A Density Condition for Interpolation on the Heisenberg Group

TL;DR

This paper establishes a precise density criterion for certain subspaces of the Heisenberg group to have the interpolation property, providing concrete examples and conditions for shift-invariant bases.

Contribution

It introduces a necessary and sufficient density condition for interpolation in multiplicity free subspaces of L^2(N), including explicit examples and basis characterization.

Findings

Density condition for interpolation on the Heisenberg group

Concrete example of an interpolating subspace for the integer lattice

Necessary and sufficient condition for shift-invariant bases

Abstract

Let $N$ be the Heisenberg group. We consider left-invariant multiplicity free subspaces of $L^2(N)$. We prove a necessary and sufficient density condition in order that such subspaces possess the interpolation property with respect to a class of discrete subsets of $N$ that includes the integer lattice. We exhibit a concrete example of a subspace that has interpolation for the integer lattice, and we also prove a necessary and sufficient condition for shift invariant subspaces to possess a singly-generated orthonormal basis of translates.