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The Fourier transform on 2-step Lie groups


In this paper, we study the Fourier transform on finite dimensional $2$-step Lie groups in terms of its canonical bilinear form (CBF) and its matrix coefficients. The parameter space of these matrix coefficients $\tilde{g}$, endowed with a distance $\rho_E$ which exchanges the regularity of a function with the decay of its Fourier matrix coefficients (cf. the Riemann-Lebesgue lemma for the classical Fourier transform) is however not complete. We compute explicitly its completion $\hat{g}$; the lack of completeness appears exactly when the CBF has nonmaximal rank. We provide an example for which partial degeneracy (partial rank loss) of the canonical form occurs, as opposed to the full degeneracy at the origin. We also compute the kernel $(w,\hat{w}) \mapsto \Theta(w,\hat{w})$ of the matrix-coefficients Fourier transform, the analogue for the $2$-step groups of the classical Fourier kernel $(x,\xi) \mapsto e^{i \langle \xi, x\rangle}$.

Key words: Fourier transform, frequency space, Heisenberg group, Hermite functions, matrix coefficients.

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Article Information

TitleThe Fourier transform on 2-step Lie groups
SourceMethods Funct. Anal. Topology, Vol. 25 (2019), no. 3, 248-272
MathSciNet   MR4016212
Milestones  Received 05/11/2018; Revised 15/05/2019
CopyrightThe Author(s) 2019 (CC BY-SA)

Authors Information

Guillaume Lévy
Centre de mathématiques Laurent Schwartz, École Polytechnique, route de Saclay, 91128 Palaiseau, France.

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Guillaume Lévy, The Fourier transform on 2-step Lie groups, Methods Funct. Anal. Topology 25 (2019), no. 3, 248-272.


@article {MFAT1210,
    AUTHOR = {Guillaume Lévy},
     TITLE = {The Fourier transform on 2-step Lie groups},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {25},
      YEAR = {2019},
    NUMBER = {3},
     PAGES = {248-272},
      ISSN = {1029-3531},
  MRNUMBER = {MR4016212},
       URL = {},


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