Not polynomials. Polyomials of degree $\ge 1$ cannot belong to $L^p(\mathbb R)$.
For analytic functions... how about using the Hermite functions? They look like polynomial times exponential, so they are analytic.
The Hermite functions are an orthonormal basis for $L^2(\mathbb R)$.
Do we have convergence of the expansion in other $L^p$ also?