Let $\{e_0, e_1, \dots, e_{N-1} \}$ be the Euclidean basis for $l^2(Z_N)$, and let $\{F_0, F_1, \dots, F_{N−1} \}$ be the Fourier basis( where $F_m(n)= \frac{1}{N} e^{2 \pi i m n/N}$ and $\hat{z}(m) = \sum_{n=0}^{N-1} z(n) e^{-2 \pi i m n/N}$).
- How to show that $\hat{e}_m(k) = e^{−2πimk/N}$ for all $k$? Notice that $\hat{e}_m$ is very nearly (up to a reflection and a normalization) an element of the Fourier basis.
- How to show that $\hat{F}_m = e_m$?