Throughout I'll use the fact that the Hardy space $H_2$ is the set of $L_2$ functions on the boundary with vanishing Fourier coefficients.
We know that the Fourier Transform is an isometric isomorphism from $\ell_2(\mathbb{Z}_+)$ to $H_2(\mathbb{D})$ given by $[a_0, a_1, ...] \mapsto f(z) = \sum_k a_k z^k$. Let's call this $\Phi_1$.
We also know that the synonymous Transform is an isometric isomorphism of $L_2(\mathbb(R)_+)$ to $H_2(\mathbb{C}_+)$ (Hardy space on right half plane) given by $$f(t)\mapsto F(i\omega) = \int_{t= 0}^\infty e^{-i\omega t} f(t)dt$$. Let's call this $\Phi_2$.
We know that $z(s) = \tfrac{1-s}{1+s}$ is a bi-holomorphism of $\mathbb{C}_+$ to $\mathbb{D}$ where $1\mapsto 0$, $\infty \mapsto 1$, $0\mapsto -1$, and $i \mapsto i$. (it is its own inverse too). Further, $T_2(f) = \tfrac{f\circ z(s)}{(1+z)^2\sqrt{pi}}$ is an isometric isomorphism of $H_2(\mathbb{D})\to H_2(\mathbb{C}_+)$.
So we have an interesting set of relationships here:
$$\ell_2(\mathbb{Z}_+) $\xrightarrow[\sim]{\Phi_1} H_2(\mathbb{D}) \xrightarrow[\sim]{T_2} H_2(\mathbb{C}_+) \xrightarrow[\sim]{\Phi_2^{-1}} L_2(\mathbb{R}_+) $$
!!
I've never seen this discussed, so I'm wondering what I'm missing-- Are there important/useful theorems derived? It seems apparent that this shows $L_2(\mathbb{R}_+)$ has a representation as an $\ell_2$ vector in a way that leverages some deep relationship between the discrete Fourier Transform ($\Phi_1$) and its continuous analogue ($\Phi_2$). Is there a way to do most analysis on $L_2(\mathbb{R}_+)$ simply using $\ell_2$ vectors? Similarly for ODE theory--does this convert every ODE with functions in $x(t) \in L_2(\mathbb{R}_+)$ into a discrete recurrence relationship $x(k)\in \ell_2$?
This has to be known, so I'm wondering why it is not used more or what I am missing.
