Invertibility of Frame Operators on Besov-Type Decomposition Spaces

José Luis Romero, Jordy Timo van Velthoven*, Felix Voigtlaender

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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Abstract

We derive an extension of the Walnut–Daubechies criterion for the invertibility of frame operators. The criterion concerns general reproducing systems and Besov-type spaces. As an application, we conclude that L2 frame expansions associated with smooth and fast-decaying reproducing systems on sufficiently fine lattices extend to Besov-type spaces. This simplifies and improves recent results on the existence of atomic decompositions, which only provide a particular dual reproducing system with suitable properties. In contrast, we conclude that the L2 canonical frame expansions extend to many other function spaces, and, therefore, operations such as analyzing using the frame, thresholding the resulting coefficients, and then synthesizing using the canonical dual frame are bounded on these spaces.

Original languageEnglish
Article number149
Pages (from-to)1-72
Number of pages72
JournalJournal of Geometric Analysis
Volume32
Issue number5
DOIs
Publication statusPublished - 2022

Keywords

  • Atomic decompositions
  • Banach frames
  • Besov-type decomposition space
  • Canonical dual frame
  • Frame operator
  • Generalized shift-invariant systems
  • Walnut–Daubechies representation

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