The Feichtinger Conjecture and Reproducing Kernel Hilbert Spaces

TL;DR

This paper establishes new equivalences of the Feichtinger conjecture involving reproducing kernel Hilbert spaces and demonstrates conditions under which Bessel sequences can be partitioned into Riesz basic sequences.

Contribution

It introduces novel equivalences of the Feichtinger conjecture using reproducing kernel Hilbert spaces and proves partitioning results for Bessel sequences in these spaces.

Findings

If Bessel sequences of normalized kernel functions in certain spaces can be finitely partitioned into Riesz sequences, then this holds for all Hilbert spaces.

In specific reproducing kernel Hilbert spaces, bounded Bessel sequences of normalized kernel functions are finite unions of Riesz basic sequences.

Abstract

We prove two new equivalences of the Feichtinger conjecture that involve reproducing kernel Hilbert spaces. We prove that if for every Hilbert space, contractively contained in the Hardy space, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences, then a general bounded Bessel sequence in an arbitrary Hilbert space can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and prove that for these spaces bounded Bessel sequences of normalized kernel functions are finite unions of Riesz basic sequences.