TITLE: Positive-definite function and Positive-definite matrix
QUESTION [5 upvotes]: I am trying to understand Positive-definite function and read the wikipedia link:
https://en.wikipedia.org/wiki/Positive-definite_function
It has a relation to Positive-definite matrix and I did not understand how clearly. 
My questions: 

is the Positive-definite function $f$ mentioned in the wikipedia link is continuous? 
It is related to to the element of a Positive-definite matrix which
are just numbers what it signifies?  
Can anyone please clarify?

REPLY [2 votes]: The key observation in characterizing the Fourier transform of a positive function $f$ is this:
\begin{align}
 \sum_{j,k=1}^{N}\hat{f}(s_j-s_k)a_j \overline{a_k} 
   & = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)\sum_{j,k=1}^{N}e^{i(s_j-s_k)t}a_j \overline{a_k} dt \\
   & = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)\left|\sum_{j=1}^{N}e^{is_jt}a_j\right|^{2}dt \ge 0.
\end{align}
So it is easy to generate positive matrices of any order from the Fourier transform of a positive function. Conversely, if $g(s)$ is a continuous (bounded?) function on $\mathbb{R}$ that is positive in the sense that
$$
             \sum_{j,k=1}^{N}g(s_j-s_k)a_j\overline{a_k} \ge 0
$$
for all points $\{s_1,s_2,\cdots,s_N\}$ and complex numbers $\{a_1,a_2,\cdots,a_N\}$, then $g$ is the Fourier transform of a positive measure $\mu$:
$$
             g(s) = \int_{-\infty}^{\infty}e^{-ist}d\mu(t).
$$
A proof of Bochner's Theorem is obtained by a clever application of Functional Analysis.