Characteristic function

In mathematics, the term "characteristic function" can refer to any of several distinct concepts:

• The indicator function of a subset, that is the function

${\displaystyle \mathbf {1} _{A}\colon X\to \{0,1\},}$

which for a given subset A of X, has value 1 at points of A and 0 at points of X − A.

• There is an indicator function for affine varieties over a finite field:[1] given a finite set of functions ${\displaystyle f_{\alpha }\in \mathbb {F} _{q}[x_{1},\ldots ,x_{n}]}$ let ${\displaystyle V=\{x\in \mathbb {F} _{q}^{n}:f_{\alpha }(x)=0\}}$ be their vanishing locus. Then, the function ${\displaystyle P(x)=\prod (1-f_{\alpha }(x)^{q-1})}$ acts as an indicator function for ${\displaystyle V}$. If ${\displaystyle x\in V}$ then ${\displaystyle P(x)=1}$, otherwise, for some ${\displaystyle f_{\alpha }}$, we have ${\displaystyle f_{\alpha }(x)\neq 0}$, which implies that ${\displaystyle f_{\alpha }(x)^{q-1}=1}$, hence ${\displaystyle P(x)=0}$.
• The characteristic function in convex analysis, closely related to the indicator function of a set:

${\displaystyle \chi _{A}(x):={\begin{cases}0,&x\in A;\\+\infty ,&x\not \in A.\end{cases}}}$

• In probability theory, the characteristic function of any probability distribution on the real line is given by the following formula, where X is any random variable with the distribution in question:

${\displaystyle \varphi _{X}(t)=\operatorname {E} \left(e^{itX}\right),}$

where ${\displaystyle \operatorname {E} }$ denotes expected value. For multivariate distributions, the product tX is replaced by a scalar product of vectors.

• The characteristic function of a cooperative game in game theory.
• The characteristic polynomial in linear algebra.
• The characteristic state function in statistical mechanics.
• The Euler characteristic, a topological invariant.
• The receiver operating characteristic in statistical decision theory.
• The point characteristic function in statistics.

References

1. Serre. Course in Arithmetic. p. 5.