A result on the composition of distributions
type: Item
instance of: scholarly article
title: A result on the composition of distributions
zbMATH Open document ID: 0936.46030
DOI: 10.1007/BF02843535
author: J. D. Nicholas
author: Brian Fisher
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 2000-05-18T00:00:00Z
review text: The authors define \[ f_n(x)= (f*\delta_n)(x)= \langle f(t); \delta_n(x- t)\rangle \] for \(n= 1,2,\dots\), where \(f\) is an arbitrary distribution in \({\mathcal D}'\), and \(\delta(x)\) the delta function. Using the definition \[ N-\lim_{n\to\infty} \int^\infty_{-\infty} F_n(f(x))\phi(x) dx= \langle h(x), \phi(x)\rangle, \] where \(\phi\) is a test function with compact support contained in \((a,b)\), \(f\) a locally summable function, and two theorems by the second author two lemmas are given to establish the main result that, the distribution \((x^r)^{-s}\) exists and \((x^r)^{-s}= x^{-rs}\), \(r\), \(s= 1,2,\dots\)\ . The proof is attractive and very neatly presented to arrive at a statement for the justification for any continuous function as \[ \lim_{n\to\infty} \Biggl|\int^\eta_{n^{-1/r}} x^{rs}[(x^r)^{-s}]_n\psi(x) dx\Biggr|= O(\eta) \] and \[ \lim_{n\to\infty} \Biggl|\int^{-n^{-1/r}}_\eta x^{rs}[(x^r)^{-s}]_n \psi(x) dx\Biggr|= O(\eta). \] Further, for any arbitrary function \(\phi(x)\) in \({\mathcal D}\) with compact support in \([-1,1]\), the authors employ Taylor's theorem with some obvious conditional steps to complete the proof of the theorem. A corollary defines the distribution \((x+i0)^{-s}\) to establish \[ (x^r+ i0)^{-s}= (x+ i0)^{-rs}. \]
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 46F30
zbMATH DE Number: 1348063
zbMATH Keywords: composition of distributions
zbMATH Keywords: neutrix limit
reviewed by: P. K. Banerji
MaRDI profile type: Publication
cites work: Introduction to the neutrix calculus
cites work: Q3308575
cites work: Q3025657
cites work: Q3724972
cites work: Q5724171
full work available at URL: https://doi.org/10.1007/bf02843535
OpenAlex ID: W2066377293
description: scientific article; zbMATH DE number 1348063