A Comparison of Two Spaces of Generalized White Noise Functionals

In the literature (see [5, 6, 8]) there are two families of spaces called Kondratiev spaces: (ℵc)±ρ and (Sc)±ρ for 0 ≤ ρ ≤ 1. We investigate the relation between the spaces and show that they are topologically isomorphic when ℵ (ℝd) ⊂ L2 (ℝd) ⊂ ℵ′ (ℝd) is the underlying Gel'fand triple for (ℵc)±ρ. In this case we also give the explicit relation between the S-transform and ℋ-transform on (ℵc)-1 and (Sc)-1, respectively.