Let X = G/ K be a Riemannian symmetric space of non compact type and rank-one. The spectral projection P f of a function f on X can be written P f = f * φ where φ is the elementary spherical function corresponding to the complex parameter λ. We characterize the image of the Schwartz space $${\mathcal S^p(X)}$$ under the spectral projection for 0 < p ≤ 2.