Dense subset for $C_0(\mathbb{R})$

Question

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 $$\max_{x\in\mathbb{R}} \left| f(x)-\sum_{k=1}^na_k\frac{(1-\cos \alpha_k x)} {x^2} \right| <\varepsilon.$$

Answer 1

The functions $f_a$ considered are (up to a multiplicative constant) the Fourier transforms of even tent functions. The real linear combinations of even tent functions yield all even continuous and piecewise affine real functions with compact support, which are dense in the even integrable real functions for the $L^1$-norm. Hence, the vector space spanned by the functions $f_a$ is dense in the subspace of all even real functions of $\mathcal{F}L^1$, which is dense in subset of all even real functions $\mathcal{C}_0$.