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More options … # 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

Online
ISSN
1869-6082
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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

## Abstract.

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 $z\in ℂ$. 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 ${\left(x+i0\right)}^{a}.{\left(x+i0\right)}^{b}$ and ${\left(x-i0\right)}^{a}.{\left(x-i0\right)}^{b}$, with $a+b\in {ℤ}_{-}$, are exceptionally unique. This fact combined with Sokhotskii–Plemelj expressions then leads to linear dependencies of the arbitrary constants occurring in products like ${\delta }^{\left(k\right)}.{\delta }^{\left(l\right)}$, ${\eta }^{\left(k\right)}.{\delta }^{\left(l\right)}$, ${\delta }^{\left(k\right)}.{\eta }^{\left(l\right)}$ and ${\eta }^{\left(k\right)}.{\eta }^{\left(l\right)}$ for all $k,l\in ℕ$ ($\eta \triangleq \frac{1}{\pi }{x}^{-1}$). This in turn gives a unique distribution for products like ${\delta }^{\left(k\right)}.{\eta }^{\left(l\right)}+{\eta }^{\left(k\right)}.{\delta }^{\left(l\right)}$ and ${\delta }^{\left(k\right)}.{\delta }^{\left(l\right)}-{\eta }^{\left(k\right)}.{\eta }^{\left(l\right)}$. 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.

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

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,

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© 2014 by Walter de Gruyter Berlin/Boston. 