Properties of delta functions of a class of observables on white noise functionals

Abstract

Let δ a be the Dirac delta function at a ∈ R and ( E ) ⊂ ( L 2 ) ⊂ ( E ) ∗ the canonical framework of white noise analysis over white noise space ( E ∗ , μ ) , where E ∗ = S ∗ ( R ) . For h ∈ H = L 2 ( R ) with h ≠ 0 , denote by M h the operator of multiplication by W h = 〈 ⋅ , h 〉 in ( L 2 ) . In this paper, we first show that M h is δ a -composable. Thus the delta function δ a ( M h ) makes sense as a generalized operator, i.e. a continuous linear operator from ( E ) to ( E ) ∗ . We then establish a formula showing an intimate connection between δ a ( M h ) as a generalized operator and δ a ( W h ) as a generalized functional. We also obtain the representation of δ a ( M h ) as a series of integral kernel operators. Finally we prove that δ a ( M h ) depends continuously on a ∈ R .