It is well known that to be able to sample a band-limited function without introducing distortion, we need to sample with twice the rate of the highest frequency. This is the Nyquist / Shannon sampling theorem which most students encounter in their first signal processing / Fourier analysis course.
However, do there exist any other theorems for other models of sampling. Say for example another type of sampling performs:
$$a[n] = \frac{1}{\Delta_t}\int_{-\Delta_t/2}^{\Delta_t/2}f(n\Delta_t+t)dt$$
What effect would that have on a sampling theorem if we wanted to derive one?
I think you can use a dimensionality argument at least if the time window is finite.
Since the signal is frequency limited there is a one-to-one mapping of the fourier transform to a periodic extension of it. And then the reverse fourier transform is back to a a discrete function (with the support of the original).
This basically means that you have a bijection from the function space to $\mathbb C^{2\Delta f\Delta T}$. So any tuple representation must at least have that many elements.
That is no matter how you sample (using a linear mapping to samples) you cannot circumvent the Nyquist criterion on the number of samples required.