I want to know if $\{\frac{(1-\cos \alpha x)} {x^2}\}_{\alpha>0}$ is dense in $C_0(\mathbb{R})=\{f\in C(\mathbb{R})|\lim_{|x|\to\infty}f(x)=0\}$? That is, for any $f\in C_0(\mathbb{R})$ and $\epsilon >0,$ there is some $\alpha_1>0,\alpha_2>0,\cdots,\alpha_n>0$ and $a_1,a_2,\cdots,a_n\in \mathbb{R}$ such that $$\max_{x\in\mathbb{R}}|f(x)-\sum_{k=1}^na_k\frac{(1-\cos \alpha_k x)} {x^2}|<\epsilon.$$

I want to know if $\left\{\frac{(1-\cos \alpha x)} {x^2}\right\}_{\alpha>0}$ is dense in $C_0(\mathbb{R})=\{f\in C(\mathbb{R})\mid \lim_{|x|\to\infty}f(x)=0\}$? That is, for any $f\in C_0(\mathbb{R})$ and $\varepsilon >0,$ there is some $\alpha_1>0,\alpha_2>0,\cdots,\alpha_n>0$ and $a_1,a_2,\cdots,a_n\in \mathbb{R}$ such that $$\max_{x\in\mathbb{R}} \left| f(x)-\sum_{k=1}^na_k\frac{(1-\cos \alpha_k x)} {x^2} \right| <\varepsilon.$$

I want to know if $\{\frac{(1-\cos \alpha x)} {x^2}\}_{\alpha\in\mathbb{R}}$ is dense in $C_0(\mathbb{R})=\{f\in C(\mathbb{R})|\lim_{|x|\to\infty}f(x)=0\}$? That is, for any $f\in C_0(\mathbb{R})$ and $\epsilon >0,$ there is some $\alpha_1,\alpha_2,\cdots,\alpha_n$ and $a_1,a_2,\cdots,a_n$ such that $$\max_{x\in\mathbb{R}}|f(x)-\sum_{k=1}^na_k\frac{(1-\cos \alpha_k x)} {x^2}|<\epsilon.$$

I want to know if $\{\frac{(1-\cos \alpha x)} {x^2}\}_{\alpha>0}$ is dense in $C_0(\mathbb{R})=\{f\in C(\mathbb{R})|\lim_{|x|\to\infty}f(x)=0\}$? That is, for any $f\in C_0(\mathbb{R})$ and $\epsilon >0,$ there is some $\alpha_1>0,\alpha_2>0,\cdots,\alpha_n>0$ and $a_1,a_2,\cdots,a_n\in \mathbb{R}$ such that $$\max_{x\in\mathbb{R}}|f(x)-\sum_{k=1}^na_k\frac{(1-\cos \alpha_k x)} {x^2}|<\epsilon.$$