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On the lower Riesz basis bound of exponential systems over an interval

Thibaud Alemany, Shahaf Nitzan

TL;DR

This work analyzes the lower Riesz basis bound A(Λ) for exponential systems on an interval by revisiting Pavlov's characterization and linking Riesz-basis properties to the boundedness of the Riesz projection in weighted L^2 spaces. It develops three complementary frameworks—generating functions, phase functions, and counting functions—to derive explicit, quantitatively sharp lower bounds for A(Λ) in terms of the separation δ(Λ) and associated weight norms, improving previously known estimates. The results yield exponential-in-y bounds in Avdonin-type settings and extend to sine-type zero sets via the Levin–Golovin theory, with concrete implications for perturbations of the integers and MZ families. By connecting these bounds to model spaces, Riesz projections, and A_2 weight theory, the paper both unifies and strengthens the toolkit for analyzing Riesz bases of exponentials on intervals. The findings have potential impact on stability analyses in Fourier-analytic sampling, signal processing, and related harmonic-analytic applications.

Abstract

We revisit Pavlov's characterization for Riesz bases of exponentials and study the corresponding lower Riesz basis bounds. In particular, this approach allows us to improve on known estimates for the bounds in Avdonin's theorem regarding average perturbations, and Levin's theorem regarding zeroes of sine-type functions.

On the lower Riesz basis bound of exponential systems over an interval

TL;DR

This work analyzes the lower Riesz basis bound A(Λ) for exponential systems on an interval by revisiting Pavlov's characterization and linking Riesz-basis properties to the boundedness of the Riesz projection in weighted L^2 spaces. It develops three complementary frameworks—generating functions, phase functions, and counting functions—to derive explicit, quantitatively sharp lower bounds for A(Λ) in terms of the separation δ(Λ) and associated weight norms, improving previously known estimates. The results yield exponential-in-y bounds in Avdonin-type settings and extend to sine-type zero sets via the Levin–Golovin theory, with concrete implications for perturbations of the integers and MZ families. By connecting these bounds to model spaces, Riesz projections, and A_2 weight theory, the paper both unifies and strengthens the toolkit for analyzing Riesz bases of exponentials on intervals. The findings have potential impact on stability analyses in Fourier-analytic sampling, signal processing, and related harmonic-analytic applications.

Abstract

We revisit Pavlov's characterization for Riesz bases of exponentials and study the corresponding lower Riesz basis bounds. In particular, this approach allows us to improve on known estimates for the bounds in Avdonin's theorem regarding average perturbations, and Levin's theorem regarding zeroes of sine-type functions.
Paper Structure (15 sections, 25 theorems, 176 equations)

This paper contains 15 sections, 25 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Riesz bases and the Riesz projection
  3. Riesz bases in the abstract setting
  4. First observations regarding Riesz bases of exponentials
  5. A reformulation in terms of model spaces
  6. A reformulation in terms of the Riesz projection
  7. Norm estimates for the Riesz projection
  8. Lower bound estimates in some Riesz basis characterizations
  9. Riesz bases bounds via generating functions
  10. Riesz bases bounds via phase functions
  11. Riesz bases bounds via counting functions
  12. Riesz basis bounds in some classical results
  13. Perturbations of the integers
  14. Sine-type functions
  15. Finite dimensions

Key Result

Lemma 2.2

Let $\Phi:=\{\phi_n\}_{n\in I}\subseteq H$. Assume that $\Phi$ is a Riesz basis in $S:=\overline{\textrm{span}}\:(\Phi)$. Given a closed subspace $L\subseteq H$ let $P_L:H\rightarrow L$ denote the orthogonal projection of $H$ onto $L$. Then the following are equivalent: Moreover, in this case we have

Theorems & Definitions (59)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition \ref{thm; exp rb and projections}$'$
  • Lemma 2.5
  • ...and 49 more