[woopsn]

Convolution with dirac delta will give you an exact sample of f(0), and in principle a whole signal could be constructed as a combination of delayed delta signals - but we can't realize an exact delta signal in most spaces, only approximations.

As a result we get finite resolution and truncation of the spectrum. So "Fourier analysis with pre-applied lowpass filter" would be analysis of sampled signals, the filter determined by the sampling kernel (delta approximator) and properties of the DFT.

But so long as the sampling kernel is good (that is the actual terminology), we can form f exactly as the limit of these fuzzy interpolations.

The term "resolution of the identity" is associated with the fact that delta doesn't exist in most function spaces and instead has to be approximated. A good sampling kernel "resolves" the missing (convolutional) identity. I like thinking of the term also in the sense that these operators behave like the identity if it were only good up to some resolution.