The convolution x^-r*x^s

Extract

In a recent paper [1], Jones extended the definition of the convolution of distributions so that further convolutions could be defined. The convolution w₁*w₂ of two distributions w₁ and w₂ was defined as the limit ofithe sequence {w_ln*w_2n}, provided the limit w exists in the sense that

for all fine functions ф in the terminology of Jones [2], where

w_1n(x) = w_l(x)τ(x/n), W_2n(x) = w₂(x)τ(x/n)

and τ is an infinitely differentiable function satisfying the following conditions:

(i) τ(x) = τ(—x),

(ii)0 ≤ τ (x) ≤ l,

(iii)τ (x) = l for |x| ≤ ½,

(iv) τ (x) = 0 for |x| ≥ 1.