[Paper Review] A short note on the Feichtinger Conjecture

Isabelle Chalendar, Emmanuel Fricain+1 more|arXiv (Cornell University)|Jun 17, 2011

Geometric Analysis and Curvature Flows10 references3 citations

TL;DR

This paper establishes the equivalence between the Feichtinger Conjecture and a weaker variant involving separated Bessel sequences of unit vectors in Hilbert spaces, and links this to the Agler–McCarthy–Seip conjecture on complete Nevanlinna–Pick spaces. The key contribution is proving that normalized reproducing kernels in certain complete Nevanlinna–Pick spaces—such as Dirichlet spaces $\mathcal{D}_\alpha$ for $\alpha \in [0,1]$ and local Dirichlet spaces $\mathcal{D}(\mu)$ with finite atomic measures—satisfy the Feichtinger Conjecture, providing new explicit examples of such sequences.

ABSTRACT

We discuss the equivalence of the Feichtinger Conjecture with a weaker variant and we show its connection with a conjecture of Agler--McCarthy--Seip concerning complete Nevanlinna--Pick reproducing kernel spaces. Some new examples of sequences of normalized reproducing kernels satisfying the Feichtinger Conjecture are consequently obtained.

Motivation & Objective

• To establish the equivalence between the Feichtinger Conjecture and its weaker variant involving separated Bessel sequences of unit vectors.
• To connect the Feichtinger Conjecture to the Agler–McCarthy–Seip conjecture on interpolating sequences in complete Nevanlinna–Pick spaces.
• To provide new classes of reproducing kernel sequences satisfying the Feichtinger Conjecture through known results on Carleson measures and kernel separation.
• To extend the validity of the Feichtinger Conjecture to broader classes of Hilbert spaces, particularly reproducing kernel Hilbert spaces with complete Nevanlinna–Pick structure.

Proposed method

• Prove equivalence between the Feichtinger Conjecture (FC) and its weaker version (WFC) using a combinatorial lemma on graph coloring with bounded degree.
• Use the fact that the Gram matrix of a sequence determines its Bessel and Riesz sequence properties, linking operator-theoretic conditions to matrix boundedness.
• Apply the theory of reproducing kernel Hilbert spaces, particularly the characterization of interpolating sequences via separation and Carleson measure conditions.
• Leverage known results on complete Nevanlinna–Pick spaces satisfying condition (*)—which implies the Agler–McCarthy–Seip conjecture—to deduce that normalized kernels in such spaces satisfy the Feichtinger Conjecture.
• Use the invariance of Bessel, Riesz, and separation properties under invertible operators to reduce the problem to equivalent Hilbert space configurations.
• Apply renormalization techniques to show that Dirichlet spaces $\mathcal{D}_\alpha$ and local Dirichlet spaces $\mathcal{D}(\mu)$ with finite atomic measures are complete Nevanlinna–Pick spaces, enabling the use of the conjecture.

Experimental results

Research questions

• RQ1 Is the Feichtinger Conjecture equivalent to its weaker variant involving separated Bessel sequences of unit vectors in Hilbert spaces?
• RQ2 Does the Agler–McCarthy–Seip conjecture on interpolating sequences in complete Nevanlinna–Pick spaces imply the Feichtinger Conjecture for normalized reproducing kernels?
• RQ3 Are normalized reproducing kernels in Dirichlet spaces $\mathcal{D}_\alpha$ for $\alpha \in [0,1]$ satisfying the Feichtinger Conjecture?
• RQ4 Do local Dirichlet spaces $\mathcal{D}(\mu)$ with finite sums of point masses satisfy the Feichtinger Conjecture for their normalized kernels?
• RQ5 Can the validity of the Feichtinger Conjecture be extended to reproducing kernel Hilbert spaces that are not known to be complete Nevanlinna–Pick spaces?

Key findings

• The Feichtinger Conjecture is equivalent to its weaker variant (WFC), which involves only separated Bessel sequences of unit vectors.
• The Agler–McCarthy–Seip conjecture implies the Feichtinger Conjecture for normalized reproducing kernels in complete Nevanlinna–Pick spaces.
• Normalized reproducing kernels in Dirichlet spaces $\mathcal{D}_\alpha$ for $\alpha \in [0,1]$ satisfy the Feichtinger Conjecture.
• Normalized reproducing kernels in local Dirichlet spaces $\mathcal{D}(\mu)$, where $\mu$ is a finite sum of point masses, satisfy the Feichtinger Conjecture.
• The conjecture holds for all known classes of reproducing kernel Hilbert spaces where interpolating sequences are characterized by separation and Carleson measure conditions.