Bounded variation

In calculus is a function of bounded variation ( bounded variation ) if their total variation (total variation ) is finite, so they do not arbitrarily strongly oscillates in a certain way. These concepts are closely related to the continuity and integration of functions.

The space of all functions of limited variation in the field is designated.

Real functions

Definition

The total variation of a real-valued function: that is defined on a closed interval is the supremum

Where this supremum is taken over all possible partitions of the interval. The specified here depends on.

Exactly the continuous functions of bounded variation are Riemann - Stieltjes integrable. It can therefore be equipped with a seminorm:

This supremum is taken over all functions with compact support and function values ​​in the interval.

The semi standard match the upper bound defining the limited variation in accordance.

Example

A simple example of an unlimited variation of the function near. It is clearly understood that the value of splitting with increasing proximity to 0 will grow ever faster towards ∞, and thus the sine of this value is determined by running an infinite number of oscillations. This displays the image on the right.

The function

Is also not of bounded variation in the interval [ 0, 1], in contrast to the function:

Here, the variation of the sine term, which increases for heavily damped by the additional power enough.

Extensions

Above definition can also be used for complex functions. In the multidimensional case, the term of the total variation of the weak derivative is defined.

Related to rectifiable paths

A function can also be interpreted as a path in the metric space. It is considered that if and is of bounded variation if a rectifiable path is, so it has a finite length.

Related to measure theory

In measure theory, the reell-/komplexwertigen functions of bounded variation are accurate to the distribution functions of signed / complex Borel measure.