Let u be a positive continuous function on [0,∞) satisfying the conditions: (i) limr→∞ r−1/2 log u(r) = ∞, (ii) infr≥0 u(r) = 1, (iii) limr→∞ r log u(r) < ∞, (iv) the function logu(x), x ≥ 0, is convex. A Gel’fand triple [E]u ⊂ (L) ⊂ [E]u is constructed by making use of the Legendre transform of u discussed in [4]. We prove a characterization theorem for generalized functions in [E]u and also f...