If there is an N-point sequence, it has a Fourier Transform. DFT of this sequence is the sampling of the Fourier Transform at N equally spaced points on the unit circle. What happens if we take some M non-uniformly spaced samples on the unit circle with M < N, M = N and M > N. Any studies done on this. If yes, what are the applications?
Uniform sampling is the definite standard as it simplifies not only mathematical derivations but also practical applications. On the other hand, there are some applications (such as array of sensors located in a nonuniform grid) in which taking uniformly distributed samples from a signal of interest is impossible. In such cases nonuniform sampling in either time or space is performed and the usual approach is to convert those nonuniform samples to uniform ones. Then follows the standard operations on those precomputed uniform samples.
You might like to have a look at Wikipedia here. There are non-uniform Fourier Transformation applications. The article also gives some fields as example that use this technique.
One more application that I am aware of is magnetic resonance imaging (MRI). In MRI, image data is sampled in Fourier space, hence you simply do a FFT of your data and you have your image - as long as the imaging sequence samples the data on a Cartesian grid. However, there are techniques that do not obey this restriction, e.g. sample the data radially. By this, the measured data is distributed on concentric circles around the DC frequency and a simple FFT is therefore not suitable. It is possible to interpolate this data to a Cartesian grid and use FFT for the imaging process (this process is called regridding). Another technique is using a non-uniform FFT algorithm - which answers your question where it is used.