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 δacomposable. 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. © 2006 Elsevier Inc. All rights reserved.