So far I have studied the STFT and how it works. Following this question: Inverse Short Time Fourier Transform algorithm described in words , I got a grasp of how it works and how to implement it.
Continuous and discrete Fourier transform are both invertible function by the mathematical standpoint, but this is not true for STFT.
The main problem I encountered is: which is the main criteria in order to obtain the perfect reconstruction of a signal by its STFT? Which is the mathematical costraint (regarding windows or any other assumption) we are using in order to achieve its invertibility?