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 $\lambda\_{min}(G\_{\gamma}) = \Theta(\frac{1}{\gamma})$



Bounding the smallest eigenvalue of a matrix generated by a positive definite function
Given any $n$ distinct points $\{x\_i/x\_i\in(0,1)\}$ which are pairwise distinct. For any $c\_i,i = 1,2,3,...n$, and not all zeros.


Using the given expression for $g(x)$ we can deduce that


$$\sum\_{i=1}^n\sum\_{j=1}^nc\_ic\_jg(x\_i-x\_j) = \sum\_{\eta\in\mathbb{Z}} \left(\frac{1}{1+\gamma\eta^2} \left|\sum\_{i=1}^n c\_i e^{2\pi i \eta x\_i}\right|^2 \right)> 0$$


as $\sum\_{i=1}^n c\_i e^{2\pi i \eta x\_i}$ does not vanish simultaneously for all $\eta\in\mathbb{Z}$ and $\frac{1}{1+\gamma\eta^2}>0\forall \eta \in\mathbb{Z}$.


Hence the matrix $G\_{\gamma}$ is positive definite.


**Estimate on $\lambda\_{min}(G\_{\gamma})$ as $\gamma\to\infty$**


Let $c = [c\_1,c\_2,...c\_n]$ be such that $\|c\|\_2 = 1$. Then $$c^TG\_{\gamma}c = \sum\_{\eta\in\mathbb{Z}} \left(\frac{1}{1+\gamma\eta^2} \left|\sum\_{i=1}^n c\_i e^{2\pi i \eta x\_i}\right|^2 \right)> 0 .$$ As $|\sum\_{i=1}^n c\_i e^{2\pi i \eta x\_i}|^2 \le n$ and we already know $\sum\_{i=1}^n c\_i e^{2\pi i \eta x\_i}$ does not vanish simultaneously for all $\eta\in\mathbb{Z}$, there exists constants $K\_1$ and $K\_2$ such that $$\frac{K\_1}{\gamma} \le c^TG\_{\gamma}c \le \frac{K\_2}{\gamma} \mbox{ }\forall c\in\mathbb{R}^m\ \setminus\{0\}^m \mbox{ and } \|c\|\_2 = 1$$


Let $e(\gamma)$ be the smallest eigenvector and as $\|e(\gamma)\|\_2 = 1$ $$\lambda\_{min}(G\_{\gamma}) = \lambda\_{min}(G\_{\gamma})e(\gamma)^Te(\gamma) = e(\gamma)^TG\_{\gamma}e(\gamma) = \Theta(\frac{1}{\gamma})$$ So


$$\lambda\_{min}(G\_{\gamma}) = \Theta(\frac{1}{\gamma})$$


