# Spanning set for $L^2(S^2)$?

It is well know that the spherical harmonics form a basis for $$H = L^2_{\mathbb{C}}(S^2)$$, the square integrable, complex-valued functions on the 2-sphere. My question is if the functions $$f_v(x) = e^{-i v \cdot x }$$ span $$H$$ where $$v \in \mathbb{R}^3$$ and $$x \in S^2$$ is viewed as a point in $$\mathbb{R}^3$$. "Span" here means that any function in $$H$$ can be written as a linear combination of the $$f_v$$. Presumably $$f_v$$ aren't a basis since $$H$$ is separable, but if the $$f_v$$ do span $$H$$ then is there some subset of them that would be a basis for $$H$$?

Assuming you mean that their span (finite linear combinations of the $$f_v$$) is dense in $$H$$ (i.e. any element can be written as an infinite linear combination of the $$f_v$$), this will be true.
One way to see this is to see that the span of the $$f_v$$ is dense in the space of continuous functions on the sphere w.r.t. to the uniform norm $$\|f\|_{\infty}=\sup_{x\in \mathbb{S}^2}|f(x)|$$ (this follows from Stone-Weierstrass).
Then, you can use that the continuous functions are dense in $$H$$ and that $$\|f\|_{L^2}\leq \|f\|_{\infty} Vol(\mathbb{S}^2)$$ to get that the span of the $$f_v$$ must be dense in $$H$$.