  In this paper we study spaces of holomorphic functions on the right
half-plane $\cal R$, that we denote by $\cal M^p_\omega$, whose growth
conditions are given in terms of a translation invariant measure $\omega$ on
the closed half-plane $\overline\cal R$. Such a measure has the form
$\omega=\nu\otimes m$, where $m$ is the Lebesgue measure on $\mathbb R$ and
$\nu$ is a regular Borel measure on $[0,+\infty)$. We call these spaces
generalized Hardy-Bergman spaces on the half-plane $\cal R$.
  We study in particular the case of $\nu$ purely atomic, with point masses on
an arithmetic progression on $[0,+\infty)$. We obtain a Paley-Wiener theorem
for $\cal M^2_\omega$, and consequentely the expression for its reproducing
kernel. We study the growth of functions in such space and in particular show
that $\cal M^p_\omega$ contains functions of order 1. Moreover, we prove that
the orthogonal projection from $L^p(\cal R,d\omega)$ into $\cal M^p_\omega$ is
unbounded for $p\neq2$.
  Furthermore, we compare the spaces $\cal M^p_\omega$ with the classical Hardy
and Bergman spaces, and some other Hardy-Bergman-type spaces introduced more
recently.
