The fourier transform in white noise calculus

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Let S* be the space of termpered distributions with standard Gaussian measure μ. Let (S) ⊂ L2(μ) ⊂ (S)* be a Gel'fand triple over the white noise space (S*, μ). The S-transform (Sφ{symbol})(ζ) = ∫S* φ{symbol}(x + ζ) dμ(x), ζ ∈ S, on L2(μ) extends to a U-functional U[φ{symbol}](ζ) = «exp(·, ζ), φ{symbol} a ̊ exp( -∥ζ∥2 2), ζ ∈ S, on (S)*. Let D consist of φ{symbol} in (S)* such that U[φ{symbol}](-iζ1T) exp[-2-1 ∫Tζ(t)2 dt], ζ ∈ S, is a U-functional. The Fourier transform of φ{symbol} in D is defined as the generalized Brownian functional φ{symbol}̌ in (S)* such that U[φ{symbol}̌](ζ) = U[φ{symbol}](-iζ1T) exp[-2-1 ∫Tζ(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 ΔG, the number operator N, and the Volterra Laplacian ΔV are obtained. In particular, (ΔG*φ{symbol})^ = -ΔG*φ{symbol}̌ and [(ΔV + N)φ{symbol}]^ = -(ΔV + N)φ{symbol}̌. Many examples of the Fourier transform are given. © 1989.

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Journal of Multivariate Analysis

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