Suppose $\mathcal{A}\subset L^p(\mathbb{R})$ is an algebra of functions with the following property:
For every compact $K\subset\mathbb{R}$, $\mathcal{A}$ is dense in $\mathcal{C}(K)$ with respect to the uniform norm $\|\cdot\|_{\infty}$, where $\mathcal{C}(K)$ is the collection of real continuous functions on $K$. The uniform norm I'm referring to is $$\|f\|_{\infty}=\sup_{t\in K}|f(t)|.$$ (See http://en.wikipedia.org/wiki/Uniform_norm.)
Can we conclude that $\mathcal{A}$ is dense in $L^p(\mathbb{R})$ (with respect to the $L^p$ norm)?
I became interested in this question while investigating a special case, the $L^2$-density of finite linear combinations of Gaussians:
$$\sum_{i=1}^n\alpha_ie^{-k_i(x-x_i)^2},\qquad\alpha_i,k_i,x_i\in\mathbb{R},k_i>0.$$
The question above occurred to me because I can imagine it being useful in cases like this to, say, verify the hypotheses of the Stone-Weierstrass theorem for a given family of functions rather than to explicitly approximate functions in $L^p$.