In algebraic geometry, there is a Deligne-Fourier transform, from the bounded derived category of $\ell$-adic sheaves on the affine line over a finite field, say $\mathbb F_q,$ to itself. This operation depends on the choice of an additive character of $\mathbb F_q$ (into $\bar{\mathbb Q}_{\ell}^*$), and it is an important technique in Laumon's simplified proof for Weil II. Under the sheaf-function correspondence this Fourier transform gives the classical Fourier transform for $\mathbb F_q.$
For instance, if $\chi$ is a multiplicative complex-valued character of $\mathbb F_q,$ then its Fourier transform is known as the Gauss sum. While $\chi$ takes values in roots of unity, which have absolute value 1 and hence weight 0, its Gauss sum takes values in weight 1 numbers.