Some Results on the Composition of Distributions

Let F be a distribution, f be a locally summable function and Fₙ = (F * δₙ)(x), where δₙ(x) is a certain sequence converging to the Dirac delta-function δ(x). The distribution $F(f)$ is defined as the neutrix limit of the sequence \ Fₙ(f) \, provided its limit h exists in the sense that ∫-∞∞ Fₙ(f(x))φ (x) dx = h(x) , φ (x) for all test functions φ(x) in D(a,b). The composition of the distributions x₋⁻ˢln x₋ and x₊ʳ is evaluated for r=0,1, … and the composition of the distributions x₋⁻ˢlnᵐ x₋ and $H(x)$ is evaluated for s,m=1,2,…\,. 2000 Mathematics Subject Classification. 46F10