.
BV functions of several variables Functions of bounded variation, BV functions, are functions whose distributional derivative is a finiteIn this context, "finite" means that its value is never infinite, i.e. it is a finite measure. Radon measure. More precisely:
Let be an open subset of . A function belonging to is said to be of bounded variation (BV function), and written
if there exists a finite vector Radon measure such that the following equality holds
that is, defines a linear functional on the space of continuously differentiable vector functions of compact support contained in : the vector measure represents therefore the distributional or weak gradient of .
BV can be defined equivalently in the following way.
Given a function belonging to , the total variation of See the entry "Total variation" for further details and more information. in is defined as
where is the essential supremum norm. Sometimes, especially in the theory of Caccioppoli sets, the following notation is used
in order to emphasize that is the total variation of the distributional / weak gradient of . This notation reminds also that if is of class (i.e. a continuous and differentiable function having continuous derivatives) then its variation is exactly the integral of the absolute value of its gradient.
The space of functions of bounded variation (BV functions) can then be defined as
The two definitions are equivalent since if then
therefore defines a continuous linear functional on the space . Since as a linear subspace, this continuous linear functional can be extended continuously and linearly to the whole by the Hahn–Banach theorem. Hence the continuous linear functional defines a Radon measure by the Riesz–Markov–Kakutani representation theorem.
Locally BV functions If the function space of locally integrable functions, i.e. functions belonging to , is considered in the preceding definitions , and instead of the one of globally integrable functions, then the function space defined is that of functions of locally bounded variation. Precisely, developing this idea for , a local variation is defined as follows,
for every set , having defined as the set of all precompact open subsets of with respect to the standard topology of finite-dimensional vector spaces, and correspondingly the class of functions of locally bounded variation is defined as
Notation There are basically two distinct conventions for the notation of spaces of functions of locally or globally bounded variation, and unfortunately they are quite similar: the first one, which is the one adopted in this entry, is used for example in references (partially), (partially), and is the following one identifies the space of functions of globally bounded variation identifies the space of functions of locally bounded variation The second one, which is adopted in references and (partially), is the following: identifies the space of functions of globally bounded variation identifies the space of functions of locally bounded variation
Basic properties Only the properties common to functions of one variable and to functions of several variables will be considered in the following, and proofs will be carried on only for functions of several variables since the proof for the case of one variable is a straightforward adaptation of the several variables case: also, in each section it will be stated if the property is shared also by functions of locally bounded variation or not. References , and are extensively used.
BV functions have only jump-type or removable discontinuities In the case of one variable, the assertion is clear: for each point in the interval of definition of the function , either one of the following two assertions is true
while both limits exist and are finite. In the case of functions of several variables, there are some premises to understand: first of all, there is a continuum of directions along which it is possible to approach a given point belonging to the domain ⊂. It is necessary to make precise a suitable concept of limit: choosing a unit vector it is possible to divide in two sets
Then for each point belonging to the domain of the BV function , only one of the following two assertions is true
or belongs to a subset of having zero -dimensional Hausdorff measure. The quantities
are called approximate limits of the BV function at the point .
V(⋅, Ω) is lower semi-continuous on L1(Ω) The functional is lower semi-continuous: to see this, choose a Cauchy sequence of BV-functions converging to . Then, since all the functions of the sequence and their limit function are integrable and by the definition of lower limit
Now considering the supremum on the set of functions such that then the following inequality holds true
which is exactly the definition of lower semicontinuity.
BV(Ω) is a Banach space By definition is a subset of , while linearity follows from the linearity properties of the defining integral i.e.
for all therefore for all , and
for all , therefore for all , and all . The proved vector space properties imply that is a vector subspace of . Consider now the function defined as
where is the usual norm: it is easy to prove that this is a norm on . To see that is complete respect to it, i.e. it is a Banach space, consider a Cauchy sequence in . By definition it is also a Cauchy sequence in and therefore has a limit in : since is bounded in for each , then by lower semicontinuity of the variation , therefore is a BV function. Finally, again by lower semicontinuity, choosing an arbitrary small positive number From this we deduce that is continuous because it's a norm.
BV(Ω) is not separable