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 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.