Abstract Holomorphic functions f H ({C} I) f ∈ H (C \ I) are represented by an “integral" over the jump of the associated distribution along the branch cut I. The “integral" is the evaluation of the Cauchy kernel in the sense of the duality between the locally convex spaces B₁ ({R}) B 1 (R) and D'₋℉, -₁ ({R}) D L 1, - 1 ′ (R). New Hilbert transforms of distributions are given.
Ortner et al. (Mon,) studied this question.