Associative and commutative distribution algebra with multipliers, and generalized solutions of nonlinear equations
type: Item
instance of: scholarly article
title: Associative and commutative distribution algebra with multipliers, and generalized solutions of nonlinear equations
zbMATH Open document ID: 0868.46028
DOI: 10.1007/BF02304423
published in: Mathematical Notes
publication date: 1996-05-14T00:00:00Z
review text: The author constructs an associative-commutative algebra \(E^*\) with unit and without divisors of zero generated by the distributions from \[ E=\text{span}\{\delta^{(m-1)}(x- c_k), P(x-c_k)^{-m} x^{m-1}; m=1,2,\dots;c_k\in\mathbb{R}; k=1,2,\dots,s\}\subset{\mathcal S}'. \] If \(S_*\) is the space of test functions of asymptotic form \[ \varphi_*(x)= \sum^\infty_{n=-\infty} \varphi_n(x)y^n,\quad y^n\to 0_+, \] \(\varphi_n\in{\mathcal S}\) with only a finite number of components different from zero (each \(\varphi_*\) can be represented as a vector \(\varphi_*(x)=(\dots,\varphi_{-n}(x),\varphi_{-n-1}(x),\dots);\) \(\varphi_{\pm\infty}=0\)), the elements of \(E^*\) are vector valued distributions on \(S_*\), \(f^*(x)= (\dots,f_n(x),f_{n+1}(x),\dots)\); \(f_n\in E\), and can also be represented in the asymptotic form \(f^*(x)=\sum^\infty_{n=-\infty} f_n(x)y^n\); \(y\to 0_+\), \(f_{-\infty}=0\). On \(E^*\) one defines the derivatives, the antiderivatives, the value at a point, the Fourier transform and, of course, an associative-commutative product which allows to solve problems connected with the product, particularly, to find the general solution of a nonlinear equation as a linear combination of elements from \(E^*\).
Mathematics Subject Classification ID: 46F05
Mathematics Subject Classification ID: 46F10
Mathematics Subject Classification ID: 47J05
zbMATH DE Number: 857382
zbMATH Keywords: algebra of distributions
zbMATH Keywords: multipliers
zbMATH Keywords: generalized solutions of nonlinear equations
zbMATH Keywords: associative-commutative algebra
zbMATH Keywords: derivatives
zbMATH Keywords: antiderivatives
zbMATH Keywords: value at a point
zbMATH Keywords: Fourier transform
zbMATH Keywords: associative-commutative product
author: V. M. Shelkovich
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02304423
OpenAlex ID: W1986487677
description: scientific article; zbMATH DE number 857382