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?
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.