Gauss sums and Fourier analysis on multiplicative subgroups of $Z_{q}$

Abstract:

Let $G(q)$ denote the multiplicative group of invertible elements in ${{\mathbf {Z}}_q}$, the ring of integers modulo $q$. Let $H \subseteq G(q)$ be a multiplicative subgroup with cosets $aH$ and $bH$. If $f:\ {\mathbf {Z}}_q \to {\mathbf {C}}$ is supported in $aH$ we show that $f$ can be recovered from the values of $\hat f$ restricted to $bH$ if and only if Gauss sums for $H$ are nonvanishing. Here $\hat f$ is the (finite) Fourier transform of $f$ with respect to the additive group ${{\mathbf {Z}}_q}$. The main result is a simple criterion for deciding when these Gauss sums are nonvanishing. If $H = G(q)$ then $f$ can be recovered from $\hat f$ restricted to $G(q)$ by a particularly elementary formula. This formula provides some inequalities and extremal functions.