I am currently dealing with discrete Fourier transform and correlation technique to construct the spectrum of a broad band signal. It's already known that if I have enough observations of the signal, in frequency domain it's a Gaussian spectrum.

I want to explore more on the frequency spectrum estimation of this kind of signal when there are not enough samples or there are samples with gaps in between sets of observation or random sampling cases. In these cases, the existing DFT or correlation techniques don't work very well.

I have come across many frequency estimation techniques such as [1]. The techniques like this can estimate the discrete frequencies present in a signal when limited data is available. However, for a continuous broad band spectrum, these techniques can fail as the computational complexity is extremely high.

I have 2 questions.

How many coherent samples in observation is enough to characterize a Gaussian distribution in frequency? Should it be estimator dependent (I suppose not). If it's not estimator dependent, how to find this number?

Is it useful to get a broader perspective on harmonic analysis based on some functional analysis? I'm very curious to get a broader perspective to such estimators. Will it be helpful for my current problem?

[1] C. Andrieu and A. Doucet, “Joint bayesian model selection and estimation of noisy sinusoids via reversible jump MCMC,” IEEE Transactions on Signal Processing, vol. 61, no. 14, pp. 3653–3655, 2013.