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Journal of Applied Analysis

Editor-in-Chief: Fechner, Włodzimierz / Ciesielski, Krzysztof

Managing Editor: Gajek, Leslaw

CiteScore 2018: 0.45

SCImago Journal Rank (SJR) 2018: 0.181
Source Normalized Impact per Paper (SNIP) 2018: 0.845

Mathematical Citation Quotient (MCQ) 2018: 0.20

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Volume 20, Issue 1


Multiplication of the distributions (x±i0)z

Ghislain R. Franssens
Published Online: 2014-04-24 | DOI: https://doi.org/10.1515/jaa-2014-0003


In previous work of the author, a convolution and multiplication product for the set of Associated Homogeneous Distributions (AHDs) with support in ℝ was defined and fully investigated. Here this definition is used to calculate the multiplication product of homogeneous distributions of the form (x±i0)z, for all z. Multiplication products of AHDs generally contain an arbitrary constant if the resulting degree of homogeneity is a negative integer, i.e., if it is a critical product. However, critical products of the forms (x+i0)a.(x+i0)b and (x-i0)a.(x-i0)b, with a+b-, are exceptionally unique. This fact combined with Sokhotskii–Plemelj expressions then leads to linear dependencies of the arbitrary constants occurring in products like δ(k).δ(l), η(k).δ(l), δ(k).η(l) and η(k).η(l) for all k,l (η1πx-1). This in turn gives a unique distribution for products like δ(k).η(l)+η(k).δ(l) and δ(k).δ(l)-η(k).η(l). The latter two products are of interest in quantum field theory and appear for instance in products of the partial derivatives of the zero-mass two-point Wightman distribution.

Keywords: Generalized function; associated homogeneous distribution; multiplication; Wightman distribution; quantum field theory

MSC: 46-02; 46F10; 46F12; 46F30

About the article

Received: 2012-01-26

Revised: 2013-01-08

Accepted: 2013-04-03

Published Online: 2014-04-24

Published in Print: 2014-06-01

Citation Information: Journal of Applied Analysis, Volume 20, Issue 1, Pages 15–27, ISSN (Online) 1869-6082, ISSN (Print) 1425-6908, DOI: https://doi.org/10.1515/jaa-2014-0003.

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