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

Question

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 $H$?

Answer 1

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 $H$.