On the Foiaş and Strătilă theorem

We extend the Foiaş and Strătilă theorem to the case of $L^2$-functions whose spectral measures are continuous and concentrated on an independent Helson set, and to ergodic actions of locally compact second countable abelian groups. We first prove it for functions satisfying Carleman’s condition for the Hamburger moment problem, without the assumption that the spectral measure is supported by a Helson set. Then we show independently that the spectral projector associated with a Helson set preserves each $L^p$-space, with an appropriate bound of the corresponding norm.