In mathematics, a positive-definite function is, depending on the context, either of two types of function.

Definition 1

Let ${\displaystyle \mathbb {R} }$ be the set of real numbers and ${\displaystyle \mathbb {C} }$ be the set of complex numbers.

A function ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$ is called positive semi-definite if for any^{[clarification needed]} real numbers x₁, …, x_n the n × n matrix

${\displaystyle A=\left(a_{ij}\right)_{i,j=1}^{n}~,\quad a_{ij}=f(x_{i}-x_{j})}$

is a positive semi-definite matrix.^{[citation needed]}

By definition, a positive semi-definite matrix, such as ${\displaystyle A}$, is Hermitian; therefore f(−x) is the complex conjugate of f(x)).

In particular, it is necessary (but not sufficient) that

${\displaystyle f(0)\geq 0~,\quad |f(x)|\leq f(0)}$

(these inequalities follow from the condition for n = 1, 2.)

A function is negative semi-definite if the inequality is reversed. A function is definite if the weak inequality is replaced with a strong (<, > 0).

Examples

If ${\displaystyle (X,\langle \cdot ,\cdot \rangle )}$ is a real inner product space, then ${\displaystyle g_{y}\colon X\to \mathbb {C} }$, ${\displaystyle x\mapsto \exp(i\langle y,x\rangle )}$ is positive definite for every ${\displaystyle y\in X}$: for all ${\displaystyle u\in \mathbb {C} ^{n}}$ and all ${\displaystyle x_{1},\ldots ,x_{n}}$ we have

${\displaystyle u^{*}A^{(g_{y})}u=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}e^{i\langle y,x_{k}-x_{j}\rangle }=\sum _{k=1}^{n}{\overline {u_{k}}}e^{i\langle y,x_{k}\rangle }\sum _{j=1}^{n}u_{j}e^{-i\langle y,x_{j}\rangle }=\left|\sum _{j=1}^{n}{\overline {u_{j}}}e^{i\langle y,x_{j}\rangle }\right|^{2}\geq 0.}$

As nonnegative linear combinations of positive definite functions are again positive definite, the cosine function is positive definite as a nonnegative linear combination of the above functions:

${\displaystyle \cos(x)={\frac {1}{2}}(e^{ix}+e^{-ix})={\frac {1}{2}}(g_{1}+g_{-1}).}$

One can create a positive definite function ${\displaystyle f\colon X\to \mathbb {C} }$ easily from positive definite function ${\displaystyle f\colon \mathbb {R} \to \mathbb {C} }$ for any vector space ${\displaystyle X}$: choose a linear function ${\displaystyle \phi \colon X\to \mathbb {R} }$ and define ${\displaystyle f^{*}:=f\circ \phi }$. Then

${\displaystyle u^{*}A^{(f^{*})}u=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}f^{*}(x_{k}-x_{j})=\sum _{j,k=1}^{n}{\overline {u_{k}}}u_{j}f(\phi (x_{k})-\phi (x_{j}))=u^{*}{\tilde {A}}^{(f)}u\geq 0,}$

where ${\displaystyle {}^{}}$Zdroj:https://en.wikipedia.org?pojem=Positive-definite_function
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