In mathematical analysis, a function of bounded variation refers to 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 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 "" which can be written as a difference "", where both "" and "" are bounded monotone.
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 finite vector 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, ordinary and partial differential equations in mathematics, physics and engineering. Considering the problem of multiplication of distributions or more generally the problem of defining general nonlinear operations on generalized functions, "function of bounded variation are the smallest algebra which has to be embedded in every space of generalized functions preserving the result of multiplication".
According to Golubov, "BV" functions of a single variable were first introduced by Camille Jordan, in the paper Harv|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 Harv|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 Harv|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. Aizik Isaakovich Vol'pert developed extensively a calculus for "BV" functions in the paper Harv|Vol'pert|1967 and in the book Harv|Hudjaev|Vol'pert|1986.
"BV" functions of one variable
Definition 1. The total variation of a real-valued function "", defined on an interval is the quantity
where the supremum is taken over the set of all partitions of the interval considered.
If 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,
Definition 2. 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 f is of bounded variation in an interval if and only if it can be written as the difference of two non decreasing functions (This is known as the Jordan decomposition.)
Through the Stieltjes integral, any function of bounded variation on a closed interval defines a bounded linear functional on . In this special case harv|Kolmogorov|Fomin|1969|pp=374-376, the Riesz representation theorem states that every bounded linear functional arises uniquely in this way. The normalised positive functionals or probability measures correspond to positive non-decreasing lower semicontinuous functions. This point of view has been important in
spectral theory harv|Riesz|Sz.-Nagy|1990, 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 finite Radon measure. More precisely:
Definition 1 Let be an open subset of . A locally integrable function is said of bounded variation (BV function), and write
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 .
An equivalent definition is the following.
Definition 2 Given a locally integrable function , the total variation of in is defined as
where is the essential supremum norm.
The space of functions of bounded variation (BV functions) can then be defined as
The two definition 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 linearily to the whole by the Hahn–Banach theorem i.e. it defines a Radon measure.
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. References Harv|Giusti|1984|pp=7-9, Harv|Hudjaev|Vol'pert|1986 and Harv|Màlek|Nečas|Rokyta|Růžička|1996 are extensively used.
"BV" functions have only jump-type singularities
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 exists and are finite. In the case of functions 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 or one of the following two assertion 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 BV(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 "L""1""loc"(Ω), 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