Sampling and interpolation on some nilpotent Lie groups (Q254050)

This paper deals with sampling spaces. The simplest example of such spaces with interpolation property over a nilpotent Lie group is provided by the well-known Whittaker, Shannon, Kotelnikov theorem, although the first example of a sampling space with interpolation property on a noncommutative nilpotent Lie group was defined over the three-dimensional Heisenberg Lie group by Currey and Mayely using the Plancherel transform. However, the question of existence of sampling spaces with interpolation property on some non-commutative nilpotent Lie groups is a challenging problem which constitutes the main focus of this paper, which is, in fact a generalization of recent results obtained for the Heisenberg group by \textit{B. Currey} and \textit{A. Mayeli} [Rocky Mt. J. Math. 42, No. 4, 1135--1151 (2012; Zbl 1271.42042)]. Indeed, the author, by using well-known facts from time-frequency analysis, provides some precise sufficient conditions for the existence of sampling spaces with the interpolation property, with respect to some discrete subset of a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra verifying certain conditions. The author also computes several explicit examples.