[Paper Review] Spectral gaps for sets and measures

Alexei Poltoratski|ArXiv.org|Aug 14, 2009

Advanced Harmonic Analysis Research15 references4 citations

TL;DR

This paper establishes a precise characterization of the maximal spectral gap size for non-trivial complex measures supported on a closed subset of the real line. Using a new metric invariant $\mathcal{C}_X$, the authors prove that the supremum of spectral gaps equals $2\pi\mathcal{C}_X$, linking harmonic analysis, Toeplitz operators, and Beurling-Malliavin theory through a duality framework grounded in inner functions and Cauchy transforms.

ABSTRACT

If $X$ is a closed subset of the real line, denote by $\GG_X$ the supremum of the size of the gap in the Fourier spectrum, taken over all non-trivial finite complex measures supported on $X$. In this paper we attempt to find $\GG_X$ in terms of metric properties of $X$.

Motivation & Objective

• To determine the supremum of spectral gap sizes over all non-trivial complex measures supported on a given closed set $X \subset \mathbb{R}$.
• To introduce a new metric invariant $\mathcal{C}_X$ that quantifies this maximal gap size in terms of geometric and analytic properties of $X$.
• To establish an exact duality between the gap problem and the kernel structure of Toeplitz operators via inner functions and Cauchy transforms.
• To unify the gap problem with classical problems in harmonic analysis, such as completeness of exponentials in $L^2(\mu)$ and Bernstein-type approximation.
• To provide a sharp, necessary and sufficient condition for the existence of a measure on $X$ with a given spectral gap size, using the invariant $\mathcal{C}_X$.

Proposed method

• Define $\mathcal{C}_X$ as a metric characteristic combining a density condition and an energy condition, reflecting the geometric and analytic structure of $X$.
• Reformulate the spectral gap problem as a problem on the kernel of Toeplitz operators with symbols involving inner functions $\theta$ and $\phi = \exp(-iax)\theta$.
• Utilize the Beurling-Malliavin theory, particularly the multiplier theorem and pseudocontinuation properties, to analyze the structure of measures and their Fourier transforms.
• Establish a duality between the gap problem and the Wiener-Kolmogorov completeness problem via $L^1$ and $L^2$ duality, linking to approximation by trigonometric polynomials.
• Apply the Clark representation and Cauchy integral theory to analyze the structure of extremal measures, showing they must be singular and supported on discrete sets when $\theta$ is meromorphic.
• Use the theory of $N[\phi]$-classes and outer functions to prove that the kernel of the Toeplitz operator is one-dimensional, implying extremality and uniqueness of the associated measure.

Experimental results

Research questions

• RQ1 What is the maximal possible spectral gap size for a non-trivial complex measure supported on a given closed set $X \subset \mathbb{R}$?
• RQ2 How can this maximal gap size be characterized purely in terms of metric and geometric properties of $X$?
• RQ3 What is the precise relationship between the spectral gap problem and the completeness of exponential systems in $L^2(\mu)$?
• RQ4 How do Toeplitz operator kernels and inner functions encode the extremal structure of measures with maximal spectral gaps?
• RQ5 Under what conditions does a measure on $X$ exist with a prescribed spectral gap, and what is the structure of such measures?

Key findings

• The maximal spectral gap size over all non-trivial complex measures supported on a closed set $X \subset \mathbb{R}$ is exactly $2\pi\mathcal{C}_X$, where $\mathcal{C}_X$ is a new metric invariant defined via density and energy conditions.
• The invariant $\mathcal{C}_X$ is finite if and only if there exists a non-trivial measure on $X$ with a spectral gap of size $2\pi\mathcal{C}_X$.
• Extremal measures achieving the maximal gap are necessarily singular and, when the associated inner function $\theta$ is meromorphic, concentrated on a discrete set.
• The kernel of the relevant Toeplitz operator is one-dimensional, which implies that the extremal measure is unique up to scalar multiplication.
• The Cauchy transform of the extremal measure $\nu$ is divisible by the inner function $\theta$, and $K\nu$ has no zeros in $\mathbb{C} \setminus \text{supp}\,\nu$ beyond the zeros of $\theta$.
• The Clark measure of the inner function $J$ associated with $\nu$ satisfies $\text{spec}_J \subset \text{spec}_I$, and the kernel $N[\bar{J}\theta]$ contains a purely outer function with no real zeros outside $\text{spec}_J$.