Rocky Mountain Journal of Mathematics

Bandlimited spaces on some 2-step nilpotent Lie groups with one Parseval frame generator

Vignon Oussa

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Let $N$ be a step two connected and simply connected non commutative nilpotent Lie group which is square-integrable modulo the center. Let $Z$ be the center of $N$. Assume that $N=P\rtimes M$ such that $P$ and $M$ are simply connected, connected abelian Lie groups, $P$ is a maximal normal abelian subgroup of $N$, $M$ acts non-trivially on $P$ by automorphisms and $\dim P/Z=\dim M$. We study bandlimited subspaces of $L^2(N)$ which admit Parseval frames generated by discrete translates of a single function. We also find characteristics of bandlimited subspaces of $L^2(N)$ which do not admit a single Parseval frame. We also provide some conditions under which continuous wavelets transforms related to the left regular representation admit discretization, by some discrete set $\Gamma\subset N$. Finally, we show some explicit examples in the last section.

Article information

Source
Rocky Mountain J. Math. Volume 44, Number 4 (2014), 1343-1366.

Dates
First available in Project Euclid: 31 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.rmjm/1414760957

Digital Object Identifier
doi:10.1216/RMJ-2014-44-4-1343

Mathematical Reviews number (MathSciNet)
MR3274352

Zentralblatt MATH identifier
1304.42080

Citation

Oussa, Vignon. Bandlimited spaces on some 2-step nilpotent Lie groups with one Parseval frame generator. Rocky Mountain J. Math. 44 (2014), no. 4, 1343--1366. doi:10.1216/RMJ-2014-44-4-1343. https://projecteuclid.org/euclid.rmjm/1414760957.


Export citation

References

  • D. Arnal, B. Currey and B. Dali, Construction of canonical coordinates for exponential Lie groups, Trans. Amer. Math. Soc. 361 (2009), 6283–6348.
  • L. Corwin and F.P. Greenleaf, Representations of nilpotent Lie groups and their applications, Cambridge Univ. Press, Cambridge, 1990.
  • H. Führ, Abstract harmonic analysis of continuous wavelet transforms, Lect. Notes Math. 1863, Springer, New York, 2005.
  • K. Gröchenig and Yurii Lyubarskii, Gabor $($super$)$ frames with Hermite functions, Math. Ann. 345 (2009), 267–286.
  • D. Han and Y. Wang, Lattice tiling and the Weyl Heisenberg frames, Geom. Funct. Anal. 11 (2001), 742–758.
  • C. Heil, A basis theory primer, Springer, New York, 2010.
  • A. Mayeli, Shannon multiresolution analysis on the Heisenberg group, J. Math. Anal. Appl. 348 (2008), 671–684.
  • S. Thangavelu, A Paley-Wiener theorem for step two nilpotent Lie groups, Rev. Math. Iber. 10 (1994), 177–187.