The Wikipedia entry on Weierstrass Transform says "The generalized Weierstrass transform provides a means to approximate a given integrable function f arbitrarily well with analytic functions."
But it doesn't cite any references nor goes deep on that. How are the Weierstrass Transform and analytic functions related? How can I use it to approximate functions? Is there any multidimensional version of the Weierstrass Transform? References would be valuable.
If $f$ is $L^1(\Bbb{R}^n)$ then $f_k = f \ast k^n e^{-\pi k^2 |x|^2}$ is entire and it converges to $f$ in $L^1(\Bbb{R}^n)$.
It follows from the density of $C^0_c$ in $L^1$ which itself follows from the definition of the Lebesgue measurable sets, as if $f\in C^0_c$ then the proof is easy.
It stays true when replacing the convergence in $L^1$ by $L^p,p <\infty$, $C^k \cap L^\infty$, the Sobolev norms, the Schwartz topology, the tempered distributions topology...