**On Functions of ** w**-bounded Type in the Half-plane**

This paper gives the basic representations of
the general theory of functions of w-bounded type in
the upper half-plane. The starting point are the canonical representations of
some Banach spaces
of holomorphic functions. For p = 2 (i.e. in the case
of Hilbert spaces) there is a theorem on the orthogonal projection from the
corresponding
to
, a Paley - Wiener type
theorem and a theorem on a natural isometry between
and the Hardy space H^{2}, which is an
integral operator along with its inversion. Then the canonical representations
of Nevanlinna - Djrbashian type classes of d-subharmonic functions are given. The functions from the
considered spaces and classes can have arbitrary growth near the finite points
of the real axis.