Multiplication product formula for associated homogeneous distributions on R
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The set of associated homogeneous distributions (AHDs) on R, ℋ′(R), consists of the distributional analogues of power-log 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.
CitationFranssens, G.R. (2011). Multiplication product formula for associated homogeneous distributions on R. , Mathematical Methods in the Applied Sciences, Vol. 34, Issue 12, 1460-1471, DOI: 10.1002/mma.1455.