If a signal is periodic, can the error of approximation by Discrete Fourier Transform be avoided when using finite number of samples?

Question

As title says, if a signal $f(t)$ is periodic, can approximation errors of approximation by discrete Fourier transform (DFT) be avoided when only finite number of samples are used?

Answer 1

No. In the case of periodic signals, the Fourier transform is a Dirac comb corresponding to the Fourier decomposition of the function with respect to its period. Most functions will have infinitely many nonzero Fourier coefficients, so any finite sampling will necessarily involve approximating.

Answer 2

Generally the answer is no, but if the signal is band-limited, it can be represented by a finite number of samples. Example: take a sine wave; it is completely characterized by 3 samples.