I need some help with proving that function $f: \mathbb{R} \to \mathbb C$, $f(t)=(\cos(t))^2$ is a positive-definite function.
I know that if $\sum_{k,l\le n}(f(t_k-t_l)z_k\overline z_l)\ge0$ then $f$ is positive-definite, but I don't know how to show this inequality for my function. Does someone know any easier way to prove it?
For every $a\in\mathbb R$, the complex exponential function $e^{iat}$ is positive-definite because $$ \sum_{k,l} e^{ia(t_k-t_l)}z_k\bar z_l = \left|\sum_{k} e^{iat_k} z_k\right|^2 \ge 0 $$ And $\cos^2 t$ can be written as a combination of complex exponentials with positive coefficients: $$ \cos^2 t = \frac12+\frac12 \cos 2t = \frac12+\frac14e^{2it}+\frac14e^{-2it} $$
The above is a special case of the fact that the Fourier transform of a positive measure is a positive-definite function; the converse is also true (Bochner's theorem).