I have a signal which is bandlimited and can be sampled at arbitrary continuous positions. The value at any position is given by an expensive computation. I need to do some further computation on various arbitrary small regions of this signal.
My plan is to take advantage of the known bandlimiting to sample the signal at or slightly above the Nyquist rate within these regions that I need to process, and then do my computation on a reconstructed approximation of the region. This reconstruction needs to be in a form that allows for evaluation at arbitrary time
t within the reconstructed range. Because of the expense of computing each sample, I need to minimise the number of samples required to reconstruct each region. The regions are sized such that they are between 8 and 16 samples wide at the Nyquist rate.
The thing that makes this different from standard usages of sinc interpolation is the small number of samples. That's the focus of this question. It's typical for sinc based reconstructions to experience problems near the edges of the signals, but when a signal is only 8 samples wide, most of it is edge.
My question concerns how to optimally reconstruct these small regions with minimal samples. I particularly would like to know whether it's possible to achieve a reasonable reconstruction with all samples being within the region I'm reconstructing. I suspect that the answer is no, because there's insufficient information to reconstruct towards the edges of the region, but I would like to have this confirmed. My experiments with various windowed sinc methods support this suspicion.
Assuming that I can't rely on only samples within the region, is there anything better I can do than taking a few extra samples on either side of the region and then continuing to use basic sinc methods?
I don't have a firm definition of a good enough reconstruction, but for the purposes of the question, let's say that at all points along the reconstruction, the error should be no more than 10%.