Spectrally reasonable measures

TL;DR

This paper explores measures with natural spectra, introduces spectrally reasonable measures as perturbations, and characterizes which measures have this property, revealing that absolutely continuous measures do, while most discrete measures do not.

Contribution

It defines spectrally reasonable measures, broadens understanding of their properties, and distinguishes between types of measures based on spectral characteristics.

Findings

Absolutely continuous measures are spectrally reasonable.

Most discrete measures are not spectrally reasonable.

The set of spectrally reasonable measures forms a broad class including absolutely continuous ones.

Abstract

In this paper we investigate the problems related to measures with a natural spectrum (equal to the closure of the set of the values of the Fourier-Stieltjes transform). Since it is known that the set of all such measures does not have a Banach algebra structure we consider the set of all suitable perturbations called spectrally reasonable measures. In particular, we exhibit a broad class of spectrally reasonable measures which contains absolutely continuous ones. On the other hand, we show that except trivial cases all discrete (purely atomic) measures do not posses this property.