Spose I have a $ h \in L^2(U) $ where $ U \subset \mathbb{R}^3 $ is open and bounded. Is it possible to approximate this by analytic functions? If so, spose now we take $ U = \mathbb{R}^3 $. Is this still possible and if so would it mean that each of your approximating analytic functions has some decay so as to make it square integrable over the whole 3 space? I think it must do if they are approximations.
Thanks!
I will focus on the one-dimensional case $U \subset \mathbb{R}$ first. If $U$ is an interval like $[0,2\pi]$ (for $L^2$ alone it does not matter if the domain is open or closed because it does not really enter the definition), then the theorem of Stone-Weierstrass is used to prove that any function $h \in L^2(U)$ can be approximated to arbitrary precision measured by the $L^2$-norm by a Fourier series which (if cut off at some high enough index) is clearly analytic and can also be real-analytic with Fourier Sine/Cosine series. One says the linear span of the Fourier basis set $\{ \exp(ikx) \}_{k \in \mathbb{Z}}$ is dense in $L^2(U)$.
In the unbounded case $U=\mathbb{R}$ the decrease of $L^2$-functions for $|x| \rightarrow \infty$ has to be taken into account. A classical orthonormal basis for the weighted space $L^2(\mathbb{R},\exp(-x^2))$ are the Hermite polynomials $H_n$. But the weight is itself analytic and so linear combinations of $\{ H_n \exp(-x^2) \}$ can be used to approximate any $L^2(\mathbb{R})$ function to arbitrary precision.
The multidimensional case should work out similarily at least for square domains $U$ with multidimensional Fourier series and Hermite polynomials of many variables respectively.