Consider a random mapping $f:[N] \to [N]$, .i.e, a function such that for each $x \in [N]$, $f(x) \in [N]$ is chosen uniformly at random.
My question is what would the (discrete time) fourier transform of $f$ look like?
Additionally, if we create a "spike" in the function by choosing some $y_0 \in [N]$ and a subset $S \subset [N]$ and then set $f(x) = y_0$ for all $x \in S$ (when $S$ is relatively "large") then does this translate to any interesting behavior in the fourier transform?
(Here $[N]$ denotes the set $\{1,2,...,N\}$.)