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I have taken a basic course or two in signal processing and I've seen the Nyquist sampling theorem - an interesting and surprising result giving sufficient conditions for reconstruction of a reasonably regular signal in terms of spectral support.

I was told there are many much deeper results in this field than this canonical theorem. Unfortunately, I find the literature rather inaccessible. Hence, I'm asking here for examples of deep results in sampling theory.

I am most interested in results which rely on other branches of mathematics, namely topology and algebraic geometry.

I understand the question is vague but also think that it is clear enough to give answers to. I don't mind making this a community wiki since it has no single correct answer.

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    $\begingroup$ I don't know about any relationship between sampling and topology or algebraic geometry. Some interesting sampling theories are: non-uniform sampling, and sparse sampling. Sparse sampling is a very active area now; basically, there are some signals that can be sampled at less than the Nyquist rate because they have some underlying structure ("sparseness") that is captured with less samples. $\endgroup$ – MBaz Dec 20 '15 at 1:45
  • $\begingroup$ You might want to read "Topological Data Analysis and Machine Learning Theory" (birs.ca/workshops/2012/12w5081/report12w5081.pdf) which touches upon the "shape of data". If data has "a shape" it means that some prior knowledge can be exploited for various purposes. That's what "compressed sensing" is based on. Also see (en.wikipedia.org/wiki/Iterative_reconstruction) and more importantly (en.wikipedia.org/wiki/Algebraic_Reconstruction_Technique) $\endgroup$ – A_A Dec 20 '15 at 23:16
  • $\begingroup$ I have edited my initial answer, I forgot to mention the topological aspects of sampling in chapters 9 and 10 of the 2015 book. $\endgroup$ – Laurent Duval Dec 21 '15 at 20:53
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As you are looking for some deep results, let me mention only two:

  • the Kadec's-$1/4$ theorem that gives a stability bound under which perturbed exponential Riesz bases remain Riesz bases (M. I. Kadec, The exact value of the Paley-Wiener constant, Dokl. Akad. Nauk SSSR, 1964, see for instance R. Young, An introduction to non-harmonic Fourier series, p. 42)
  • the theory called "finite rate of innovation" (Sampling Signals With Finite Rate of Innovation, IEEE Trans. Signal Proc., 2012, M. Vetterli et al.) which essentially generalizes Nyquist-Shannon theory to non band-limited signals, requiring "twice the number of degrees-of-freedom" per unit interval, like the two-points per period in the classical cases, for some parametric signals (finite number of spikes, piece-wise polynomials, etc.).

I like the genericity and the simple bounds for both examples.

You can probably find more ideas in Classical and approximate sampling theorems; studies in the $L^p(R)$ source and the uniform norm, J. Approx. Theor., 2005, P. L. Butzer et al. and Sampling Theory, a Renaissance: Compressive Sensing and Other Developments, 2015, G. E. Pfander editor. In the latter, you might like Chapter 9 - Sampling in Euclidean and non-Euclidean Domains: A Unified Approach (sphere, Hyperbolic Space), and Chapter 10 - A Sheaf-Theoretic Perspective on Sampling, where you get some topology.

There are all sorts of exiting developments in zero- or level-crossing, single-bit sampling, which I do not know enough to be more precise: round-off, jitter, truncation, saturation and aliasing errors are real problems.

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