Gauss sums and Fourier analysis on multiplicative subgroups of $Z_{q}$
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- by Harold G. Diamond, Frank Gerth and Jeffrey D. Vaaler PDF
- Trans. Amer. Math. Soc. 277 (1983), 711-726 Request permission
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.References
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Additional Information
- © Copyright 1983 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 277 (1983), 711-726
- MSC: Primary 12C25; Secondary 12C20
- DOI: https://doi.org/10.1090/S0002-9947-1983-0694384-1
- MathSciNet review: 694384