Unitary analytic representations of the conformal group are relized on Hilbert spaces of holomoprhic or antiholomorphic functions over a tube domain in complex Minkowski space. The distributional boundary values of these functions are tempered distributions on real Minkowski space. The representations are characterized by an integral scale dimension labeln and two spin labelsj ₁ andj ₂. The connection between the dimensionn and the degree of singularity of the tempered distribution is investigated. We propose an application to inclusive reactions of elementary particles.