A Factorization Theorem for the Fourier Transform of a Separable Locally Compact Abelian Group

  • Louis Auslander
Part of the Mathematics and Its Applications book series (MAIA, volume 18)


By a separable locally compact Abelian group, abbreviated SLCA group, we will mean a locally compact Abelian group whose topology comes from a separable metric. Let A be a SLCA group and let Δ be a subgroup of A with a discrete induced topology and such that A/Δ is compact. Let  be the dual group of A and let
$$T:{L^2}\left( A \right) \to {L^2}(\hat A)$$
be the Fourier transform. In this paper we will show that there is a natural factorization of 7 into the product of three unitary operators. When one specializes this factorization to the case where A = Z/r1r2, Z the integers one obtains the Cooley-Tukey algorithm [5]; when A is a finite Abelian group one obtains the results in [3], and for more general groups one obtains the results of A. Weil in [6,7].


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  1. [1]
    L. Auslander and R. Tolimieri, Abelian harmonic analysis, theta functions and function algebras on a nilmanifold, Lecture Notes in Math. vol. 436, Springer-Verlag 1975Google Scholar
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    L. Auslander and R. Tolimieri, Is computing into the finite Fourier transform pure or applied mathematics? Bull. (New Series) of the A.M.S. 1 (1979) 847–897MathSciNetzbMATHGoogle Scholar
  3. [3]
    L. Auslander, R. Tolimieri and S. Winograd, Hecke’s theorem in quadratic reciprocity, finite nilpotent groups and the Cooley-Tukey algorithm. Advances in Math. 43 (1982) 122–172MathSciNetzbMATHGoogle Scholar
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    A. Weil, L’Integration dans les groupes topologue. Hermann and CieGoogle Scholar
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    A. Weil, Sur certaines groupes d’operateurs unitaires, Acta Math. 111 (1964) 143–211MathSciNetzbMATHGoogle Scholar

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© D. Reidel Publishing Company 1984

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  • Louis Auslander

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