Additive function
Additive , subadditive, and superadditive functions are mathematical objects. There are certain classes of functions . Linear maps are special additive functions.
definition
A function is called additive if it has the functional equation
Fulfills. If the definition and target area are Abelian groups , one also speaks of - linearity .
Sub and super additive functions
If there is a semigroup with the link , then a mapping is called subadditive if the following applies to all and from :
- .
The figure is called superadditive if the following applies to all and off :
- .
Examples
- According to the triangle inequality , norms and amounts are always subadditive.
- Sublinear functions are subadditive.
- Linear maps are additive.
properties
- A mapping is additive if and only if it is both subadditive and superadditive.
- If it is an additive function, then for every finite number of elements :
- The same applies to sub- and superadditivity.
Definition in number theory
With number theoretic functions one regards a link to the multiplication. A number theoretic function is called additive if the equation
for all coprime and applies. If this even applies to all and , the function is called strictly additive .
There is a similar restriction of additivity (to disjoint instead of arbitrary unions) in measure theory.
See also
Individual evidence
- ↑ Prasanna Sahoo, Thomas Riedel: Mean Value Theorem and Functional Equations . 1998, ISBN 978-981-02-3544-4 , pp. 1 .
- ↑ ^{a } ^{b} Josip E. Peajcariaac, YL Tong: Convex functions, partial orderings, and Statistical Applications . Academic Press, 1992, ISBN 978-0-12-549250-8 , pp. 8 .