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?)

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 $$W_\varphi(\mathbb{Z}):=\overline{\operatorname{span}}^{L^2(\mathbb{R})}\left\{\varphi(\cdot-k):k\in\mathbb{Z}\right\}.$$ What relation we have in between the spaces $ W_{\varphi}(\mathbb{Z})$ and $V_{\varphi}(\mathbb{Z})$? (i.e. are these definitions equivalent?)

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 $$W_\varphi(\mathbb{Z}):=\overline{span}^{L^2(\mathbb{R})}\left\{\varphi(\cdot-k):k\in\mathbb{Z}\right\}.$$ What relation we have in between the spaces $ W_{\varphi}(\mathbb{Z})$ and $V_{\varphi}(\mathbb{Z})$? (i.e. are these definitions equivalent?)

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?)

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 $$W_\varphi(\mathbb{Z}):=\overline{span}^{L_2(\mathbb{R})}\left\{\varphi(\cdot-k):k\in\mathbb{Z}\right\}.$$ What relation we have in between the spaces $ W_{\varphi}(\mathbb{Z})$ and $V_{\varphi}(\mathbb{Z})$? (i.e. are these definitions equivalent?)

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 $$W_\varphi(\mathbb{Z}):=\overline{span}^{L^2(\mathbb{R})}\left\{\varphi(\cdot-k):k\in\mathbb{Z}\right\}.$$ What relation we have in between the spaces $ W_{\varphi}(\mathbb{Z})$ and $V_{\varphi}(\mathbb{Z})$? (i.e. are these definitions equivalent?)