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Variation of a Function

Let be a real valued function defined on an interval . The least upper bound of the sums of the type

where runs over all partitions of , is called the (total) variation of over . For instance, . This concept was introduced by C.Jordan [1] .

Function is called of bounded (or finite) variation if there is a number such that .  

If is real valued then roughly saying is of bounded variation means that the graph of has finite arc length.

Theorem: If a real valued function defined on
a) has bounded derivation here, or
b) is monotone here, or
c) has a finite number of local extrema here and is continuous here,
then is of bounded variation.

The sum, the difference and the product of two functions of bounded variation is again a function of bounded variation. If the modulus of the denominator is larger than a positive constant then the quotient of two such function is also of bounded variation.

Theorem (Jordan decomposition of functions of bounded variation): A real valued function is of bounded variation if and only if can be represented in the form , where and are monotone increasing functions on .

To prove this put , and .

It follows that if is of bounded variation, then is bounded and its discontinuities are all jump discontinuities (i.e. of the so called first kind ) and are countable in number.

All these basic properties of functions of bounded variation were already found by Jordan [1] , [2] .

Jordan was led to the concept of functions of bounded variation in the context of generalization of the Dirichlet criterion for the convergence of Fourier series of piecewise monotone functions:

Theorem (Dirichlet-Jordan): Let ->R be a bounded function.
1) If is of bounded variation on an interval about for some , then the Fourier series of evaluated at converges to .
2) If is continuous and of bounded variation, then the Fourier series of converges uniformly to .

In 1904 Lebesgue  [3]  proved the following result:

Theorem (Lebesgue decomposition of a function of bounded variation): If is a function of bounded variation on the interval then it can be represented in the form
,
where is a absolutely continuous function, is a singular function, and is a jump function.

The Lebesgue decomposition is unique in some cases, e.g. if . ( [4] , [5] ).

If is absolutely continuous on then its total variation is

A multidimensional extension requires an appropriate generalization of total variation for functions of several variables is needed. There more possibilities how to do this.

Let be a real valued function defined on , .  Given a set of indices , let denote the value of at the point in whose th coordinate is equal to if and is equal to otherwise. For an -dimensional subinterval of , let   be the alternating sum of the values of at the vertices of . The variation  in the sense of Vitali of on is defined by
,
where the supremum is taken over all partitions of into -dimensional subintervals.

This notion was introduced by Vitali [6] , and later rediscovered by  Lebesgue  [7]  and Fréchet [8] .

If all the involved partial derivatives are continuous on then

An analog of the above Jordan decomposition theorem says that a function is of bounded Vitali variation if and only if it can be represented in the form where for both functions and all the sums are non-negative.

For non-empty , let denote the function on obtained by fixing the th argument of to whenever , and letting the other arguments vary. Function   is said to be of  bounded Hardy-Krause variation if is of bounded Vitali variation, for all non-empty .

This type of variation was introduced by Hardy [9]  in the case in connection with his investigation of convergence of double Fourier expansion of functions of two variables.  

For functions of a single variable, the notions of bounded Vitali variation and bounded Hardy-Krause variation coincide, and reduce to the usual definition of bounded variation. In multivariate case there are functions that are of bounded Vitali variation but not of bounded Hardy-Krause variation. Let be not of bounded variation. Define by . Then for all rectangles since does not vary with . Thus defined is of bounded Vitali variation, but not of bounded Hardy-Krause variation.

There is a similar analog to the Jordan decomposition theorem also for functions of bounded Hardy-Krause variation as above.

References

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