Motivation: Euler-Maclaurin formula uses calculus to estimate discrete sums. I wonder what one can do by reverse engineering. A concrete problem I ran into is the following.
Question: Let $\chi: \mathbb{R} \to \mathbb{R}$ be the characteristic function supported on $[0,1]$. Can one construct a sequence of polynomial or analytic functions $f_{n}: \mathbb{R} \to \mathbb{R}$ such that $f_{n} \to \chi$ in $L^p$-norm for some $p \in \mathbb{N}$?
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 $L^2(\mathbb R)$. 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?
