Several Asymptotic Products of Particular Distributions

The problem of defining products of distributions is a difficult and not completely understoodproblem, studied from several points of views since Schwartz established the theory of distributionsaround 1950. Many fields, such as wave propagation or quantum mechanics, require suchmultiplications. The product of an infinitely differentiable functionφ(x)and distribution4kδ(x)inRnis well defined by(φ(x)4kδ(x), ψ) = (δ(x),4k(φψ)),since4k(φψ)∈ D(Rn). Using an induction, we derive an interesting formula for4k(φ(x)ψ(x))and hence we are able to write out an explicit expression of the productφ(x)4kδ(x). In particular,we imply the productXs4kδ(x)with a few applications in further simplifying existing distributionalproducts. Furthermore, we obtain an asymptotic expression forδ(r−a)in terms of4kδ(x), which isequivalent to the well-known Pizzetti’s formula. Several asymptotic products includingφ(x)δ(r−1),Xsδ(r−1)as well as the more generalizedφ(x)δ(k)(r−1)are calculated and presented as infinitelyseries.