Let E be a real Hilbert space and A a densely defined linear operator
on E satisfying certain conditions. Let E subset of E subset of E be
the Gel'fand triple arising From E and A. Let mu denote the standard G
aussian measure on E and let (L(2)) = L(2)(mu). The Wiener-Ito decomp
osition theorem for (L(2)) and the second quantization operator Gamma(
A) can be used to introduce a Gel'fand triple (E) subset of (L(2)) su
bset of (E). The elements in (E)* and (E) are called Hida distributio
ns and test functions, respectively. A Hida distribution phi is define
d to be finite dimensional if there exists a finite dimensional subspa
ce V of E such that cp belongs to the (E)-closure of polynomials in [
., e(1)], [., e(2)],..., [., e(k)], where the ej's span V. In this cas
e, delta is said to be based on V. A test function phi is said to be f
inite dimensional if phi is an element of (E) and there exists a finit
e dimensional subspace V of E such that phi is based on V. Several cha
racterization theorems for the finite dimensional Hida distributions a
nd test functions are obtained. Approximation theorems of Hida distrib
utions and test functions by finite dimensional Hida distributions and
test functions, respectively, are proved. The characterization theore
ms are based on the Gel'fand triple H(R(k)) subset of H-0(R(k)) subset
of (H(R(k)) arising from the standard Gaussian measure on R(k) and t
he operator e(-1L), where L = Delta - Sigma(j=1)(k) u(j) partial deriv
ative/partial derivative u(j). Properties anti characterizations of el
ements in H(R(k))(Rk) and H((R(k))) are also obtained. The classical
Fourier transform on the space L(R(k)) of tempered distributions is e
xtended to the space H(R(k)). The generalized Ito formula is proved f
or F(B(t)) with F is an element of H(R(k)). (C) 1995 Academic Press,
Inc.