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Advances in Pure and Applied Mathematics

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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


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=𝔷𝔟𝔞, 𝔷 is the center of 𝔫, 𝔷=Zn-2dZn-2d-1Z1, 𝔟=YdYd-1Y1, 𝔞=XdXd-1X1. Next, assume 𝔷𝔟 is a maximal commutative ideal of 𝔫, [𝔞,𝔟]𝔷, and det ([Xi,Yj])1i,jd is a non-trivial homogeneous polynomial defined over the ideal [𝔫,𝔫]𝔷. We do not assume that [𝔞,𝔞] is generally trivial. We obtain some precise description of band-limited spaces which are sampling subspaces of L2(N) with respect to some discrete set Γ. The set Γ 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 (f(γ))γΓ. We also provide an explicit formula for the corresponding sinc-type functions. Several examples are also computed in the paper.

Keywords: Sampling; interpolation; nilpotent Lie groups; representations

MSC: 22E25; 22E27

About the article

Received: 2013-08-19

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, DOI: https://doi.org/10.1515/apam-2013-0028.

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© 2014 by Walter de Gruyter Berlin/Boston. Copyright Clearance Center

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