Is the DTFT of a sampled Gaussian a positive function?

Question

I have an infinite sequence $x_{n}$ for $n \in \mathcal{Z}$ which is a sampled Gaussian function $x_{n} = \exp(-n^2/a)$ with a > 0. I need to check whether its DTFT $x(\theta) = \sum_{n \in \mathcal{Z}} x_{n} \exp(-i n \theta)$ is positive for all $\theta$, or sufficiently for all $\theta \in [0, 2 \pi)$ since it is periodic. This function is real and can be expressed $$x(\theta) = 1 + 2\sum_{n = 1}^{\infty} \exp(-n^2/a) \cos(n \theta)$$ due to the symmetry of the Gaussian function. I suspect that it is positive but I'm not sure. Can anyone help me prove or disprove it? Thanks!

Answer 1

Let $y(t)$ be a continuos gaussian, let $u(t)=\sum \delta(t-Tk)$ be a train of pulses (Dirac comb). The values of your discrete function correspond to the product $x(t)=y(t) u(t)$, and the Fourier transform is the convolution - $$ X(w) = Y(w) \star U(W) =\sum_k Y(w-k/T) $$

But $Y(w)$ (Fourier transform of a centered Gaussian) is another Gaussian, hence positive. Hence, $X(w)$ is positive.