Journal of Noncommutative Geometry


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Volume 2, Issue 2, 2008, pp. 215–261
DOI: 10.4171/JNCG/20

Published online: 2008-06-30

Fourier analysis on the affine group, quantization and noncompact Connes geometries

Victor Gayral[1], José M. Gracia-Bondí­a[2] and Joseph C. Várilly[3]

(1) University of Copenhagen
(2) Universidad Complutense de Madrid
(3) Universidad de Costa Rica

We find the Stratonovich–Weyl quantizer for the nonunimodular affine group of the line. A noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of Connes, is obtained from it. The corresponding Wigner functions reproduce the time-frequency distributions of signal processing. The same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by Kirillov.

Keywords: Scalar Fourier transforms, affine symmetry, nonunital spectral triples, Kirillov orbit method, Stratonovich–Weyl quantization

Gayral Victor, Gracia-Bondí­a José, Várilly Joseph: Fourier analysis on the affine group, quantization and noncompact Connes geometries. J. Noncommut. Geom. 2 (2008), 215-261. doi: 10.4171/JNCG/20