System bandwidth and the existence of generalized shift-invariant frames

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Highlights

  • First systematic study of the existence question of tight frame generators for lattice systems.

  • Examples highlighting subtlety of the problem, e.g., an infinite lattice system with non-existence of tight frame generators.

  • Introduction of system bandwidth as a criterion, including an analysis of its properties and limitations.

  • A generalization of the previously established local integrability conditions, via the theory of almost periodic functions.

Abstract

We consider the question whether, given a countable family of lattices (Γj)jJ in a locally compact abelian group G, there exist functions (gj)jJ such that the resulting generalized shift-invariant system (gj(γ))jJ,γΓj 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.

MSC

primary
42C15
secondary
43A60

Keywords

Almost periodic
Bandwidth
Calderón sum
Frame
Generalized shift-invariant system

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