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.
and the formation of the difference in the -th component as
.
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 :
.
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.
properties
The difference operator is linear, that is, it holds
Furthermore is
Also applies to
The formation of the difference between the components is therefore interchangeable.
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 .