Bounded Variation

Bounded variation of a function over the closed interval means that there exists an such that

(1)

for all (Hunter 2014, §3.A.3).

For locally integrable functions, the distributional space of bounded variation is denoted and consists of integrable functions whose distributional derivative is a finite signed measure. Its seminorm on is

(2)

Here, consists of continuously differentiable functions with compact support in . This seminorm depends only on the almost-everywhere class of . It is the essential variation, the infimum of the pointwise variations over representatives equal to almost everywhere. It need not equal the pointwise partition variation of an arbitrary representative. For a representative that is an absolutely continuous function, it equals .

On the interval , the function (purple) is of bounded variation, but (red) is not, with both functions defined to be 0 at . More generally, a function is locally of bounded variation in a domain if is locally integrable, , and for all open subsets , with compact closure in , and all smooth vector fields with compact support in ,

(3)

where div denotes divergence and is a constant which only depends on the choice of and .

Such functions form the space . They may not be differentiable, but by the Riesz representation theorem, the derivative of a -function is a regular Borel measure , interpreted as a vector-valued measure in higher dimensions. Its total variation measure is denoted . Functions of bounded variation also satisfy a compactness theorem.

Given a sequence of functions in , such that

(4)

for every relatively compact open set , there is a subsequence which converges to a function in the topology of . Moreover, the limit satisfies

(5)

On a bounded convex domain with Lipschitz boundary, functions of bounded variation satisfy a version of the Poincaré inequality (Ancona et al. 2016).