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Frame bound, spectral gap and Plus space

Zheng-Yi Lu

Abstract

In this paper, we investigate the relationship between frame bounds and spectral gaps. By introducing the notion of \emph{essential minimum(maximal) spectral gap}, we provide a local characterization of Landau's theorem \cite{Lan67}. As an application, we resolve the spectrality additive measures of Lebesgue type, conclusively answering an open question on the spectrality of Plus spaces originally raised by Lai, Liu, Prince \cite{LLP21} and further studied by Ai, Lu, Zhou \cite{ALZ23} and Kolountzakis, Wu \cite{KW25}.

Frame bound, spectral gap and Plus space

Abstract

In this paper, we investigate the relationship between frame bounds and spectral gaps. By introducing the notion of \emph{essential minimum(maximal) spectral gap}, we provide a local characterization of Landau's theorem \cite{Lan67}. As an application, we resolve the spectrality additive measures of Lebesgue type, conclusively answering an open question on the spectrality of Plus spaces originally raised by Lai, Liu, Prince \cite{LLP21} and further studied by Ai, Lu, Zhou \cite{ALZ23} and Kolountzakis, Wu \cite{KW25}.

Paper Structure

This paper contains 6 sections, 16 theorems, 99 equations.

Table of Contents

  1. Introduction
  2. Frame and spectral gap
  3. Spectral measure and Plus space
  4. Proof of Theorems \ref{['thm1.1']} and \ref{['thm1.3']}
  5. The estimation of the essential maximal spectral gap
  6. Spectrality of the additive measure of Lebesgue type

Key Result

Theorem 1.1

Let $\mu$ be a Borel probability measure on $\Bbb R$. Suppose that there exists $C>0$ such that $|\widehat{\mu}(\xi)|\leq C|\xi|^{-1},\;\forall\xi\in\Bbb R\setminus\{0\}.$ If a countable set $\Lambda\subset\Bbb R$ satisfies then

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 1.6
  • Theorem 1.7
  • proof : Proof of Theorem \ref{['thm1.1']}
  • proof : Proof of Remark 1.2
  • proof : Proof of Theorem \ref{['thm1.3']}
  • ...and 21 more