Frame bound, spectral gap and Plus space

TL;DR

This paper explores the connection between frame bounds and spectral gaps, introduces a new spectral gap concept, and resolves an open question about the spectrality of Plus spaces, advancing understanding in harmonic analysis.

Contribution

It introduces the notion of essential spectral gaps and provides a local characterization of Landau's theorem, solving an open problem on Plus space spectrality.

Findings

Characterization of Landau's theorem via spectral gaps

Resolution of the spectrality of Plus spaces

Introduction of essential spectral minimum and maximum gaps

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}.