Not polynomials.  Polyomials of degree $\ge 1$ cannot belong to $L^p(\mathbb R)$.  

For analytic functions... how about using the [Hermite functions][1]?  They look like polynomial times exponential, so they are analytic.

>[![from Wikipedia][2]][2]

The Hermite functions are an orthonormal basis for $L^2(\mathbb R)$.  

Do we have convergence of the expansion in other $L^p$ also?


  [1]: https://en.wikipedia.org/wiki/Hermite_polynomials#Hermite_functions
  [2]: https://i.sstatic.net/YM2eE.jpg