Weighted exponential approximation and non-classical orthogonal spectral measures

TL;DR

This paper investigates the minimal frequency width needed for function approximation in harmonic analysis, characterizes measure perturbations that do not alter this width, and applies findings to spectral measures of Schrödinger operators, answering a longstanding question.

Contribution

It introduces a perturbative analysis of spectral measure stability and demonstrates the absence of local restrictions on orthogonal spectral measures for 1D Schrödinger operators.

Findings

Characterized asymptotic smallness of measure perturbations affecting spectral width

Proved stability of spectral width under certain measure perturbations

Showed no local restrictions on spectral measures of 1D Schrödinger operators

Abstract

A long-standing open problem in harmonic analysis is: given a non-negative measure $\mu$ on $\mathbb R$, find the infimal width of frequencies needed to approximate any function in $L^2(\mu)$. We consider this problem in the "perturbative regime", and characterize asymptotic smallness of perturbations of measures which do not change that infimal width. Then we apply this result to show that there are no local restrictions on the structure of orthogonal spectral measures of one-dimensional Schrodinger operators on a finite interval. This answers a question raised by V.A.Marchenko.