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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

Online
ISSN
1435-5337
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Volume 28, Issue 2

# Sampling and interpolation on some nilpotent Lie groups

Vignon Oussa
Published Online: 2014-09-09 | DOI: https://doi.org/10.1515/forum-2014-0035

## Abstract

Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra 𝔫 such that $𝔫=𝔞\oplus 𝔟\oplus 𝔷$, $\left[𝔞,𝔟\right]\subseteq 𝔷$, the algebras $𝔞,𝔟,𝔷$ are abelian, $𝔞=ℝ\text{-}\mathrm{span}\left\{{X}_{1},{X}_{2},...,{X}_{d}\right\}$, and $𝔟=ℝ\text{-}\mathrm{span}\left\{{Y}_{1},{Y}_{2},...,{Y}_{d}\right\}$. Also, we assume that $det{\left[\left[{X}_{i},{Y}_{j}\right]\right]}_{1\le i,j\le d}$ is a non-vanishing homogeneous polynomial in the unknowns ${Z}_{1},...,{Z}_{n-2d}$ where $\left\{{Z}_{1},...,{Z}_{n-2d}\right\}$ 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].

MSC: 22E25; 22E27

Revised: 2014-07-07

Published Online: 2014-09-09

Published in Print: 2016-03-01

Citation Information: Forum Mathematicum, Volume 28, Issue 2, Pages 255–273, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741,

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