System bandwidth and the existence of generalized shift-invariant frames

Hartmut Führ, Jakob Lemvig*

*Corresponding author for this work

Research output: Contribution to journalJournal articleResearchpeer-review

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Abstract

We consider the question whether, given a countable family of lattices (Γj)j∈J in a locally compact abelian group G, there exist functions (gj)j∈J such that the resulting generalized shift-invariant system (gj(⋅−γ))j∈J,γ∈Γj is a tight frame of L2(G). This paper develops a new approach to the study of generalized shift-invariant system via almost periodic functions, based on a novel unconditional convergence property. From this theory, we derive characterizing relations for tight and dual frame generators, we introduce the system bandwidth as a measure of the total bandwidth a generalized shift-invariant system can carry, and we show that the so-called Calderón sum is uniformly bounded from below for generalized shift-invariant frames. Without the unconditional convergence property, we show, counter intuitively, that even orthonormal bases can have arbitrary small system bandwidth. Our results show that the question of existence of frame generators for a general lattice system is rather subtle and depends on analytical and algebraic properties of the lattice system.

Original languageEnglish
JournalJournal of Functional Analysis
Volume276
Issue number2
Pages (from-to)563-601
ISSN0022-1236
DOIs
Publication statusPublished - 15 Jan 2019

Keywords

  • Almost periodic
  • Bandwidth
  • Calderón sum
  • Frame
  • Generalized shift-invariant system

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