General characterization theorems and intrinsic topologies in white noise analysis

الأصل · EN

Let u be a positive continuous function on [0, ∞) satisfying the conditions: (i) ᵣ→∞ r⁻¹/² u(r)=∞, (ii) ᵣ≥ ₀ u(r)=1, (iii) ᵣ→ ∞ r⁻¹ u(r)<∞, (iv) the function u(x²), x≥ 0, is convex. A Gel'fand triple []ᵤ ⊂ (L²) ⊂ []ᵤ* is constructed by making use of the Legendre transform of u discussed in akk3. We prove a characterization theorem for generalized functions in []ᵤ* and also for test functions in []ᵤ in terms of their S-transforms under the same assumptions on u. Moreover, we give an intrinsic topology for the space[]ᵤ of test functions and prove a characterization theorem for measures. We briefly mention the relationship between our method and a recent work by Gannoun et al.ghor. Finally, conditions for carrying out white noise operator theory and Wick products are given.