Bin sizes for non-uniform discrete Fourier transforms

For a non-uniform discrete Fourier transforms, do the specified frequencies – i.e., $$f_k$$ in

– refer to the midpoint of the bin or the lower bound? I read the answer here, but that stated that the frequency bins for NUFFT are always evenly spaced which is not always the case for NUDFT (e.g., NUDFT-I and NUDFT-III – see here).

If they are the lower bound, does that mean $$f_k$$ includes frequencies $$[f_k, f_{k+1})$$?

• The bandwidth (not brick wall bandwidths) of a bin consisting of sinusoids is governed by the window function. short windows have wider bandwidth, long windows, narrower bandwidth. how you populate your bins is independent of that. It is more typical for nonuniform fourier to refer to transforms to be nonuniform in time sampling..
– user28715
Nov 9 '19 at 1:30

The notion of a "frequency bin" is somewhat misleading. In general a sine wave of a single frequency will show up in ALL frequency "bins" of the Time discrete Fourier Transform. It will generate the highest value where $$f_k$$ is closest to the signal frequency and but the values at any other Fourier frequency are not zero, just smaller.
The only exception is if the signal frequency is an exact integer multiple of $$f_k$$ and not windowing is applied.