Coulomb gauge fixing and "normalizability" The Setup
Let Greek indices be summed over $0,1,\dots, d$ and Latin indices over $1,2,\dots, d$.  Consider a vector potential $A_\mu$ on $\mathbb R^{d,1}$ defined to gauge transform as
$$
  A_\mu\to A_\mu'=A_\mu+\partial_\mu\theta
$$
for some real-valued function $\theta$ on $\mathbb R^{d,1}$.  The usual claim about Coulomb gauge fixing is that the condition
$$
  \partial^i A_i = 0
$$
serves to fix the gauge in the sense that $\partial^iA_i' = 0$ only if $\theta = 0$.  The usual argument for this (as far as I am aware) is that $\partial^i A'_i =\partial^iA_i + \partial^i\partial_i\theta$, so the Coulomb gauge conditions on $A_\mu$ and $A_\mu'$ give $\partial^i\partial_i\theta=0$, but the only sufficiently smooth, normalizable (Lesbegue-integrable?) solution to this (Laplace's) equation on $\mathbb R^d$ is $\theta(t,\vec x)=0$ for all $\vec x\in\mathbb R^d$.
My Question
What, if any, is the physical justification for the smoothness and normalizability constraints on the gauge function $\theta$?
EDIT 01/26/2013
Motivated by some of the comments, I'd like to add the following question:  are there physically interesting examples in which the gauge function $\theta$ fails to be smooth and/or normalizable?  References with more details would be appreciated.  Lubos mentioned that perhaps monopoles or solitons could be involved in such cases; I'd like to know more!
Cheers!
 A: A quick answer, if I may. 
You need $\theta$ to be smooth since you want to derive it. So mathematics imposes you to choose $\theta$ smooth. 
Now the trick: choosing $\theta$ to be smooth means you can always impose $\mathbf{A}$ to be smooth, and use several patches related to each other by a gauge transform. Then you should always discuss smooth vector potential... do you ? Well, you should, if you want to make proper math. But physicists usually don't care about that, and choose a singular vector potential to prove that the field configuration hosts a monopole. The prototype example is the vortex associated with the U(1) Lie group / algebra. See for instance the paper by Dirac: 

Dirac, P. A. M. Quantised Singularities in the Electromagnetic Field. Proc. R. Soc. London. Ser. A 133, 60–72 (1931)

where the vector potential is singular at the north or south pole. Note that the theory of connection on fiber bundle was still to be discovered at that time ! The correct mathematical picture came late in physics, as far as I know at least. Here a beautiful reading

Wu, T. T. & Yang, C. N. Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D 12, 3845–3857 (1975)

where they choose two parameterisation of the circle: one for the south and one for the north pole, these two parameterisations of the vector potential being related to the other one by a gauge transformation.
How about normalisable then? Well, I never heard about that, and mainly one defines everything in compact space(s), where it scarcely makes sense to impose norm.
A: It means that the gauge ambiguity is practically removed in the Coulomb gauge if you deal with a "nice" $\mathbf{A}$ (which is your purpose).
However, it does not mean you only deal with the radiation (propagating solutions). Transversal $\mathbf{A}$ is different from zero for a uniformly moving charge too.
A: You don't actually need smoothness in general, you just need a $C^2$ transition function so that the Laplacian is well-defined - without that, you don't have a well-defined source field, which clearly kind of goes against the whole point of introducing gauge fields in the first place.
In a classical context, the normalizability is there to make Coulomb gauge a unique gauge fixing in the physically realistic situation where the charges are spatially bounded. There are at least three reasons to  motivate this particular choice of complete gauge fixing:


*

*It satisfies the physical intuition from locality that if the sources are localized, then the gauge fields should naturally fall off to zero at spatial infinity.

*It often allows us to integrate by parts and drop the surface terms, which is a trick that's constantly used in E&M.

*It gives by far the simplest Green's function for the gauge field, leading to the simple Coulomb-like equation $\varphi({\bf x}) = \int d^3x' \frac{\rho({\bf x'})}{|{\bf x} - {\bf x}'|}$, and similarly for the vector potential and the electric current.


I'm less familiar with the quantum situation, but I know that there are all kinds of subtleties with large gauge transformations in nonabelian gauge theory, instantons, etc. Here, the non-uniqueness of Coulomb gauge requires a more careful consideration of the boundary conditions.
