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Journal of Mathematical Physics

A composite Fourier-wavelet transform and square-integrable group representations

S. Twareque Ali, L. V. Denisov

Published Aug 1, 1994• 2 citations
Publisher

Abstract

A formula for a composite Fourier-wavelet transform is suggested, from which the noncommutative Fourier and wavelet transforms can be obtained as particular examples, as well as several other Fourier-type transforms. It is shown how this composite transform can be used to extend certain results, originally obtained for square-integrable group representations, to more general measures, arising in quantum probability theory.

DOI
10.1063/1.530849

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A formula for a composite Fourier-wavelet transform is suggested, from which the noncommutative Fourier and wavelet transforms can be obtained as particular examples, as well as several other Fourier-type transforms. It is shown how this composite transform can be used to extend certain results, originally obtained for square-integrable group representations, to more general measures, arising in quantum probability theory.
Starting from square-integrable wave functions on a Lie group, we build an invertible Fourier transform mapping them on wave functions on the dual of the Lie algebra. This is a group-theoretic version of the map from position space to momentum space, with generally noncommuting momenta owing to the group structure. As a result, the multiplication of momentum-dependent functions involves star products, which makes the construction of noncommutative Fourier series much more involved than that of their commutative cousin. This is especially true when compact subgroups are present, in which case we carefully take into account quotients of the operator algebra, and the resulting normalization issues. We show that our formalism provides an isometry of Hilbert spaces, and use it to derive a noncommutative Poisson summation formula for any compact Lie group. This is a key preliminary for the computation of Wigner functions and path integrals for quantum systems on group manifolds.