We present a theory of general two-point functions and of generalized
free fields in d-dimensional de Sitter space-time which closely parall
els the corresponding Minkowskian theory. The usual spectral condition
is now replaced by a certain geodesic spectral condition, equivalent
to a precise thermal characterization of the corresponding ''vacuum''
states. Our method is based on the geometry of the complex de Sitter s
pace-time and on the introduction of a class of holomorphic functions
on this manifold, called perikernels, which reproduce mutatis mutandis
the structural properties of the two-point correlation functions of t
he Minkowskian quantum field theory. The theory contains as basic elem
entary case the linear massive field models in their ''preferred'' rep
resentation. The latter are described by the introduction of de Sitter
plane waves in their tube domains which lead to a new integral repres
entation of the two-point functions and to a Fourier-Laplace type tran
sformation on the hyperboloid. The Hilbert space structure of these th
eories is then analysed by using this transformation. In particular we
show the Reeh-Schlieder property. For general two-point functions, a
substitute to the Wick rotation is defined both in complex space-time
and in the complex mass variable, and substantial results concerning t
he derivation of Kallen-Lehmann type representation are obtained.