[Paper Review] Redundancy for localized and Gabor frames

Radu Bălan, Pete Casazza+1 more|ArXiv.org|Apr 28, 2009

Mathematical Analysis and Transform Methods25 references3 citations

TL;DR

This paper establishes a quantitative notion of redundancy for infinite frames using the redundancy function derived from localization theory. It proves that for $\ell^1$-localized frames and Gabor frames with generators in $M^1(\mathbb{R}^d)$, any frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one, resolving a key open problem in frame theory and supporting a general redundancy definition from prior work.

ABSTRACT

Redundancy is the qualitative property which makes Hilbert space frames so useful in practice. However, developing a meaningful quantitative notion of redundancy for infinite frames has proven elusive. Though quantitative candidates for redundancy exist, the main open problem is whether a frame with redundancy greater than one contains a subframe with redundancy arbitrarily close to one. We will answer this question in the affirmative for $\ell^1$-localized frames. We then specialize our results to Gabor multi-frames with generators in $M^1(\R^d)$, and Gabor molecules with envelopes in $W(C,l^1)$. As a main tool in this work, we show there is a universal function $g(x)$ so that for every $ε>0$, every Parseval frame $\{f_i\}_{i=1}^M$ for an $N$-dimensional Hilbert space $H_N$ has a subset of fewer than $(1+ε)N$ elements which is a frame for $H_N$ with lower frame bound $g(ε/(2\frac{M}{N}-1))$. This work provides the first meaningful quantative notion of redundancy for a large class of infinite frames. In addition, the results give compelling new evidence in support of a general definition of reudndancy given in [7].

Motivation & Objective

• To develop a meaningful quantitative measure of redundancy for infinite frames, addressing a long-standing open problem in frame theory.
• To show that frames with redundancy greater than one contain subframes with redundancy arbitrarily close to one, satisfying a key desideratum for redundancy.
• To validate the redundancy function as a legitimate measure by demonstrating it satisfies four essential properties, including the ability to reduce redundancy toward one.
• To extend the framework to Gabor frames with $M^1(\mathbb{R}^d)$ generators and Gabor molecules with $W(C,\ell^1)$ envelopes.

Proposed method

• Introduces a universal function $g(x)$ that bounds the lower frame bound of a subframe of a Parseval frame in terms of the relative size of the subframe.
• Uses $\ell^1$-localization between a frame $\mathcal{F}$ and a reference frame $\mathcal{E}$ to relate the redundancy of $\mathcal{F}$ to the density of the localization map $a: I \to G$.
• Defines the redundancy function $R(p; \mathcal{F})$ via a $p$-limit over increasing subsets of the index set, capturing asymptotic average redundancy.
• Applies the theory to Gabor frames by showing that when the generator $g \in M^1(\mathbb{R}^d)$, the frame is $\ell^1$-localized with respect to a Gabor frame $\mathcal{E}$, enabling the use of the redundancy function.
• Establishes that $R(p; \mathcal{F}) = D(p,a) \cdot R(p; \mathcal{E})$ for $\ell^2$-localized frames, and extends this to $\ell^1$-localized frames for the main results.
• Uses the existence of a sequence of reference frames $\mathcal{E}_n$ with $\liminf R(p; \mathcal{E}_n) = 1$ to construct subframes of $\mathcal{F}$ with redundancy $\leq 1+\varepsilon$.

Experimental results

Research questions

• RQ1 Does every frame with redundancy greater than one contain a subframe with redundancy arbitrarily close to one?
• RQ2 Can a universal quantitative redundancy measure be defined for infinite frames, particularly for Gabor and localized frames?
• RQ3 Does the redundancy function satisfy the four key properties $P_1$ through $P_4$ for $\ell^1$-localized frames?
• RQ4 Is the redundancy function consistent with the general definition proposed in [5] for frames without a natural density?
• RQ5 How does the redundancy of a Gabor frame with $M^1(\mathbb{R}^d)$ generator relate to its subframe redundancy?

Key findings

• For any $\varepsilon > 0$, every Parseval frame $\{f_i\}_{i=1}^M$ in an $N$-dimensional Hilbert space has a subset of fewer than $(1+\varepsilon)N$ elements that forms a frame with lower frame bound $g(\varepsilon / (2M/N - 1))$.
• For $\ell^1$-localized frames $\mathcal{F}$ with respect to a reference frame $\mathcal{E}$, and for every $\varepsilon > 0$, there exists a subframe $\mathcal{F}[J_\varepsilon]$ such that $R(p; \mathcal{F}[J_\varepsilon]) \leq (1+\varepsilon)R(p; \mathcal{E})$ for all $p \in \mathbb{N}^*$.
• For Gabor frames $\mathcal{G}(g;\Lambda)$ with $g \in M^1(\mathbb{R}^d)$, and for every $\varepsilon > 0$, there exists a subframe $\mathcal{G}' = \mathcal{G}(g;J_\varepsilon)$ such that $R(p; \mathcal{G}') \leq 1+\varepsilon$ for all $p \in \mathbb{N}^*$.
• The redundancy function satisfies $R(p; \mathcal{F}) \geq 1$ for all $p \in \mathbb{N}^*$, and $R(p; \mathcal{F}) = 1$ if $\mathcal{F}$ is a Riesz basis.
• For two $\ell^1$-localized frames $\mathcal{F}_1$ and $\mathcal{F}_2$ with the same localization map, the redundancy of their disjoint union satisfies $R(p; \mathcal{F}_1 \dot{\sqcup} \mathcal{F}_2) = R(p; \mathcal{F}_1) + R(p; \mathcal{F}_2)$.
• The redundancy function satisfies all four properties $P_1$ through $P_4$, confirming its suitability as a quantitative redundancy measure for $\ell^1$-localized and Gabor frames.