DSP is all about when you can approximate discrete systems with continuous ones and vice versa. Sampling theorems. Nyquist rates, Compressive sampling, nonuniform signal sampling, stochastic signal sampling, signatures of rough paths, etc.

There are a few ways to frame this. Traditionally we talk about Shannon sampling theorems, Nyquist rates and so on. To be frank, I havenβt actually read Shannon, because the setup is not useful for the types of problems I face in my work, although Iβm sure it boils down to some similar results.

The received-wisdom version of the Shannon theorem is that you can reconstruct a
signal if you know it has frequencies in it that are βtoo highβ. Specifically,
if you sample a continuous time signal at intervals of \(T\) seconds, then you had
better have no frequencies of period shorter than \(2T\).^{1} If you do much non-trivial signal processing, (in my case I
constantly need to do things like multiplying signals) it rapidly becomes
impossible to maintain bounds on the support of the spectrogram
(TODO explain this with diagrams).

This doesnβt tell us much about more bizarre non-uniform sampling regimes, mild violations of frequency constraints, or whether other sets of (perhaps more domain-appropriate) constraints on our signals will lead to a sensible reconstruction theory.

Letβs talk about the modern, abstract and fashionable Hilbert-space framing of this problem
This way is general, and based on projections between Hilbert spaces.
Nice works in this tradition are, e.g.
(Vetterli, Marziliano, and Blu 2002) that observes that you donβt care about *Fourier*
spectrogram support, but rather the *rate of degrees of freedom* to construct a
coherent sampling theory.
Also accessible is (M. Unser 2000), which constructs the problem
of discretising signals as a minimal-loss projection/reconstruction problem.

More recently you have fancy persons such as Adcock and Hansen unifying compressed sensing and signal sampling (Adcock et al. 2014; Adcock and Hansen 2016) with more or less the same framework, so Iβll dive into their methods here.

TODO.

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Iβm playing fast-and-loose with definitions here β the spectrum in this context is the continuous Fourier spectrogram.β©οΈ

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