Linear algebraic foundations of the operational calculi
type: Item
instance of: scholarly article
title: Linear algebraic foundations of the operational calculi
zbMATH Open document ID: 1223.44003
author: Luis Verde-Star
author: Gabriel Bengochea
published in: Advances in Applied Mathematics
publication date: 2011-07-08T00:00:00Z
review text: Let \([p_k; k\in\mathbb Z[\) be a group under the operation \(p_kp_n=p_{k+n}\) and let \(\mathfrak F\) be the algebra of formal Laurent series generated by the elements \(p_k\) over the complex numbers. The elements of \(\mathfrak F\) are of the form \(\sum a_kp_k\), where the coefficients \(a_k\) are complex numbers and only a finite number of them with \(k<0\) are nonzero. The left-shift on \(\mathfrak F\) is the operator \(S_{-1}\) that corresponds to the multiplication with \(p_{-1}\). The modified shift \(L\) is defined by \(Lp_k=p_{k-1}\), if \(k\neq 0\) and \(Lp_0=0\). With the operational calculus developed here the authors are finding solutions of \(w(L)f=g\), where \(w\) is a polynomial, \(g\) is a given element in \(\mathfrak F\), and \(f\) is the unknown. If \(g\) can be written as an element of \(\mathfrak F\) , a solution in the abstract space \(\mathfrak F\) can be written as \(d_wg\), where \(d_w\) is the solution of \(w(L)d_w=p_0\) Examples of this method are presented for the case of \(L\) being a difference operator \(\Delta\) that acts on sequences, the case of \(L\) being a differential operator \(a(t)D+b(t)\), and the case of \(L\) being the fractional differential operator \(D^\alpha\).
Mathematics Subject Classification ID: 44A45
Mathematics Subject Classification ID: 44A40
Mathematics Subject Classification ID: 13F25
Mathematics Subject Classification ID: 44A55
Mathematics Subject Classification ID: 26A33
zbMATH DE Number: 5919051
zbMATH Keywords: operational calculus
zbMATH Keywords: convolutions
zbMATH Keywords: formal Laurent series
zbMATH Keywords: Mikusiński calculus
zbMATH Keywords: Heaviside calculus
zbMATH Keywords: difference operator
zbMATH Keywords: differential operator
MaRDI profile type: Publication
full work available at URL: v1
OpenAlex ID: W3097395052
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DOI: 10.1016/J.AAM.2010.08.001
DOI: 10.22541/AU.160435654.47546708/V1
description: scientific article; zbMATH DE number 5919051