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# Advances in Pure and Applied Mathematics

Editor-in-Chief: Trimeche, Khalifa

Editorial Board Member: Aldroubi, Akram / Anker, Jean-Philippe / Aouadi, Saloua / Bahouri, Hajer / Baklouti, Ali / Bakry, Dominique / Baraket, Sami / Ben Abdelghani, Leila / Begehr, Heinrich / Beznea, Lucian / Bezzarga, Mounir / Bonami, Aline / Demailly, Jean-Pierre / Fleckinger, Jacqueline / Gallardo, Leonard / Ismail, Mourad / Jarboui, Noomen / Jouini, Elyes / Karoui, Abderrazek / Kamoun, Lotfi / Kobayashi, Toshiyuki / Maday, Yvon / Marzougui, Habib / Mili, Maher / Mustapha, Sami / Ovsienko, Valentin / Peigné, Marc / Pouzet, Maurice / Radulescu, Vicentiu / Schwartz, Lionel / Sifi, Mohamed / Zaag, Hatem / Zarati, Said

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CiteScore 2016: 0.36

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1869-6090
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Volume 5, Issue 1 (Mar 2014)

# Sinc-type functions on a class of nilpotent Lie groups

Vignon Oussa
Published Online: 2014-01-14 | DOI: https://doi.org/10.1515/apam-2013-0028

## Abstract.

Let N be a simply connected, connected nilpotent Lie group with the following assumptions. Its Lie algebra $𝔫$ is an n-dimensional vector space over the reals. Moreover, $𝔫=𝔷\oplus 𝔟\oplus 𝔞$, $𝔷$ is the center of $𝔫$, $𝔷=ℝ{Z}_{n-2d}\oplus ℝ{Z}_{n-2d-1}\oplus \cdots \oplus ℝ{Z}_{1}$, $𝔟=ℝ{Y}_{d}\oplus ℝ{Y}_{d-1}\oplus \cdots \oplus ℝ{Y}_{1}$, $𝔞=ℝ{X}_{d}\oplus ℝ{X}_{d-1}\oplus \cdots \oplus ℝ{X}_{1}$. Next, assume $𝔷\oplus 𝔟$ is a maximal commutative ideal of $𝔫$, $\left[𝔞,𝔟\right]\subseteq 𝔷$, and $\mathrm{det}{\left(\left[{X}_{i},{Y}_{j}\right]\right)}_{1\le i,j\le d}$ is a non-trivial homogeneous polynomial defined over the ideal $\left[𝔫,𝔫\right]\subseteq 𝔷$. We do not assume that $\left[𝔞,𝔞\right]$ is generally trivial. We obtain some precise description of band-limited spaces which are sampling subspaces of ${L}^{2}\left(N\right)$ with respect to some discrete set $\Gamma$. The set $\Gamma$ is explicitly constructed by fixing a strong Malcev basis for $𝔫$. We provide sufficient conditions for which a function f is determined from its sampled values on ${\left(f\left(\gamma \right)\right)}_{\gamma \in \Gamma }$. We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.

MSC: 22E25; 22E27

Revised: 2013-12-20

Accepted: 2013-12-20

Published Online: 2014-01-14

Published in Print: 2014-03-01

Funding Source: Czech Ministry of Education

Award identifier / Grant number: ERC CZ LL1203

Citation Information: Advances in Pure and Applied Mathematics, ISSN (Online) 1869-6090, ISSN (Print) 1867-1152,

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© 2014 by Walter de Gruyter Berlin/Boston.