Shift-invariant spaces We can define a shift-invariant space as
$$V_{\varphi}(\mathbb{Z}):=\left\{\sum_{k\in\mathbb{Z}}c_k\varphi(\cdot-k):(c_k)\in \ell_2\right\},$$
where convergence of the series is taken to be in $L^2(\mathbb{R})$.  Is this a closed subspace of $L^2(\mathbb{R})$ or for what conditions on $\varphi$, this is a closed subspace of $L^2(\mathbb{R})$?
On the other hand, let $$W_\varphi(\mathbb{Z}):=\overline{\mathrm{span}}^{L^2(\mathbb{R})}\left\{\varphi(\cdot-k):k\in\mathbb{Z}\right\}.$$
What relation we have in between the spaces $ W_{\varphi}(\mathbb{Z})$ and $V_{\varphi}(\mathbb{Z})$? (i.e. are these definitions equivalent?)
 A: If $V_\phi$ were a closed subspace, then $V_{\phi}$ would be of second category in itself by the Baire Category theorem. Define $V_n =\left\{\sum_{|k|\leq n}c_k\phi(\cdot-k): (c_k)\in\ell^2\right\}$. Then $V_n$ are finite dimensional and $\displaystyle V_{\phi} = \bigcup_{n\in\mathbb{N}} V_n$. Thus, $V_\phi$ is of first category in itself unless $V_\phi = V_n$ for some $n\in\mathbb{N}$. Consequently, $V_{\phi}$ is a closed subspace iff it is finite dimensional. The non/existence of $\phi\in L^2(\mathbb{R})$ is left to the reader.

edit: I misread the original question: I took $V_\phi$ as the algebraic span of $\{\phi(\cdot-k):k\in\mathbb{Z}\}$, my apologies for the oversight. Please find below the answer that replaces the one above.
Let $\Phi(\gamma) = \sum_{k\in\mathbb{Z}}|\hat{\phi}(\gamma-k)|^2$ where $\hat{\phi}$ denotes the Fourier transform of $\phi$. Suppose for almost every $\gamma\in\mathbb{R}$ $$A\leq \Phi(\gamma)\leq B$$ for some $A,B>0$.
Then,  $f\in V_\phi$ iff $\hat{f}(\gamma) = p(\gamma)\hat{\phi}(\gamma)$ for some 1-periodic $p\in L^2(\mathbb{T})$.
\begin{eqnarray}
\|f\|_2^2 = \|\hat{f}\|_2^2 
= \sum_{k\in\mathbb{Z}}\int_k^{k+1} |p(\gamma)\hat{\phi}(\gamma)|^2\ d\gamma
= \int_0^1 |p(\gamma)|^2\Phi(\gamma)\ d\gamma
\end{eqnarray}
Consequently, $V_{\phi}$ is isomorphic to $L^2(\mathbb{T})$, so it is closed, and so $W_\phi\subseteq V_\phi$. Clearly $W_\phi\supseteq V_\phi$, so $W_\phi=V_\phi$.
If $\Phi\notin L^{\infty}(\mathbb{R})$, then the series $\sum_{k\in\mathbb{Z}}c_k\phi(\cdot-k)$ may not converge for some $(c_k)\in\ell^2$, i.e., the condition $``\Phi\in L^{\infty}(\mathbb{R})"$ is necessary for the well-definition of $V_\phi$.
