Extension theorems for integral representations of solutions of a functional equation
type: Item
instance of: scholarly article
title: Extension theorems for integral representations of solutions of a functional equation
zbMATH Open document ID: 0704.39004
author: Elsie Sterbin Gottlieb
published in: Elemente der Mathematik
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/141473
review text: Consider: \((1)\quad F(x+1)=g(x)F(x),\) \((2)\quad F(x)=\int^{\infty}_{0}e^{-t}\Phi (t,x-1)dt,\) \((3)\quad \Phi_ t(t,x)=g(x)\Phi (t,x-1)\) and \((4)\quad \Phi (t,-x)=1/\Phi (t,x).\) The classical example where \(F(x)=\Gamma (x),\) \(g(x)=x\) and \(\Phi (t,x)=t^ x\) suggests that solutions of the functional equation (1) may be found among functions of the form (2), where the kernel \(\Phi\) (t,x) satisfies the functional differential equation (3) and the reflection equation (4). The principal result of this paper is the following extension theorem. Let \(g: {\mathbb{R}}_+\to {\mathbb{R}}_+\) be given, and let \(\Phi^*\) be a positive valued function defined on \({\mathbb{R}}_+\times ((-1,0)\cup (0,1)).\) Assume that \((\Phi^*)_ t\) exists and that \(\Phi^*\) satisfies (3) and (4) for \(t>0\) and \(0<x<1.\) Assume that, for each x in (0,1), \((i)\quad \Phi^*(t,x)={\mathcal O}(e^{pt})\) as \(t\to \infty\) for some p depending on x, and \((ii)\quad \Phi^*(t,x)\to 0\) as \(t\to 0^+\). Then there is an extension \(\Phi\) : \({\mathbb{R}}_+\times (- 1,\infty)\) of \(\Phi^*\) which is of the form \(\Phi (t,x)=G(x)t^{\nu (x)}\) and which satisfies (3) for \(t>0\), \(x>0\). Moreover, the function F defined by (2) satisfies (1) for \(x>0\).
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 39B22
zbMATH DE Number: 4155097
zbMATH Keywords: integral representations
zbMATH Keywords: difference equation
zbMATH Keywords: functional equation
zbMATH Keywords: functional differential equation
zbMATH Keywords: reflection equation
zbMATH Keywords: extension
reviewed by: Francis W. Carroll
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4155097