Multiplication product formula for associated homogeneous distributions on R

✍ Scribed by Ghislain R. Franssens

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 191 KB
• Volume: 34
• Category: Article
• ISSN: 0170-4214
• DOI: 10.1002/mma.1455

✦ Synopsis

The set of associated homogeneous distributions (AHDs) on R, H (R), consists of the distributional analogues of powerlog functions with domain in R. This set contains the majority of the (one-dimensional) distributions one typically encounters in physics applications.

The recent work done by the author showed that the set H (R) admits a closed convolution structure (H (R), * ). By combining this structure with the generalized convolution theorem, a distributional multiplication product was defined, resulting in also a closed multiplication structure (H (R),.). In this paper, the general multiplication product formula for this structure is derived. Multiplication of AHDs on R is associative, except for critical triple products. These critical products are shown to be non-associative in a simple and interesting way. The non-associativity is necessary and sufficient to circumvent Schwartz's impossibility theorem on the multiplication of distributions.