The continuous $d$-dim Fourier transform and series are unified in abstract harmonic analysis under one notion. Namely, given an LCA (locally compact, Hausdorff, abelian) group $G$, equipped with its Haar measure (left/right, I don't think it matters since $G$ is abelian) $\mu$, the Fourier transform of $f\in L^1(G,\mu)$ is a function on $\hat{G} = \text{Hom}(G,S^1)$ given by
$$\hat{f}(\xi)=\int_G f(g)\bar{\xi}(g)\text{d}\mu(g)$$
For $G=(\mathbb{R}^d,+)$ we get the Fourier transform and for $G=(\mathbb{R}^d/\mathbb{Z}^d,+)$ we get the Fourier series.
The Nyquist–Shannon sampling theorem says that for the Fourier transform, if $\hat{f}$ is band-limited i.e. $\text{supp}(\hat{f})\subseteq \left [-\frac{1}{2},\frac{1}{2} \right ]^d$ then $f$ can be recovered from its samples on $\mathbb{Z}^d$ via $$f(x)=\sum_{p\in\mathbb{Z}^d}f(p)\text{sinc}(x-p)$$ The Fourier series case has a similar yet different analogue. If $\hat{f}(n)$ is zero beside some finite set $I\subseteq\mathbb{Z}^d$ then the coefficients can be recovered with a system of $|I|$ equations for $|I|$ correspondent samples. The reason this is not as powerful as the continious case is that the number of samples for the recovery process is on-par with the information $\hat{f}$ withholds. I reckon this is a different phenomenon of less interest.
I was wondering if this concept can be generalized to more cases other than this? probably with some additional assumptions on $G$'s structure that could guarantee us a general result? I would really like that the role $\left[-\frac{1}{2},\frac{1}{2} \right ]^d$ takes is reminiscent of a subgroup of $\hat{G}\cong\mathbb{R}^d$ but that is obviously not the case.