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Sampling and interpolation on some nilpotent Lie groups

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Veröffentlicht/Copyright: 9. September 2014
Forum Mathematicum
Aus der Zeitschrift Forum Mathematicum Band 28 Heft 2

Abstract

Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra 𝔫 such that 𝔫=𝔞⊕𝔟⊕𝔷, [𝔞,𝔟]⊆𝔷, the algebras 𝔞,𝔟,𝔷 are abelian, 𝔞=ℝ- span {X1,X2,...,Xd}, and 𝔟=ℝ- span {Y1,Y2,...,Yd}. Also, we assume that det[[Xi,Yj]]1≤i,j≤d is a non-vanishing homogeneous polynomial in the unknowns Z1,...,Zn-2d where {Z1,...,Zn-2d} is a basis for the center of the Lie algebra. Using well-known facts from time-frequency analysis, we provide some precise sufficient conditions for the existence of sampling spaces with the interpolation property, with respect to some discrete subset of N. The result obtained in this work can be seen as a direct application of time-frequency analysis to the theory of nilpotent Lie groups. Several explicit examples are computed. This work is a generalization of recent results obtained for the Heisenberg group by Currey and Mayeli in [Rocky Mountain J. Math. 42 (2012), no. 4, 1135–1151].

Keywords: Sampling; interpolation; nilpotent Lie groups; representations
MSC: 22E25; 22E27

Many thanks go to the anonymous referee for a very thorough reading of this paper. His suggestions, remarks and corrections greatly improved the quality of the work.

Received: 2014-2-23
Revised: 2014-7-7
Published Online: 2014-9-9
Published in Print: 2016-3-1

© 2016 by De Gruyter

Artikel in diesem Heft

  1. Frontmatter
  2. Ring class invariants over imaginary quadratic fields
  3. The structure of p-local compact groups of rank 1
  4. Sampling and interpolation on some nilpotent Lie groups
  5. Topology of moduli spaces of free group representations in real reductive groups
  6. Finite symmetries of S4
  7. Variations of Minkowski's theorem on successive minima
  8. A one-relator group with long lower central series
  9. On transfinite nilpotence of the Vogel–Levine localization
  10. Time inhomogeneous generalized Mehler semigroups and skew convolution equations
  11. Specifying the Auslander–Reiten translation for complexes of modules
  12. Second order elliptic operators with diffusion coefficients growing as |x|α at infinity
  13. On the fourth derivative test for exponential sums
https://doi.org/10.1515/forum-2014-0035
Schlagwörter für diesen Artikel
Sampling; interpolation; nilpotent Lie groups; representations

Artikel in diesem Heft

  1. Frontmatter
  2. Ring class invariants over imaginary quadratic fields
  3. The structure of p-local compact groups of rank 1
  4. Sampling and interpolation on some nilpotent Lie groups
  5. Topology of moduli spaces of free group representations in real reductive groups
  6. Finite symmetries of S4
  7. Variations of Minkowski's theorem on successive minima
  8. A one-relator group with long lower central series
  9. On transfinite nilpotence of the Vogel–Levine localization
  10. Time inhomogeneous generalized Mehler semigroups and skew convolution equations
  11. Specifying the Auslander–Reiten translation for complexes of modules
  12. Second order elliptic operators with diffusion coefficients growing as |x|α at infinity
  13. On the fourth derivative test for exponential sums
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