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Concept
Total variation
Summary
In mathematics, the total variation identifies several slightly different concepts, related to the (local or global) structure of the codomain of a function or a measure. For a real-valued continuous function f, defined on an interval [a, b] ⊂ R, its total variation on the interval of definition is a measure of the one-dimensional arclength of the curve with parametric equation x ↦ f(x), for x ∈ [a, b]. Functions whose total variation is finite are called functions of bounded variation.
The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper . He used the new concept in order to prove a convergence theorem for Fourier series of discontinuous periodic functions whose variation is bounded. The extension of the concept to functions of more than one variable however is not simple for various reasons.
The total variation of a real-valued (or more generally complex-valued) function , defined on an interval is the quantity
where the supremum runs over the set of all partitions of the given interval.
Let Ω be an open subset of Rn. Given a function f belonging to L1(Ω), the total variation of f in Ω is defined as
where
is the set of continuously differentiable vector functions of compact support contained in ,
is the essential supremum norm, and
is the divergence operator.
This definition does not require that the domain of the given function be a bounded set.
Following , consider a signed measure on a measurable space : then it is possible to define two set functions and , respectively called upper variation and lower variation, as follows
clearly
The variation (also called absolute variation) of the signed measure is the set function
and its total variation is defined as the value of this measure on the whole space of definition, i.e.
uses upper and lower variations to prove the Hahn–Jordan decomposition: according to his version of this theorem, the upper and lower variation are respectively a non-negative and a non-positive measure.
Official source
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