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Approximate oblique duality for fusion frames

Jorge P. Díaz, Sigrid B. Heineken, Patricia M. Morillas

TL;DR

This work introduces the concept of approximate oblique dual fusion frame, and in particular of approximate oblique dual fusion frame system, and provides methods for obtaining them and gives the relation to approximate oblique dual frames.

Abstract

Fusion frames are a convenient tool in applications where we deal with a large amount of data or when a combination of local data is needed. Oblique dual fusion frames are suitable in situations where the analysis for the data and its subsequent synthesis have to be implemented in different subspaces of a Hilbert space. These procedures of analysis and synthesis are in general not exact, and also there are circumstances where the exact dual is not available or it is necessary to improve its properties. To resolve these questions we introduce the concept of approximate oblique dual fusion frame, and in particular of approximate oblique dual fusion frame system. We study their properties. We give the relation to approximate oblique dual frames. We provide methods for obtaining them. We show how to construct other duals from a given one that give reconstructions errors as small as we want.

Approximate oblique duality for fusion frames

TL;DR

This work introduces the concept of approximate oblique dual fusion frame, and in particular of approximate oblique dual fusion frame system, and provides methods for obtaining them and gives the relation to approximate oblique dual frames.

Abstract

Fusion frames are a convenient tool in applications where we deal with a large amount of data or when a combination of local data is needed. Oblique dual fusion frames are suitable in situations where the analysis for the data and its subsequent synthesis have to be implemented in different subspaces of a Hilbert space. These procedures of analysis and synthesis are in general not exact, and also there are circumstances where the exact dual is not available or it is necessary to improve its properties. To resolve these questions we introduce the concept of approximate oblique dual fusion frame, and in particular of approximate oblique dual fusion frame system. We study their properties. We give the relation to approximate oblique dual frames. We provide methods for obtaining them. We show how to construct other duals from a given one that give reconstructions errors as small as we want.
Paper Structure (11 sections, 21 theorems, 10 equations)

This paper contains 11 sections, 21 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Oblique projections and left inverses
  5. Frames
  6. Fusion frames
  7. Approximate oblique dual fusion frames and fusion frame systems
  8. Oblique dual fusion frames from approximate ones
  9. Uniqueness and approximate consistent reconstruction
  10. Approximate oblique dual families
  11. Better approximation

Key Result

Theorem 2.6

Casazza-Kutyniok (2004) Given $(\mathbf{W},\mathbf{w})$, let $\mathcal{F}_{i}$ be a frame for $W_{i}$ with frame bounds $\alpha_{i}, \beta_{i}$ such that $0 < \alpha ={\rm inf}_{i \in I}\alpha_{i} \leq {\rm sup}_{i \in I}\beta_{i}= \beta < \infty$. The following assertions are equivalents: If $(\mathbf{W},\mathbf{w})$ is a fusion frame for $\mathcal{W}$ with fusion frame bounds $\gamma$ and $\del

Theorems & Definitions (52)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Definition 2.7
  • Definition 3.1
  • Remark 3.2
  • Lemma 3.3
  • ...and 42 more