Assuming we have a discrete signal $ { \left\{ x \left[ n \right] \right\}}_{n = 1}^{N} $.
Which has a Discrete Fourier Transform / Series.
Now, assume I'd like to estimate its Discrete Fourier Series coefficient yet some samples of $ x \left[ n \right] $ are missing (The indices are known).
How could that be done efficiently without computing the Pseudo Inverse of the adapting Fourier Series matrix?
Given $ \left\{ x \left[ n \right] \right\}_{n \in M} $ where $ M $ is the set of indices given for the samples of $ x \left[ n \right] $.
The trivial solution (Which it would be great to have a faster more efficient solution is what I'm looking for) would be:
$$ \arg \min_{y} \frac{1}{2} \left\| \hat{F}^{T} y - x \right\|_{2}^{2} $$
Where $ \hat{F} $ is formed by subset of columns of the DFT Matrix $ F $ matching the given indices of the samples, $ x $ is the vector of the given samples and $ y $ is the vector of the estimated DFT of the full data of $ x \left[ n \right] $.
The solution is then given by the Pseudo Inverse (Least Squares Solution):
$$ y = { ( \hat{F} \hat{F}^{T} ) }^{-1} \hat{F} x $$
In practice, the matrix will be very poorly conditioned hence solution must be generated using the LS Solution using the SVD.
The full code is available on my StackExchange Signal Processing Q17734 GitHub Repository (Look at the SignalProcessing\Q17734 folder).
Result of the code:
