Discrete Fourier transform has a variety of applications. It’s job is changing a basis of input vector. At this point, the basis of output vector after transformation is a set of complex n-th root of unities.
So what is the original basis of the input vector? If it is not possible to answer without particular examples, it would be helpful for me to add the examples.
The original basis is spanned by Kronecker Deltas in the discrete case, or Dirac Deltas in the continuous case.
Discrete: $$f(i)=\sum_j f(j)\delta_{ij}$$ (this is simply the standard basis $[1,0,0,0,...]$, $[0,1,0,0,...]$,...)
Continuous: $$f(x)=\int_{x'} f(x')\delta(x-x')dx'$$