2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/61232The asymptotic behavior of holomorphic families of generalized eigenfunctions on a reductive symmetric space is studied. The family parameter is a complex character on the split component of a parabolic subgroup. The main result asserts that the family vanishes if a particular asymptotic coefficient does. This allows an induction of relations between families that will be applied in forthcoming work on the Plancherel and the Paley-Wiener theorem.115 pages, LaTeXRepresentation Theory22E30; 22E45text