Mathematical justification for zero padding? This question asks what's the point of zero padding. The accepted answer is certainly very insightful, but I don't understand a big chunk of it:


  Zero padding allows one to use a longer FFT, which will produce a
  longer FFT result vector.
  
  A longer FFT result has more frequency bins that are more closely
  spaced in frequency. But they will be essentially providing the same
  result as a high quality Sinc interpolation of a shorter
  non-zero-padded FFT of the original data.
  
  This might result in a smoother looking spectrum when plotted without
  further interpolation.
  
  Although this interpolation won't help with resolving or the
  resolution of and/or between adjacent or nearby frequencies, it might
  make it easier to visually resolve the peak of a single isolated
  frequency that does not have any significant adjacent signals or noise
  in the spectrum. Statistically, the higher density of FFT result bins
  will probably make it more likely that the peak magnitude bin is
  closer to the frequency of a random isolated input frequency sinusoid,
  and without further interpolation (parabolic, et.al.).


What exactly is the meaning of "resolve" and "resolution" here, and how, mathematically, is it apparent that zero padding does not increase resolution.
Dually, how is apparent mathematically that zero padding means interpolation?
Resolution (in the context of spectral analysis) is the ability to distinguish two or more closely spaced sinusoidal components in a spectrum. If you can't resolve them, this means that you only see one maximum instead of two or more. Resolution is determined by the window length, i.e., by the number of time domain samples (given a certain sampling rate). Windowing will smear out or broaden any narrow band components in the spectrum. Remember that windowing, i.e., multiplication in the time domain, corresponds to convolution in the frequency domain, and this convolution with the spectrum of a standard window is a sort of averaging operation in the frequency domain, hence broadening any (frequency domain) impulses.

Resolution can only be improved by adding more data points, i.e., more information, to your analysis window. Zero-padding obviously doesn't add any information. What zero-padding does is increase the sampling density of the discretet-time Fourier transform (DTFT) of the windowed signal. For a length $N$ signal, the DTFT is given by

$$\tilde{X}(\omega)=\sum_{n=0}^{N-1}x[n]e^{-jn\omega}\tag{1}$$

The discrete Fourier transform is

$$X[k]=\sum_{n=0}^{N-1}x[n]e^{-j2\pi nk/N}\tag{2}$$

Comparing $(1)$ and $(2)$, we see that the DFT is just a sampled version of the DTFT:

$$X[k]=\tilde{X}\left(\frac{2\pi k}{N}\right)\tag{3}$$

Zero-padding increases the number of equidistant samples taken from the DTFT $\tilde{X}(\omega)$.

A detailed analysis of the fact that zero-padding corresponds to interpolation in the frequency domain is given in this answer.