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A frequency space for the Heisenberg group

2019; Association of the Annals of the Fourier Institute; Volume: 69; Issue: 1 Linguagem: Inglês

10.5802/aif.3246

1777-5310

Hajer Bahouri, Jean-Yves Chemin, Raphaël Danchin,

Digital Filter Design and Implementation

We revisit the Fourier analysis on the Heisenberg group ℍ d . Starting from the so-called Schrödinger representation and taking advantage of the projection with respect to the Hermite functions, we look at the Fourier transform of an integrable function f, as a function f ^ ℍ on the set ℍ ˜ d = def ℕ d ×ℕ d ×ℝ∖{0}. After observing that f ^ ℍ is uniformly continuous on ℍ ˜ d equipped with an appropriate distance d ^, we extend the definition of f ^ ℍ to the completion ℍ ^ d of ℍ ˜ d . This new point of view provides a simple and explicit description of the Fourier transform of integrable functions, when the “vertical” frequency parameter tends to 0. As an application, we prepare the ground for computing the Fourier transform of functions on ℍ d that are independent of the vertical variable.