Characterization of Hida Measures in white noise analysis

The main purpose of this work is to prove Theorem 4.4, so-called, the characterization theorem of Hida measures (generalized measures). As examples of such measures, we shall present the Poisson noise measure and the Grey noise measure in Example 4.5 and 4.6, respectively.Comment: Preprint, July 199

Roles of Log-concavity, log-convexity, and growth order in white noise analysis

In this paper we will develop a systematic method to answer the questions $(Q1)(Q2)(Q3)(Q4)$ (stated in Section 1) with complete generality. As a result, we can solve the difficulties $(D1)(D2)$ (discussed in Section 1) without uncertainty. For these purposes we will introduce certain classes of growth functions $u$ and apply the Legendre transform to obtain a sequence which leads to the weight sequence \{\a(n)\} first studied by Cochran et al. \cite{cks}. The notion of (nearly) equivalent functions, (nearly) equivalent sequences and dual Legendre functions will be defined in a very natural way. An application to the growth order of holomorphic functions on \ce_c will also be discussed.Comment: To appear in Infinite Dimensional Analysis, Quantum Probability and Related Topics 4 (2001). Universidade da Madeira CCM preprint 37 (1999

Finite dimensional hida distributions

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