Duality questions for operators, spectrum and measures

TL;DR

This paper investigates spectral duality for measures in Euclidean space, connecting differential operators, tiling properties, and affine systems, and reviews historical and recent developments in the field.

Contribution

It provides a unified framework for spectral duality of Borel measures, linking classical tiling problems with modern spectral theory and iterated function systems.

Findings

Review of spectral duality results from 1974 to present

Connection between tiling properties and orthogonal exponential bases

Framework for analyzing pairs of measures in spectral duality

Abstract

We explore spectral duality in the context of measures in $\br^n$, starting with partial differential operators and Fuglede's question (1974) about the relationship between orthogonal bases of complex exponentials in $L^2(\Omega)$ and tiling properties of $\Omega$, then continuing with affine iterated function systems. We review results in the literature from 1974 up to the present, and we relate them to a general framework for spectral duality for pairs of Borel measures in $\br^n$, formulated first by Jorgensen and Pedersen.