Spectrally reasonable measures II
Volume 270 / 2023
Studia Mathematica 270 (2023), 285-300
MSC: Primary 43A10; Secondary 43A25.
DOI: 10.4064/sm220704-22-11
Published online: 26 January 2023
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 [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.
