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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 $L^p$.

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    I have intuition that in your case it is true that the set is dense, although I have tried attempt at a proof and some problems argue in the opposite direction ; I don't think this is true in the general case.2011-05-18
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    It is definitely true in the specific case, that part was homework (last week's).2011-05-18
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    Yeah, I hope so, but your specific case looks "very nice".2011-06-02

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