  We develop a general theory of higher semiadditive Fourier transforms that
includes both the classical discrete Fourier transform for finite abelian
groups at height $n=0$, as well as a certain duality for the $E_n$-(co)homology
of $\pi$-finite spectra, established by Hopkins and Lurie, at heights $n\ge 1$.
We use this theory to generalize said duality in three different directions.
First, we extend it from $\mathbb{Z}$-module spectra to all (suitably finite)
spectra and use it to compute the discrepancy spectrum of $E_n$. Second, we
lift it to the telescopic setting by replacing $E_n$ with $T(n)$-local higher
cyclotomic extensions, from which we deduce various results on affineness,
Eilenberg--Moore formulas and Galois extensions in the telescopic setting.
Third, we categorify their result into an equivalence of two symmetric monoidal
$\infty$-categories of local systems of $K(n)$-local $E_n$-modules, and relate
it to (semiadditive) redshift phenomena.
