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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Algebraic structures for $ \bigoplus \sum \sb{n\geq 1}L\sp{2}(Z/n)$ compatible with the finite Fourier transform


Authors: L. Auslander and R. Tolimieri
Journal: Trans. Amer. Math. Soc. 244 (1978), 263-272
MSC: Primary 22E25; Secondary 14K20, 43A80
MathSciNet review: 506619
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Abstract: Let $ {Z / n}$ denote the integers $ \bmod\,n$ and let $ {\mathcal{F}_n}$ denote the finite Fourier transform on $ {L^2}({Z / n})$. We let $ \oplus \Sigma {{\mathcal{F}_n}} = F$ operate on $ \oplus \Sigma {L^2}({Z / n})$ and show that $ \oplus \Sigma {L^2}({Z / n})$ can be given a graded algebra structure (with no zero divisors) such that $ \mathcal{F}(fg) = \mathcal{F}(f)\mathcal{F}(g)$. We do this by establishing a natural isomorphism with the algebra of theta functions with period i. In addition, we find all algebra structures on $ \oplus \Sigma {L^2}({Z / n})$ satisfying the above condition.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1978-0506619-4
PII: S 0002-9947(1978)0506619-4
Keywords: Finite Fourier transform, theta functions, Riemann surfaces
Article copyright: © Copyright 1978 American Mathematical Society