A Laplace-like method for solving linear difference equations
type: Item
instance of: scholarly article
title: A Laplace-like method for solving linear difference equations
zbMATH Open document ID: 0545.39001
DOI: 10.1007/BF02192658
author: A. Gameiro Pais
published in: Aequationes Mathematicae
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137011
review text: For a given positive number \(\tau\) let \(I_ n\) be the right-open interval \([n\tau,(n+1)\tau),\) \(n\in {\mathbb{Z}}\). Then for \(t\in {\mathbb{R}}\) let \(t_ k\) be the element of \(I_ k\) such that \(| t_ k-t|\) is an integer multiple of \(\tau\). On the set F of functions f:\({\mathbb{R}}\to {\mathbb{C}}\) which vanish to the left of some point (not the same for all mappings) an operation * (discrete convolution) is defined by \[ (f*g)(t)=\sum^{\infty}_{k=-\infty}f(t_{n-k})g(t_ k),\quad t\in I_ n. \] It is shown that with \(+\) in the obvious meaning \(<F,+,*>\) is a commutative ring with complex operators. One of the most important properties of this ring is that it possesses elements a,b such that for a given function \(f\in F\) (which vanishes for every \(t<0) f(t+\tau)- f(t)=a*f-b*f_ 0\) for every \(t\geq 0\), where \(f_ 0(t)=f(t)\) if \(t\in I_ 0\) and 0 elsewhere. From this the applicability to linear difference equations with constant coefficients can be realized. Thus the difference equation is reducable to an algebraic equation. Two examples are given and the connection with the operational methods of Mikusinski and Fenyö is pointed out.
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 44A40
Mathematics Subject Classification ID: 39A70
zbMATH DE Number: 3867883
zbMATH Keywords: discrete convolution
zbMATH Keywords: ring of functions
zbMATH Keywords: transformation of difference equation to algebraic equation
zbMATH Keywords: linear difference equations with constant coefficients
zbMATH Keywords: Mikusinski
zbMATH Keywords: Fenyö
reviewed by: Dietmar Dorninger
MaRDI profile type: Publication
OpenAlex ID: W1984486200
description: scientific article; zbMATH DE number 3867883