Sampling and Interpolation on Some Nilpotent Lie Groups
نویسندگان
چکیده
Let N be a non-commutative, simply connected, connected, two-step nilpotent Lie group with Lie algebra n such that n = a⊕ b⊕ z, [a, b] ⊆ z, the algebras a, b, z are abelian, a = R-span {X1, X2, · · · , Xd} , and b = R-span {Y1, Y2, · · · , Yd} . Also, we assume that det [[Xi, Yj ]]1≤i,j≤d is a non-vanishing homogeneous polynomial in the unknowns Z1, · · · , Zn−2d where {Z1, · · · , Zn−2d} 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 [3].
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