Definition. A Heisenberg group for a symplectic vector space (V, ω) is the Lie group with the underlying manifold V ×R and the multiplication (u, s)(v, t) = (u+ v, s+ t+ ω(u, v)/2) where u, v ∈ V and s, t ∈ R. The map t 7→ (0, t) is a Lie group homomorphism from R to the Heisenberg group. Its image coincides with the center of the Heisenberg group. The dimension of the Heisenberg group equals 2...

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1. Fourier transform on Rn 1 2. Fourier analysis on the Heisenberg group 2 2.1. Representations of the Heisenberg group 2 2.2. Group Fourier transform 3 2.3. Convolution and twisted convolution 5 3. Hermite and Laguerre functions 6 3.1. Hermite polynomials 6 3.2. Laguerre polynomials 9 3.3. Special Hermite functions 9 4. Group Fourier transform of radial functions on the Heisenberg group 12 Ref...