Say I have a signal I have sampled at N points, and I want to calculate the Discrete Fourier transform of these N points. Formally
$$y_{k} = \frac{1}{\sqrt{N}}\sum_{j=0}^{N-1}x_{j}\omega_{N}^{2\pi i jk } \;\;\;\;\;\;\;\;\;\;\; \forall\; k \in \{0,1,2,...,N\} $$
But let's say due to some restrictions, I can calculate the Fourier transform only on some m < N points, formally
$$y'_{k} = \frac{1}{\sqrt{M}}\sum_{j=0}^{M-1}x_{j}\omega_{M}^{2\pi i jk } \;\;\;\;\;\;\;\;\;\;\; \forall\; k \in \{0,1,2,...,M\} $$
Then, my question is would it be possible to calculate $y_{k}$ using $y'_{k}$
EDIT :
According to Stelios's comment below - operating on only first M samples is identical to loosing a part of the signal.
But lets say, you have stored the original $N$ samples in memory, and you can pick the $M$ samples from the N in any of the $N C_{M}$ ways possible.
(Could it be possible to recover the original fourier transform by exploiting the structure of the stored $N$ samples? )