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

Abstract

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.

37 pages, Latex, uses TikZ package to draw 3 figures. Two new subsections, main results unchanged

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