[Paper Review] An open problem concerning operator representations of frames

Ole Christensen, Marzieh Hasannasab|arXiv (Cornell University)|May 1, 2017

Mathematical Analysis and Transform Methods7 references3 citations

TL;DR

This paper investigates whether overcomplete Gabor frames and other overcomplete frames in $L^2(\mathbb{R})$ can be represented as $\{T^k\varphi\}_{k=0}^\infty$ for a bounded linear operator $T$ and some $\varphi \in L^2(\mathbb{R})$. It shows that such a representation is impossible if the frame elements are ordered such that consecutive elements from the same orthonormal basis are grouped, due to surjectivity constraints on $T$, and poses an open problem on whether such representations exist for overcomplete Gabor frames.

ABSTRACT

Recent research has shown that the properties of overcomplete Gabor frames and frames arising from shift-invariant systems form a precise match with certain conditions that are necessary for a frame in $L^2(\mathbf R)$ to have a representation $\{T^k φ\}_{k=0}^\infty$ for some bounded linear operator $T$ on $L^2(\mathbf R)$ and some $φ\in L^2(\mathbf R).$ However, for frames of this type the existence of such a representation has only been confirmed in the case of Riesz bases. This leads to several open questions connecting dynamical sampling, coherent states, frame theory, and operator theory. The key questions can either be considered in the general functional analytic context of operators on a Hilbert space, or in the specific situation of Gabor frames in $L^2(\mathbf R).$

Motivation & Objective

• To determine whether overcomplete Gabor frames in $L^2(\mathbb{R})$ can be represented as $\{T^k\varphi\}_{k=0}^\infty$ for a bounded linear operator $T$ and some $\varphi \in L^2(\mathbb{R})$.
• To investigate the conditions under which a frame in a separable Hilbert space can be expressed as $\{T^k\varphi\}_{k=0}^\infty$ with a bounded operator $T$.
• To explore the role of frame element ordering in the existence of such operator representations, particularly comparing indexing over $\mathbb{N}_0$ versus $\mathbb{Z}$.
• To examine the structural constraints on frames that would allow such a representation, especially in relation to the kernel of the synthesis operator and frame excess.
• To identify open problems in dynamical sampling, coherent states, and operator theory by connecting them to frame representations via iterated operators.

Proposed method

• Analyzes the frame representation $\{T^k\varphi\}_{k=0}^\infty$ using the synthesis operator $U$ and its kernel $N_U$, linking frame excess to $\dim(N_U)$.
• Applies the condition that for a bounded operator $T$ to generate a frame via $\{T^k\varphi\}_{k=0}^\infty$, $T$ must be surjective, derived from the requirement that the image of $T$ must span the Hilbert space.
• Uses a decomposition of the frame into two orthonormal bases $\{e_k\}$ and $\{\varepsilon_k\}$, and defines index sets $I_1, I_2, I_3, I_4$ to classify transitions between basis elements in the frame sequence.
• Employs a proof by contradiction to show that if $I_2 = \emptyset$ or $I_4 = \emptyset$, then $T$ cannot be bounded, as it would fail to be surjective.
• Compares the $\mathbb{N}_0$-indexed case with the $\mathbb{Z}$-indexed case, noting that while $\mathbb{Z}$-indexing allows bounded representations for certain frame orderings (e.g., alternating bases), $\mathbb{N}_0$-indexing does not.
• Considers the canonical dual frame and frame operator $S = UU^*$ to analyze frame properties and their relation to operator representations.

Experimental results

Research questions

• RQ1 Can overcomplete Gabor frames in $L^2(\mathbb{R})$ be represented as $\{T^k\varphi\}_{k=0}^\infty$ for a bounded linear operator $T$ and some $\varphi \in L^2(\mathbb{R})$?
• RQ2 What structural conditions on the ordering of frame elements prevent the existence of a bounded operator $T$ such that $\{T^k\varphi\}_{k=0}^\infty$ generates the frame?
• RQ3 Does the indexing set $\mathbb{N}_0$ versus $\mathbb{Z}$ significantly affect the possibility of such operator representations, especially for shift-invariant or Gabor frames?
• RQ4 Is there a general characterization of frames (in a separable Hilbert space) that admit a representation $\{T^k\varphi\}_{k=0}^\infty$ with a bounded operator $T$?
• RQ5 Can the union of two orthonormal bases, when ordered with consecutive elements from the same basis, still admit such a representation, and if not, why?

Key findings

• A frame $\{f_k\}_{k=1}^\infty$ formed as the union of two orthonormal bases $\{e_k\}$ and $\{\varepsilon_k\}$ cannot be represented as $\{T^k\varphi\}_{k=0}^\infty$ with a bounded operator $T$ if the frame elements are ordered such that $I_2 = \emptyset$ or $I_4 = \emptyset$, where $I_2$ and $I_4$ denote indices where consecutive elements belong to the same orthonormal basis.
• The operator $T$ must be surjective for such a representation to exist, and if $I_2 = \emptyset$ and $I_4 = \emptyset$, the image of $T$ would be contained in a proper closed subspace, contradicting surjectivity.
• For the $\mathbb{Z}$-indexed case, such a representation is possible even for alternating orderings of the two orthonormal bases, as demonstrated by Example 2.4, highlighting a key difference between $\mathbb{N}_0$ and $\mathbb{Z}$ indexing.
• The result implies that the existence of a bounded operator representation depends critically on frame element ordering, and that certain orderings (e.g., grouping same-basis elements) are incompatible with such representations.
• The paper establishes that only frames with 'infinite excess in all directions'—as formalized by the kernel of the synthesis operator—can potentially admit such representations, and that overcomplete frames like Gabor systems may satisfy these conditions.
• The open problem remains unresolved: it is unknown whether overcomplete Gabor frames, which satisfy the necessary conditions for such a representation, actually admit a $\{T^k\varphi\}_{k=0}^\infty$ representation with bounded $T$.