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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 $\lambda_{min}(G_{\gamma}) = \Theta(\frac{1}{\gamma})$

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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

$$\lambda_{min}(G_{\gamma}) = \Theta(\frac{1}{\gamma})$$

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  • $\begingroup$ what is $c_i, c_j$ in the sum $\sum_{j=1}^n\sum_{i+1}^nc_ic_jg(x_i-x_j)$? and how did you get the RHS of the sum? Could you eleaborate? $\endgroup$ – vidyarthi Jul 20 '20 at 16:19
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    $\begingroup$ @vidyarthi : any $c = [c_1,c_2,...c_n] \in \mathbb{R}^n \setminus \{0\}^n$. I will expand the summation later when I get time. Its just an interchange of order of summations and an algebraic manipulation. $\endgroup$ – Rajesh D Jul 20 '20 at 16:22
  • $\begingroup$ also write $2\cos{\theta} = e^{2\pi i\theta} + e^{-2\pi i\theta}$ $\endgroup$ – Rajesh D Jul 20 '20 at 17:12
  • $\begingroup$ yes, that is clear, but still, how $$\sum_{i=1}^n\sum_{j=1}^nc_ic_jg(x_i-x_j)=\sum_{i=1}^n\sum_{j=1}^nc_ic_j\sum_{\eta\in\mathbb{Z}}\frac1{1+\gamma\eta^2}cos(x_i-x_j)=2\left(\sum_{\eta\in\mathbb{Z}}\frac1{1+\gamma\eta^2}\left|\sum_{i=1}^nc_ie^{2\pi i\eta x_i}\right|^2\right)$$? Do you have all $c_i,c_j$ positive? $\endgroup$ – vidyarthi Jul 20 '20 at 17:26
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    $\begingroup$ @vidyarthi no need for positive. You need to work a bit with pen and paper. Hint : $\sum_{i=1}^3\sum_{j=1}^3 c_ic_j = c_1^2 + c_2^2 + c_3^2 + 2c_1c_2 + 2c_2c_3 + 2c_3c_1 = (c_1 + c_2 + c_3)^2$ $\endgroup$ – Rajesh D Jul 20 '20 at 17:33

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