[Paper Review] Shift-invariant spaces on SI/Z Lie groups

Bradley Currey, Azita Mayeli+1 more|arXiv (Cornell University)|May 30, 2012

Mathematical Analysis and Transform Methods7 references3 citations

TL;DR

This paper introduces a periodization framework for shift-invariant spaces on SI/Z Lie groups—simply connected nilpotent Lie groups with irreducible unitary representations square-integrable modulo the center—using group Fourier transforms and range functions. It characterizes shift-invariant subspaces via measurable range functions and establishes frame and Riesz basis criteria through fiber-wise analysis, generalizing Euclidean results to non-abelian settings, with applications to the Heisenberg group and higher-step nilpotent groups.

ABSTRACT

Given a simply connected nilpotent Lie group having unitary irreducible representations that are square-integrable modulo the center (SI/Z), we develop a notion of periodization on the group Fourier transform side, and use this notion to give a characterization of shift-invariant spaces in $L^2(N)$ in terms of range functions. We apply these results to study the structure of frame and Reisz families for shift-invariant spaces. We illustrate these results for the Heisenberg group as well as for other groups with SI/Z representations.

Motivation & Objective

• To extend the theory of shift-invariant spaces from abelian groups to a broad class of non-abelian Lie groups, specifically SI/Z groups.
• To develop a periodization operator on the group Fourier transform side for SI/Z groups, enabling characterization of shift-invariant subspaces.
• To generalize the range function framework from Euclidean spaces to non-abelian settings using representation theory and Plancherel theory.
• To characterize frames and Riesz bases in shift-invariant subspaces of $L^2(N)$ via fiber-wise conditions on the range function.
• To illustrate the theory on concrete examples, including the Heisenberg group and a 3-step nilpotent group, demonstrating its applicability.

Proposed method

• Define SI/Z groups as simply connected nilpotent Lie groups where all maximal coadjoint orbits correspond to representations square-integrable modulo the center.
• Use the coadjoint orbit correspondence and Plancherel measure to identify the unitary dual $\hat{N}$ with $\mathfrak{n}^*/N$, focusing on $\hat{N}_{\text{max}} = \text{SI/Z}$.
• Construct a periodization operator on the group Fourier transform side using the action of the discrete subgroup $\Gamma \subset N$ modulo the center $Z$, leading to a fibered decomposition over $\mathbb{T} = Z^*$.
• Define a transform $T: L^2(N) \to L^2(\mathbb{T}, \mathcal{L})$ with values in sequences of Hilbert-Schmidt operators, linking $L^2(N)$ to a fiber bundle over the circle.
• Characterize $\Gamma$-shift-invariant subspaces $S \subset L^2(N)$ via a measurable range function $J(\sigma) \subset \mathcal{L}$ such that $T(S)$ consists of sections in $J(\sigma)$ almost everywhere.
• Establish frame and Riesz basis criteria by transferring the problem to fiber-wise analysis: the system $\{L_\gamma \phi\}$ forms a frame (or Riesz basis) in $S$ iff the system $\{\tilde{\pi}_\sigma(k) T\phi(\sigma)\}$ does in $J(\sigma)$ a.e. $\sigma \in \mathbb{T}$.

Experimental results

Research questions

• RQ1 How can the concept of periodization be generalized from abelian groups to non-abelian SI/Z Lie groups to characterize shift-invariant subspaces?
• RQ2 What is the appropriate analog of the range function in the non-abelian setting of SI/Z groups, and how is it linked to the group Fourier transform?
• RQ3 How do frame and Riesz basis properties in shift-invariant subspaces of $L^2(N)$ relate to fiber-wise properties of the range function?
• RQ4 Can the theory be applied to explicit examples such as the Heisenberg group and higher-step nilpotent groups?
• RQ5 What structural features of SI/Z groups allow for a Plancherel-type decomposition that supports periodization and range function characterization?

Key findings

• The paper establishes a bijective correspondence between $\Gamma$-shift-invariant subspaces $S \subset L^2(N)$ and measurable range functions $J: \mathbb{T} \to \text{closed subspaces of } \mathcal{L}$, where $\mathcal{L} = l^2(\mathbb{Z}, \mathcal{HS}(L^2(\mathbb{R}^2)))$.
• A $\Gamma$-shift-invariant subspace $S$ is characterized by the condition that $T(S) = \{ a \in L^2(\mathbb{T}, \mathcal{L}) : a(\sigma) \in J(\sigma) \text{ a.e.} \}$, with $T$ being the transform defined via the group Fourier representation.
• The system $\{L_\gamma \phi : \gamma \in \Gamma, \phi \in \mathcal{A}\}$ forms a frame (or Riesz basis) in $S$ if and only if the system $\{\tilde{\pi}_\sigma(k) T\phi(\sigma) : k \in \mathbb{Z}^4 \}$ forms a frame (or Riesz basis) in $J(\sigma)$ for almost every $\sigma \in \mathbb{T}$, with the same frame bounds.
• For the Heisenberg group, the range function $J(\sigma)$ is invariant under the action of $\tilde{\pi}_\sigma(k_1,k_2,0,0,0)$, and the transform $T$ maps $L^2(N)$ to $L^2(\mathbb{T}, l^2(\mathbb{Z}, \mathcal{HS}(L^2(\mathbb{R})))$.
• In the 3-step nilpotent example, the Plancherel measure on $\mathbb{R}^*$ is proportional to $\lambda^2 d\lambda$, and the representation $\pi_\lambda$ acts on $L^2(\mathbb{R}^2)$ via metaplectic-type operators.
• The paper shows that $S = T^{-1}(M_J)$ is $\Gamma$-shift-invariant but not left-invariant, as demonstrated by the failure of $L_{(1/2,1/2,0,0,0,0)}\phi$ to lie in $S$ for $\phi \in S$, confirming the non-abelian structure of the shift action.