# Spectrally reasonable measures II

## TL;DR

This paper advances the understanding of spectrally reasonable measures on certain locally compact Abelian groups, providing a full characterization and exploring spectral properties of measures with non-natural spectra.

## Contribution

It offers a complete characterization of spectrally reasonable measures for specific groups and examines measures with non-natural spectra.

## Key findings

- Characterization of spectrally reasonable measures on the circle and real line
- Analysis of spectral properties of measures with non-natural spectra
- Extension of previous work on measures with natural spectra

## Abstract

A measure on a locally compact Abelian group is said to have a natural spectrum if its spectrum is equal to the closure of the range of the Fourier-Stieltjes transform. In this paper we continue the study of spectrally reasonable measures (measures perturbing any measure with a natural spectrum to a measure with a natural spectrum) initiated in \cite{ow}. Particularly, we provide a full characterization of such measures for certain class of locally compact Abelian groups which includes the circle and the real line. We also elaborate on the spectral properties of measures with non-natural but real spectra constructed by F. Parreau.