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.
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Victor Gayral, José M. Gracia-Bondía, Joseph C. Várilly, Fourier analysis on the affine group, quantization and noncompact Connes geometries. J. Noncommut. Geom. 2 (2008), no. 2, pp. 215–261DOI 10.4171/JNCG/20