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 $L^2(\mathbb R)$. So we get approximations for all elements of $L^2$, not merely for $\chi$.  The approximations for $\chi$ are:
$$
G_n:=\sum_{k=0}^n \langle \chi,\psi_k\rangle \psi_k ,
$$
so $G_n$ is a polynomial of degree $n$ times $e^{-x^2/2}$.

----


Do we have convergence of $G_n$ to $\chi$ in other $L^p$ as well?


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