I have a personal interest in this question, but I don't know if it is unfounded. Suppose I have a continuous function x (t), which is sampled using an average sliding window, which performs a continuous uniform integration. The start of the scan is not fixed, that is, the signal could be scanned at any time. Is it possible to generate a function that is resistant to this procedure and can be recovered with an acceptable error from the samples obtained? What are the limitations in the generation of this function? Thank you!
A rectangular prefilter is a sinc(x) in the frequency domain, thus quite far from the text-book brick-wall filter presented in standard discrete sampling.
Still, if your (constructed) continous waveform contains no frequency components at >= samplingrate/2 and you can compensate for the high frequency roll off, I would think that near perfect reconstruction is possible.
By multiplying the spectrum of your continous waveform with that of the fourier transformed rect (sinc), you should be able to see how far from classical generic sampling you are.
Incidentally, I think that both audio A/D (time domain) and camera sensors (spatial domain) really does something close to a sample-period integrator/sample-and-hold (quite rect-like) prior to sampling. Presumably because this maximize the signal-related energy that can easily be harvested above the noise floor?