  In this paper we give two complete characterizations of the Poletsky-
Stessin- Hardy spaces in the complex plane: First in terms of their boundary
values as a weighted subclass of the usual $L^p$ class with respect to the
arclength measure on the boundary. Second we completely describe functions in
these spaces by having a harmonic majorant with a certain growth condition and
we prove some basic results about these spaces. In particular, we prove
approximation results in such spaces and extend the classical result of
Beurling which describes the invariant subspaces of the shift operator.
Additionally we provide non-trivial examples.
