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

Let δa be the Dirac delta function at a∈R and (E)⊂(L2)⊂∗(E) the canonical framework of white noise analysis over white noise space (E∗,μ), where E∗=S∗(R). For h∈H=L2(R) with h≠0, denote by Mh the operator of multiplication by Wh=〈⋅,h〉 in (L2). In this paper, we first show that Mh is δa-composable. Thus the delta function δa(Mh) 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 δa(Mh) as a generalized operator and δa(Wh) as a generalized functional. We also obtain the representation of δa(Mh) as a series of integral kernel operators. Finally we prove that δa(Mh) depends continuously on a∈R.