Positive definite function
Graduate
Consider a function $V : D \to \reals$ whose domain $D$ contains $0$. The function $V$ is positive definite if $V(0) = 0$ and $V(x) > 0$ for all $x \neq 0$ in $D$.
Examples
Examples
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$f(x) = x^2$ on $\reals$
High school
$x \mapsto x^2$ on $\reals$ is positive definite because it is positive away from $0$ and zero at $0$.
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$r \mapsto \arctan r$ on $[0, \infty)$
Graduate
$r \mapsto \arctan r$ is of class $\mathcal{K}$ because it is continuous, increasing, and zero at $0$, and it is positive definite. It is not of class $\mathcal{K}_{\infty}$, because it is bounded by $\pi / 2$.
Notes
A symmetric matrix $P$ is positive definite if $x^{\top} P x > 0$ for all $x \neq 0$, which says exactly that the function $V(x) = x^{\top} P x$ is positive definite. Positive definite functions in harmonic analysis (Bochner’s theorem) are something else.