Difference operator

In mathematics, a difference operator is an operator that is used to generalize the difference of a function in several variables. This allows properties such as the monotony of a real function of a variable to be generalized to functions of several variables. Another area of ​​application of difference operators is stochastics and measure theory , where abstract volume concepts are defined with their help.

definition

A real-valued function of several real variables is given

${\displaystyle F:\mathbb {R} ^{n}\to \mathbb {R} }$

Then the difference operator for is defined as ${\displaystyle a^{1}=(a_{1}^{1},\dots ,a_{n}^{1}),a^{2}=(a_{1}^{2},\dots ,a_{n}^{2})}$

${\displaystyle \Delta _{a^{1}}^{a^{2}}F:=\sum _{i_{1},\dots ,i_{n}\in \{1,2\}}(-1)^{i_{1}+\dots +i_{n}}F(a_{1}^{i_{1}},\dots ,a_{n}^{i_{n}})}$

and the formation of the difference in the -th component as ${\displaystyle \nu }$

${\displaystyle {}_{\nu }\Delta _{\alpha }^{\beta }F:=F(x_{1},\dots ,x_{\nu -1},\beta ,x_{\nu +1},\dots ,x_{n})-F(x_{1},\dots ,x_{\nu -1},\alpha ,x_{\nu +1},\dots ,x_{n})}$.

Explanation

By replacement of the individual components of the two vectors is a cuboid in with generated corners. The function values ​​at these corners are then given a sign depending on the original vector of the components and then added, for example for : ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle 2^{n}}$ ${\displaystyle n=2}$

${\displaystyle \Delta _{a^{1}}^{a^{2}}F=F(a_{1}^{2},a_{2}^{2})-F(a_{1}^{1},a_{2}^{2})-F(a_{1}^{2},a_{2}^{1})+F(a_{1}^{1},a_{2}^{1})}$.

The formation of the difference in the -th component is constant in the -th entry, but is mostly still understood as a function to enable the further use of difference operators. ${\displaystyle \nu }$ ${\displaystyle \nu }$ ${\displaystyle \mathbb {R} ^{n}}$

properties

The difference operator is linear, that is, it holds

${\displaystyle \Delta _{a^{1}}^{a^{2}}(F+G)=\Delta _{a^{1}}^{a^{2}}F+\Delta _{a^{1}}^{a^{2}}G}$

Furthermore is

${\displaystyle \Delta _{a^{1}}^{a^{2}}F=\sum _{i_{1}=1}^{2}(-1)^{i_{1}}\cdot \sum _{i_{2}=1}^{2}(-1)^{i_{2}}\cdot \dots \cdot \sum _{i_{n}=1}^{2}(-1)^{i_{n}}F(a_{1}^{i_{1}},\dots ,a_{n}^{i_{n}})={}_{1}\Delta _{a_{1}^{1}}^{a_{1}^{2}}\cdots {}_{n}\Delta _{a_{n}^{1}}^{a_{n}^{2}}F}$

Also applies to ${\displaystyle \mu \neq \nu }$

${\displaystyle {}_{\mu }\Delta _{\alpha }^{\beta }{}_{\nu }\Delta _{\alpha }^{\beta }F={}_{\nu }\Delta _{\alpha }^{\beta }{}_{\mu }\Delta _{\alpha }^{\beta }F}$

The formation of the difference between the components is therefore interchangeable.

use

Using the difference operator, for example, the monotony of a function can be generalized: A function is called rectangular monotonic if ${\displaystyle F\colon \mathbb {R} ^{n}\to \mathbb {R} }$

${\displaystyle a\leq b\implies \Delta _{a}^{b}F\geq 0}$

applies. This is to be understood as components, i.e. for all indices. Based on this, such functions can then be examined further. ${\displaystyle a\leq b}$ ${\displaystyle a\leq b\iff a_{i}\leq b_{i}}$

In addition, difference operators are in the measure theory and the stochastic for defining dimensions on the means of multivariate distribution functions used. ${\displaystyle \mathbb {R} ^{n}}$

literature

• Jürgen Elstrodt : Measure and integration theory . 6th, corrected edition. Springer-Verlag, Berlin Heidelberg 2009, ISBN 978-3-540-89727-9 , doi : 10.1007 / 978-3-540-89728-6 .
• Norbert Kusolitsch: Measure and probability theory . An introduction. 2nd, revised and expanded edition. Springer-Verlag, Berlin Heidelberg 2014, ISBN 978-3-642-45386-1 , p. 65-72 , doi : 10.1007 / 978-3-642-45387-8 .