Sampling and interpolation on some nilpotent Lie groups

Abstract

Abstract Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra 𝔫 such that 𝔫 = 𝔞 ⊕ 𝔟 ⊕ 𝔷 ${\mathfrak {n=a\oplus b\oplus z}}$ , [ 𝔞 , 𝔟 ] ⊆ 𝔷 ${[ \mathfrak {a},\mathfrak {b}] \subseteq \mathfrak {z}}$ , the algebras 𝔞 , 𝔟 , 𝔷 ${\mathfrak {a},\mathfrak {b,z}}$ are abelian, 𝔞 = ℝ - span { X 1 , X 2 , ... , X d } ${\mathfrak {a}=\mathbb {R}\mathrm {\hbox{-}span}\lbrace X_{1},X_{2},\ldots ,X_{d}\rbrace }$ , and 𝔟 = ℝ - span { Y 1 , Y 2 , ... , Y d } ${\mathfrak {b}=\mathbb {R}\mathrm {\hbox{-}span}\lbrace Y_{1},Y_{2},\ldots ,Y_{d}\rbrace }$ . Also, we assume that det [ [ X i , Y j ] ] 1 ≤ i , j ≤ d ${\det [ [ X_{i} ,Y_{j}] ] _{1\le i,j\le d}}$ is a non-vanishing homogeneous polynomial in the unknowns Z 1 , ... , Z n - 2 d ${Z_{1},\ldots ,Z_{n-2d}}$ where { Z 1 , ... , Z n - 2 d } ${\lbrace Z_{1},\ldots ,Z_{n-2d}\rbrace }$ 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].