Understanding Discrete Fourier Transform

Question

I have a small query regarding the discrete Fourier transforms. If I understand correctly, then what we do is convert a polynomial to its point value representation, with n points for a polynomial that goes up to the power of n-1. But why must we evaluate it at the nth roots of unity? Wouldn't any other n points uniquely identify this polynomial AND be much simpler?

Answer 1

Wouldn't any other n points uniquely identify this polynomial AND be much simpler?

No to both. 1) There's no guarantee that n arbitrary points would work and 2) it wouldn't be simpler. Turn the question around: why do you object to the roots of unity?