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Pseudo- Riesz Bases

Deborpita Biswas, Mishko Mitkovski

TL;DR

This work generalizes near-Riesz bases by introducing pseudo-Riesz bases for Bessel sequences, defined as those that can be transformed into a Riesz basis by finite edits. It establishes a synthesis-operator, Fredholm-theoretic framework characterizing pseudo-Riesz bases, and develops the theory for associated pseudo-frames and pseudo-Riesz sequences, including dual/codual considerations. The authors prove perturbation results in the spirit of Paley-Wiener and Bari, leveraging Kato's stability theorem to show that controlled perturbations preserve the pseudo-Riesz structure. Overall, the paper broadens stability and reconstruction tools for non-frame sequences in Hilbert spaces, offering a flexible pathway to construct and manipulate quasi-Riesz systems in practice.

Abstract

In~\cite{holub1994bases} Holub introduced the concept of near-Riesz bases, as frames that can be considered Riesz bases for computational purposes or that exhibit certain desirable properties of Riesz bases. In this paper, we introduce a generalization of near-Riesz bases that includes sequences which are not necessarily frames. We demonstrate that this broader class of sequences retains many of the desirable properties of near-Riesz bases and establish fundamental perturbation results for this new class.

Pseudo- Riesz Bases

TL;DR

This work generalizes near-Riesz bases by introducing pseudo-Riesz bases for Bessel sequences, defined as those that can be transformed into a Riesz basis by finite edits. It establishes a synthesis-operator, Fredholm-theoretic framework characterizing pseudo-Riesz bases, and develops the theory for associated pseudo-frames and pseudo-Riesz sequences, including dual/codual considerations. The authors prove perturbation results in the spirit of Paley-Wiener and Bari, leveraging Kato's stability theorem to show that controlled perturbations preserve the pseudo-Riesz structure. Overall, the paper broadens stability and reconstruction tools for non-frame sequences in Hilbert spaces, offering a flexible pathway to construct and manipulate quasi-Riesz systems in practice.

Abstract

In~\cite{holub1994bases} Holub introduced the concept of near-Riesz bases, as frames that can be considered Riesz bases for computational purposes or that exhibit certain desirable properties of Riesz bases. In this paper, we introduce a generalization of near-Riesz bases that includes sequences which are not necessarily frames. We demonstrate that this broader class of sequences retains many of the desirable properties of near-Riesz bases and establish fundamental perturbation results for this new class.
Paper Structure (14 sections, 13 theorems, 28 equations)

This paper contains 14 sections, 13 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frames
  4. Riesz sequences
  5. Riesz bases
  6. Synthesis operator
  7. Fredholm operators
  8. Near-Riesz bases
  9. Pseudo-frames
  10. Pseudo-Riesz sequences
  11. Pseudo- Riesz bases
  12. Perturbation Results
  13. Paley-Wiener-type perturbations
  14. Bari-type perturbations

Key Result

Theorem 2.1

Let $\{f_n\}_{n=1}^{\infty}$ be a frame for $\mathcal{H}$. Then the following statements are equivalent.

Theorems & Definitions (25)

  • Theorem 2.1: heil2023
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Definition 4.1
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 15 more