What is the discrete Fourier transform of the sets $(1,0,0,0)$ and $(0,1,0,0)$?

Question

What is the discrete Fourier transform of the sets $(1,0,0,0)$ and $(0,1,0,0)$?

I am unable to understand the progression from a continuous Fourier transform to a discrete Fourier transform.

Regards.

Answer 1

Up to scaling with the sequence length, the DFT in this case is the coefficient tuple of the cubic polynomial, that has the given tuple of values as the values $(p(1),p(i),p(-1),p(-i))$ on the circle points $i^k$, $k=0,1,2,3$.

Now you may note that in $p(1)\pm p(-1)$ and $p(i)\pm p(-i)$ the relation of values and coefficients becomes more concentrated. This is the basis for the FFT.