Let A,ℬ denote two families of
functions a,b : X → Y. A function F : Z ⊆ Y → Y is said to operate in (A,ℬ)
provided that for each a ∈ A with range (a) ⊆ Z we have F(a) ∈ℬ. Let G denote a
locally compact Abelian group. In this paper we characterize the functions which
operate in two cases:
A = Φr(G) = positive definite functions on G with ϕ(e) = r and
ℬ = Oi.d.,s(G) = infinitely divisible positive definite functions on G with
ϕ(e) = s.
