The fourier transform in white noise calculus

Abstract

Let S ∗ be the space of termpered distributions with standard Gaussian measure μ. Let ( S ) ⊂ L 2(μ) ⊂ ( S ) ∗ be a Gel'fand triple over the white noise space ( S ∗, μ). The S-transform ( Sϕ)(ζ) = ∫ S ∗ ϕ( x + ζ) dμ( x), ζ ∈ S , on L 2( μ) extends to a U-functional U[ϕ](ζ) = « exp(·, ζ), ϕ a ̊ exp( −∥ζ∥ 2 2 ) , ζ ∈ S , on ( S ) ∗. Let D consist of ϕ in ( S ) ∗ such that U[ ϕ](− iζ1 T ) exp[−2 −1 ∫ T ζ( t) 2 dt], ζ ∈ S , is a U-functional. The Fourier transform of ϕ in D is defined as the generalized Brownian functional ϕ̌ in ( S ) ∗ such that U[ϕ̌](ζ) = U[ϕ](−iζ1 T) exp[−2 −1 ∫ T ζ( t) 2 dt], ζ ∈ S . Relations between the Fourier transform and the white noise differentiation ∂ t and its adjoint ∂ t ∗ are proved. Results concerning the Fourier transform and the Gross Laplacian Δ G , the number operator N, and the Volterra Laplacian Δ V are obtained. In particular, (Δ G ∗ϕ) ^ = −Δ G ∗ϕ̌ and [(Δ V + N)ϕ] ^ = −(Δ V + N)ϕ̌. Many examples of the Fourier transform are given.