A covariant phase space observable is uniquely characterized by a positive operator of trace one and, in turn, by the Fourier-Weyl transform of this operator. We study three properties of such observables, and characterize them in terms of the zero s
et of this transform. The first is informational completeness, for which it is necessary and sufficient that the zero set has dense complement. The second is a version of informational completeness for the Hilbert-Schmidt class, equivalent to the zero set being of measure zero, and the third, known as regularity, is equivalent to the zero set being empty. We give examples demonstrating that all three conditions are distinct. The three conditions are the special cases for $p=1,2,infty$ of a more general notion of $p$-regularity defined as the norm density of the span of translates of the operator in the Schatten-$p$ class. We show that the relation between zero sets and $p$-regularity can be mapped completely to the corresponding relation for functions in classical harmonic analysis.
The moment operators of a semispectral measure having the structure of the convolution of a positive measure and a semispectral measure are studied, with paying attention to the natural domains of these unbounded operators. The results are then appli
ed to conveniently determine the moment operators of the Cartesian margins of the phase space observables.
