Spectrally reasonable measures II

Journal article, 2023

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 [P. Ohrysko and M. Wojciechowski, St. Petersburg Math. J. 28 (2017)]. In particular, we provide a full characterization of such measures for a 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.