Fourier analysis on the affine group, quantization and noncompact Connes geometries. (English) Zbl 1148.43005
The Stratonovich-Weyl quantizer for the non-unimodular affine group of the line is determined. This provides a noncommutative product of functions on the half-plane, underlying a noncompact spectral triple in the sense of A. Connes [Commun. Math. Phys. 182, No. 1, 155–176 (1996; Zbl 0881.58009)].
It is shown that the corresponding Wigner functions reproduce the time-frequency distributions of signal processing, and that the same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by A. A. Kirillov [Lectures on the orbit method. Graduate Studies in Mathematics 64. Providence, RI: Am. Math. Soc. (2004; Zbl 1229.22003)].
It is shown that the corresponding Wigner functions reproduce the time-frequency distributions of signal processing, and that the same construction leads to scalar Fourier transformations on the affine group, simplifying and extending the Fourier transformation proposed by A. A. Kirillov [Lectures on the orbit method. Graduate Studies in Mathematics 64. Providence, RI: Am. Math. Soc. (2004; Zbl 1229.22003)].
Reviewer: Vladimir Balan (Bucureşti)
MSC:
| 43A30 | Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc. |
| 43A85 | Harmonic analysis on homogeneous spaces |
| 58B34 | Noncommutative geometry (à la Connes) |
| 81S30 | Phase-space methods including Wigner distributions, etc. applied to problems in quantum mechanics |
Keywords:
Scalar Fourier transforms; affine symmetry; nonunital spectral triples; Kirillov orbit method; Stratonovich-Weyl quantization; Moyal product; Fourier-Moyal kernelsReferences:
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