Quantitative bounds for unconditional pairs of frames

Peter Balazs, Daniel Freeman (Korresp. Autor*in), Roxana Popescu, Michael Speckbacher

Veröffentlichungen: Beitrag in FachzeitschriftArtikelPeer Reviewed

Abstract

We formulate a quantitative finite-dimensional conjecture about frame multipliers and prove that it is equivalent to Conjecture 1 in [22]. We then present solutions to the conjecture for certain classes of frame multipliers. In particular, we prove that for all C>0 and N∈N the following is true: Let (x j) j=1 N and (f j) j=1 N be sequences in a finite dimensional Hilbert space which satisfy ‖x j‖=‖f j‖ for all 1≤j≤N and ‖∑j=1Nε j〈x,f j〉x j‖≤C‖x‖,for all x∈ℓ 2 M and |ε j|=1. If the frame operator for (f j) j=1 N has eigenvalues λ 1≥…≥λ M and β>0 is such that λ 1≤βM −1j=1 Mλ j then (f j) j=1 N has Bessel bound 27β 2C. The same holds for (x j) j=1 N.

OriginalspracheEnglisch
Aufsatznummer127874
FachzeitschriftJournal of Mathematical Analysis and Applications
Jahrgang531
Ausgabenummer1
Frühes Online-Datum2024
DOIs
PublikationsstatusVeröffentlicht - 1 März 2024

ÖFOS 2012

  • 101032 Funktionalanalysis

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