Wannier functions on a ring Let's say I have a single particle hamiltonian in a periodic potential, for example a 1D lattice such that:
$$H = -\frac{\partial_x^2}{2m} + V(x) $$
with $ V(x+a) = V(x)$ where $a$ is the lattice spacing between the atoms or sites.
It is known by Bloch's theorem that a solution to such a system will have the form 
$$\psi_{k}(x)=e^{ikx}u_k(x)$$ 
where $u_k(x+a)=u_k(x)$. 
My questions is about the boundary conditions. If we take 
$$\psi(x+Na) = \psi(x)$$
we get, if $N$ is large enough, a lot of different values for $k$ in the first Brillouin zone: 
$$k=\frac{2\pi n}{N}  \text{ with }-\frac{\pi}{a}<k<\frac{\pi}{a},$$
so we get a band of possible states.
In this case we can define Wannier functions which using Fourier over the wave-functions:
$$\phi(x-R) = \sum_k e^{-ik R} \psi_k(x)$$
where the summation is over all the $k$'s in the first Brillouin zone.
But if I take the B.C 
$$\psi(x+a) = \psi(x)$$
I get a single value for the momentum in each Brillouin zone 
$$k = 0, \pm 2\pi , \pm 4\pi,...$$
Is it still possible to define a Wannier function for such a state? I mean what will the Fourier be like if we have a single possible value of $k$??
 A: From the wikipedia article linked in your question, we see that Wannier functions can also be defined as
$$ \underbrace{\phi_{\textbf{R}}(\textbf{x})}_{\textbf{R} \text{ Bravais lattice vector}} \equiv
\frac{1}{\sqrt{N}}
\underbrace{\sum_{\textbf{k}}}_{\substack{\text{summed over first}\\\text{Brillouin zone}}}
e^{-i\textbf{k}\cdot\textbf{R}}
\psi_{\textbf{k}}(\textbf{x}) $$
which is the same as "your" $\phi$ because thanks to the periodicity properties of Block wave functions it is easily seen that
$$ \underbrace{\phi_{\textbf{R}}(\textbf{x})}_{\text{"mine" }\phi}
=
\underbrace{\phi(\textbf{x}-\textbf{R})}_{\text{"your" }\phi} $$
Once we note this, it is easier to realize that going from $\psi_\textbf{k}$ to $\phi_\textbf{R}$ it's just a way to change the basis set of functions used to describe the system: instead of characterizing the states with the crystal momentum, we characterize them throught the Bravais lattice vectors.
We can do this because the number of crystal momentum vectors $\textbf{k}$ it's the same as the number of Bravais lattice points $\textbf{R}$ (which is $N$).
Back to your question: I don't think it makes much sense to define a "Wannier function" of a particular $\psi_\textbf{k}$ because Wannier functions are defined as a "redistribution" of all Bloch wave functions, as you can also see rewriting them in functional form as
$$ \underbrace{\phi}_{\text{"your" }\phi} =
\frac{1}{\sqrt{N}}\sum_\textbf{k} \psi_\textbf{k} $$
But even if you could do such thing, hence for some reason allowing your system to have a single crystal momentum vector $\textbf{k}$, you would have from the definition that
$$ \phi_\textbf{R}(\textbf{x}) = e^{-i\textbf{k}\cdot\textbf{R}} \psi_\textbf{k}(\textbf{x}), \qquad \text{in "my" notation} $$
$$ \phi(\textbf{x}) = \psi_\textbf{k}(\textbf{x}), \qquad \text{in "your" notation} $$
which as you can probably see is really not that useful.
