As the green ball travels on the graph of the given function, the length of the path travelled by that ball's projection on the y-axis, shown as a red ball, is the total variation of the function.

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 xf(x), for x ∈ [a, b].

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Historical noteEdit

The concept of total variation for functions of one real variable was first introduced by Camille Jordan in the paper (Jordan 1881).[1] 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.

DefinitionsEdit

Total variation for functions of one real variableEdit

Definition 1.1. 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.

Total variation for functions of n > 1 real variablesEdit

Definition 1.2. 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  , and   is the essential supremum norm. Note that this definition does not require that the domain   of the given function be a bounded set.

Total variation in measure theoryEdit

Classical total variation definitionEdit

Following Saks (1937, p. 10), 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

 

Definition 1.3. 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.

 

Modern definition of total variation normEdit

Saks (1937, p. 11) 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. Using a more modern notation, define

 
 

Then   and   are two non-negative measures such that

 
 

The last measure is sometimes called, by abuse of notation, total variation measure.

Total variation norm of complex measuresEdit

If the measure   is complex-valued i.e. is a complex measure, its upper and lower variation cannot be defined and the Hahn–Jordan decomposition theorem can only be applied to its real and imaginary parts. However, it is possible to follow Rudin (1966, pp. 137–139) and define the total variation of the complex-valued measure   as follows

Definition 1.4. The variation of the complex-valued measure   is the set function

 

where the supremum is taken over all partitions   of a measurable set   into a countable number of disjoint measurable subsets.

This definition coincides with the above definition   for the case of real-valued signed measures.

Total variation norm of vector-valued measuresEdit

The variation so defined is a positive measure (see Rudin (1966, p. 139)) and coincides with the one defined by 1.3 when   is a signed measure: its total variation is defined as above. This definition works also if   is a vector measure: the variation is then defined by the following formula

 

where the supremum is as above. Note also that this definition is slightly more general than the one given by Rudin (1966, p. 138) since it requires only to consider finite partitions of the space  : this implies that it can be used also to define the total variation on finite-additive measures.

Total variation of probability measuresEdit

The total variation of any probability measure is exactly one, therefore it is not interesting as a means of investigating the properties of such measures. However, when μ and ν are probability measures, the total variation distance of probability measures can be defined as   where the norm is the total variation norm of signed measures. Using the property that  , we eventually arrive at the equivalent definition

 

and its values are non-trivial. The factor   above is usually dropped (as is the convention in the article total variation distance of probability measures). Informally, this is the largest possible difference between the probabilities that the two probability distributions can assign to the same event. For a categorical distribution it is possible to write the total variation distance as follows

 

It may also be normalized to values in   by halving the previous definition as follows

 [2]

Basic propertiesEdit

Total variation of differentiable functionsEdit

The total variation of a   function   can be expressed as an integral involving the given function instead of as the supremum of the functionals of definitions 1.1 and 1.2.

The form of the total variation of a differentiable function of one variableEdit

Theorem 1. The total variation of a differentiable function  , defined on an interval  , has the following expression if   is Riemann integrable

 

The form of the total variation of a differentiable function of several variablesEdit

Theorem 2. Given a   function   defined on a bounded open set  , with   of class  , the total variation of   has the following expression

 

Here   denotes the  -norm.

ProofEdit

The first step in the proof is to first prove an equality which follows from the Gauss–Ostrogradsky theorem.

LemmaEdit

Under the conditions of the theorem, the following equality holds:

 
Proof of the lemmaEdit

From the Gauss–Ostrogradsky theorem:

 

by substituting  , we have:

 

where   is zero on the border of   by definition:

 
 
 
 
 
Proof of the equalityEdit

Under the conditions of the theorem, from the lemma we have:

 

in the last part   could be omitted, because by definition its essential supremum is at most one.

On the other hand we consider   and   which is the up to   approximation of   in   with the same integral. We can do this since   is dense in  . Now again substituting into the lemma:

 

This means we have a convergent sequence of   that tends to   as well as we know that  . q.e.d.

It can be seen from the proof that the supremum is attained when

 

The function   is said to be of bounded variation precisely if its total variation is finite.

Total variation of a measureEdit

The total variation is a norm defined on the space of measures of bounded variation. The space of measures on a σ-algebra of sets is a Banach space, called the ca space, relative to this norm. It is contained in the larger Banach space, called the ba space, consisting of finitely additive (as opposed to countably additive) measures, also with the same norm. The distance function associated to the norm gives rise to the total variation distance between two measures μ and ν.

For finite measures on ℝ, the link between the total variation of a measure μ and the total variation of a function, as described above, goes as follows. Given μ, define a function   by

 

Then, the total variation of the signed measure μ is equal to the total variation, in the above sense, of the function φ. In general, the total variation of a signed measure can be defined using Jordan's decomposition theorem by

 

for any signed measure μ on a measurable space  .

ApplicationsEdit

Total variation can be seen as a non-negative real-valued functional defined on the space of real-valued functions (for the case of functions of one variable) or on the space of integrable functions (for the case of functions of several variables). As a functional, total variation finds applications in several branches of mathematics and engineering, like optimal control, numerical analysis, and calculus of variations, where the solution to a certain problem has to minimize its value. As an example, use of the total variation functional is common in the following two kind of problems

See alsoEdit

NotesEdit

  1. ^ According to Golubov & Vitushkin (2001).
  2. ^ Gibbs, Alison; Francis Edward Su (2002). "On Choosing and Bounding Probability Metrics" (PDF). p. 7. Retrieved 8 April 2017.

Historical referencesEdit

ReferencesEdit

External linksEdit

One variable

One and more variables

Measure theory

ApplicationsEdit