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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Gauss sums and Fourier analysis on multiplicative subgroups of $ Z\sb{q}$


Authors: Harold G. Diamond, Frank Gerth and Jeffrey D. Vaaler
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
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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.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0694384-1
Keywords: Gauss sums, finite Fourier transforms
Article copyright: © Copyright 1983 American Mathematical Society