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A Survey on Constructing Parseval Fusion Frames via Scaling Weights

Ehsan Ameli, Ali Akbar Arefijamaal, Fahimeh Arabyani Neyshaburi

TL;DR

The paper addresses constructing Parseval fusion frames by scaling weights, extending the notion of scalability to the fusion-frame setting. It develops a comprehensive framework linking weight-scaling to operator equations, provides necessary and sufficient conditions for weight-scalability of fusion Riesz bases, 1-excess, and certain $k$-excess fusion frames, and furnishes concrete examples to illustrate when scalability is possible or impossible. The results offer a systematic method to obtain Parseval fusion frames without altering subspaces, enabling straightforward reconstruction and resilience in signal-processing applications. This framework enhances design guidelines for fusion-frame-based sensing and processing by clarifying when and how weight-scaling can yield Parseval frames while preserving subspace structure.

Abstract

The construction of Parseval fusion frames is highly desirable in a wide range of signal processing applications. In this paper, we study the problem of modifying the weights of a fusion frame in order to generate a Parseval fusion frame. To this end, we extend the notion of the scalability to the fusion frame setting. We then proceed to characterize scalable fusion Riesz bases and $1$-excess fusion frames. Furthermore, we provide the necessary and sufficient conditions for the scalability of certain $k$-excess fusion frames, $k\geq 2$. Finally, we present several pertinent examples to confirm the obtained results.

A Survey on Constructing Parseval Fusion Frames via Scaling Weights

TL;DR

The paper addresses constructing Parseval fusion frames by scaling weights, extending the notion of scalability to the fusion-frame setting. It develops a comprehensive framework linking weight-scaling to operator equations, provides necessary and sufficient conditions for weight-scalability of fusion Riesz bases, 1-excess, and certain -excess fusion frames, and furnishes concrete examples to illustrate when scalability is possible or impossible. The results offer a systematic method to obtain Parseval fusion frames without altering subspaces, enabling straightforward reconstruction and resilience in signal-processing applications. This framework enhances design guidelines for fusion-frame-based sensing and processing by clarifying when and how weight-scaling can yield Parseval frames while preserving subspace structure.

Abstract

The construction of Parseval fusion frames is highly desirable in a wide range of signal processing applications. In this paper, we study the problem of modifying the weights of a fusion frame in order to generate a Parseval fusion frame. To this end, we extend the notion of the scalability to the fusion frame setting. We then proceed to characterize scalable fusion Riesz bases and -excess fusion frames. Furthermore, we provide the necessary and sufficient conditions for the scalability of certain -excess fusion frames, . Finally, we present several pertinent examples to confirm the obtained results.

Paper Structure

This paper contains 7 sections, 20 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and Notations
  3. Weight-Scalable Fusion Riesz Bases and 1-excess Fusion Frames
  4. Fusion Riesz bases
  5. 1-excess fusion frames
  6. Weight-Scalable 2-excess Fusion Frames
  7. Weight-Scalability: General Case

Key Result

Theorem 2.2

frame of subspace Let $\{W_{i}\}_{i \in I}$ be a family of closed subspaces of $\mathcal{H}$, $\omega _{i} >0$ and $\{f_{i,j}\}_{j \in J_{i}}$ be a frame (Riesz basis) for $W_{i}$ with frame bounds $A_{i}$ and $B_{i}$ such that Then the following conditions are equivalent:

Theorems & Definitions (41)

  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • proof
  • Example 3.4
  • Proposition 3.5
  • ...and 31 more