Convolution product formula for associated homogeneous distributions on R

The set of Associated Homogeneous Distributions (AHDs) on R, ??′(R), consists of distributional analogues of power‐log functions with domain in R. This set contains the majority of the (one‐dimensional) distributions typically encountered in physics applications. In earlier work of the author it was shown that ??′(R) admits a closed convolution structure, provided that critical convolution products are defined by a functional extension process. In this paper, the general convolution product formula is derived. Convolution of AHDs on R is found to be associative, except for critical triple products. Critical products are shown to be non‐associative in a minimal and interesting way. Copyright © 2010 John Wiley & Sons, Ltd.