Let $g:\mathbb{T}\to\mathbb{R}$ and is given as $$g(x) = \sum\limits_{\eta\in\mathbb{Z}}\frac{1}{1+\gamma \eta^2}\cos{2\pi\eta x}$$

Consider the matrix $$G_{\gamma} = [g(x_i-x_j)]_{1\le i,j\le n}$$

where $x_1,x_2,...x_n \in (0,1)$ and pairwise distinct.

Due to Bochner's theorem, $g(x)$ is a positive semi definite function and hence $G_{\gamma}$ is a psd matrix.

Let $\lambda_{min}(G_{\gamma})$ denote the smallest eigenvalue of $G_{\gamma}$.

I want to show that $G_{\gamma}$ is infact positive definite and as $\gamma\to\infty$ $$\lambda_{min}(G_{\gamma}) = \Theta(\frac{1}{\gamma})$$

As a sanity check, I have verified this using numerical computations on some examples.

**Motivation** : I want to come up with a similar looking formula in a generic dimension, that is for $\mathbb{T}^m$. If I am able to prove this for $m=1$ dimension, then I will understand the mechanics of it so that might help me to come up with a $g:\mathbb{T}^m \to \mathbb{R}$ such that $\lambda_{min}(G_{\gamma}) = \Theta(\frac{1}{\gamma})$