Limited variation

Examples of functions of unlimited variation

Examples of functions of limited variation

In analysis , a function is of limited variation ( limited fluctuation ) if its total variation ( total fluctuation ) is finite, that is, if it does not oscillate indefinitely in a certain way. These terms are closely related to the continuity and the integrability of functions.

The space of all functions of limited variation in the field is denoted by. ${\displaystyle \Omega }$ ${\displaystyle BV(\Omega )}$

Real functions

definition

The total variation of a real-valued function defined on a closed interval is the supremum ${\displaystyle f\colon [a,b]\to \mathbb {R} }$

${\displaystyle \sup _{P}\sum _{i}|f(x_{i+1})-f(x_{i})|,}$

where this supremum is formed over all possible partitions of the interval . The one given here depends on. ${\displaystyle P=\{x_{1},\dotsc ,x_{n}\mid x_{1}<\dotsb <x_{n}\}}$ ${\displaystyle [a,b]}$ ${\displaystyle n}$ ${\displaystyle P}$

Riemann-Stieltjes integrates precisely the continuous functions of limited variation . Therefore, a semi-standard can be used: ${\displaystyle BV[a,b]}$

${\displaystyle n(f)=\sup _{\varphi }\int f\,{\frac {\mathrm {d} \varphi }{\mathrm {d} x}}}$.

This supremum is formed over all functions with a compact carrier and function values ​​in the interval . ${\displaystyle \mathrm {\varphi } }$ ${\displaystyle [-1,1]}$

The semi-norm is consistent with the supremum that defines the restricted variation.

example

Example of unlimited variation

A simple example of an unbounded variation is the function near . It is clear that the value of the quotient for will grow faster and faster towards ∞ as it approaches 0, and that the sine of this value will run through an infinite number of oscillations. This is shown in the picture on the right. ${\displaystyle \textstyle y=\sin({\frac {1}{x}})}$ ${\displaystyle x=0}$ ${\displaystyle {\tfrac {1}{x}}}$ ${\displaystyle x\to 0}$

The function

${\displaystyle f(x)={\begin{cases}0,&{\text{falls }}x=0\\x\sin(1/x),&{\text{falls }}x\neq 0\end{cases}}}$

is also not of limited fluctuation in the interval [0, 1], in contrast to the function:

${\displaystyle g(x)={\begin{cases}0,&{\text{falls }}x=0\\x^{2}\sin(1/x),&{\text{falls }}x\neq 0\end{cases}}}$.

Here the variation of the sine term, which increases for strongly, is dampened enough by the additional power. ${\displaystyle x\to 0}$

Extensions

This definition can also be used for complex-valued functions .

BV functions in several variables

Functions of limited variation, or functions, are functions whose distributional derivatives are finite vector-valued Radon measures . More accurate: ${\displaystyle BV}$

definition

Let be an open subset of . A function is of limited variation or element of if its distributional derivative is a finite, signed, vector-valued Radon measure. That is, it exists such that ${\displaystyle \Omega }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle u\in L^{1}(\Omega )}$ ${\displaystyle BV(\Omega )}$ ${\displaystyle Du\in {\mathcal {M}}(\Omega ,\mathbb {R} ^{n})}$

${\displaystyle \int _{\Omega }u(x)\,\operatorname {div} {\varphi }(x)\mathrm {d} x=-\int _{\Omega }\langle \varphi ,Du(x)\rangle \qquad {\text{für alle }}{\varphi }\in C_{c}^{1}(\Omega ,\mathbb {R} ^{n})}$

applies.

Connection with rectifiable ways

A continuous function can also be understood as a path in metric space . It holds that is of limited variation if and only if there is a rectifiable path , i.e. if it has a finite length. ${\displaystyle f\colon [a,b]\to \mathbb {R} }$ ${\displaystyle \mathbb {R} }$ ${\displaystyle f}$ ${\displaystyle f}$

Connection with the theory of measure

In the measure theory , the real- / complex-valued functions of bounded variation are exactly the distribution functions of signed / complex Borel dimensions on . ${\displaystyle \mathbb {R} }$

literature

• Jürgen Elstrodt : Measure and integration theory . 5th edition. Springer, Berlin 2007, ISBN 978-3-540-49977-0 .
• Gerald Teschl : Topics in Real and Functional Analysis . 2011 ( free online version ).
• Luigi Ambrosio , Nicola Fusco and Diego Pallara : Functions of Bounded Variation and Free Discontinuity Problems . Oxford 2000.