Let E be a real Hilbert space and A a densely defined linear operator on E satisfying certain conditions. Let E ⊂ E ⊂ E* be the Gel′fand triple arising from E and A. Let μ denote the standard Gaussian measure on E* and let (L2) = L2(μ). The Wiener-Itô decomposition theorem for (L2) and the second quantization operator Γ(A)* can be used to introduce a Gel′fand triple (E) ⊂ (L2) ⊂ (E)*. The elements in (E)* and (E) are called Hida distributions and test functions, respectively. A Hida distribution φ is defined to be finite dimensional if there exists a finite dimensional subspace V of E such that φbelongs to the (E)*-closure of polynomials in〈·, e1〉, 〈·, e2〉., 〈·, ek〉, where the ej′s span V. In this case, δ is said to be based on V. A test function φ is said to be finite dimensional if φ ∈ (E) and there exists a finite dimensional subspace V of E such that φ is based on V. Several characterization theorems for the finite dimensional Hida distributions and test functions are obtained. Approximation theorems of Hida distributions and test functions by finite dimensional Hida distributions and test functions, respectively, are proved. The characterization theorems are based on the Gel′fand triple H(Rk) ⊂ H0(Rk) ⊂ (H*(Rk) arising from the standard Gaussian measure on Rk and the operator e-tL, where L = Δ - ∑kj=1uj ∂/∂uj. Properties and characterizations of elements in H(Rk)(Rk) and H*(Rk) are also obtained. The classical Fourier transform on the space S*(Rk) of tempered distributions is extended to the space H*(Rk). The generalized Itô formula is proved for F(B(t)) with F ∈ H*(Rk). © 1995 Academic Press Limited.
Publication Source (Journal or Book title)
Journal of Functional Analysis
Kubo, I., & Kuo, H. (1995). Finite dimensional hida distributions. Journal of Functional Analysis, 128 (1), 1-47. https://doi.org/10.1006/jfan.1995.1022