If your goal is still to recover the Fourier transform of scenario 3 then I suggest you try the EDFT program, written in Matlab code and available on fileexchange . To calculate the DFT, first replace M missing samples in the "gap" by NaN ('Not a Number' in Matlab) and then run command:
F = edft(your data with NaN);
After this you can also calculate the inverse Fourier transform as:
Y = real(ifft(F));
and ensure that EDFT can fill the "gap" with interpolated data in Y.