Cubic and Hexic Integral Transforms for Locally Compact Abelian Groups

C. R. Math. Rep. Acad. Sci. Canada Vol. 37 (4) 2015, pp. 121-130
(Received: 2014-10-10 , Revised: 2014-10-10)

Sam Walters,Department of Mathematics and Statistics, University of Northern British Columbia, Prince
George, BC, V2N 4Z9, Canada; e-mail:


We prove that for locally compact, compactly generated self-dual Abelian groups \(G\), there are canonical unitary integral operators on \(L^2(G)\) analogous to the Fourier transform but which have orders 3 and 6. To do this, we establish the existence of a certain projective character on \(G\) whose phase multiplication with the FT gives rise to the Cubic transform (of order 3). (Thus, although the Fourier transform has order 4, one can “make it” have order 3 (or 6) by means of a phase factor!)

Soit \(G\) un groupe localement compact, engendré par un sousensemble compact, et isomorphe à son groupe dual. On construit des operateurs intégrals unitaires canoniques qui sont analogues à la transformée de Fourier, mais qui sont d’ordres trois et six.

Keywords: C*-algebra, Fourier transform, Gaussian sums, Hilbert space, L2 spaces, Locally compact Abelian groups, characters, cyclic groups, integral transforms, projective char- acter, self-dual groups, unitary operators

AMS Subject Classification: Harmonic analysis and almost periodicity, General properties and structure of LCA groups, Compact groups, General properties and structure of locally compact groups, $C^*$-algebras and $W$*-algebras in relation to group representations, General properties and structure of real Lie groups, Integral representations; integral operators; integral equations methods, Integral operators, Classifications of $C^*$-algebras; factors, Automorphisms, K-theory and operator algebras -including cyclic theory 11K70, 22B05, 22C05, 22D05, 22D25, 22E15, 31A10, 45P05, 46L35, 46L40, 46L80

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