Solution of a functional equation and applications
type: Item
instance of: scholarly article
title: Solution of a functional equation and applications
zbMATH Open document ID: 0804.39008
author: Yahan Yang
published in: Journal of Functional Analysis
publication date: 1995-01-17T00:00:00Z
review text: The problem of the vibration of the cochlea leads to a functional equation fo the form \((*)\) \(f(x + i) = d(x) f(x - i)\), \(x \in\mathbb{R}\), where \(d(x)\) is a given function and \(f\), a function to be found, is analytic in the strip \(-1<y<1\) in the \(z\)-plane, \(z = x + iy\). In this paper the author shows how to solve equation \((*)\) for a large class of functions \(d\) by solving the Dirichlet problem for the Laplace equation in a strip. First the case that \(d(x)\) is positive and satisfies the growth condition \(| \ln d(x) | < M (1 + | x |)^ m\), \(M,m\) constants, is treated and then it is shown how to reduce the case of complex \(d(x)\) to the case when \(d\) is positive. The given solution method is applied to treat \(d\) given by \[ d(x) = \gamma_ 0 \cdot e^{\beta_ 0 i} \cdot | x | \cdot e^{-2 \pi \alpha (x)i}, \] where \(\gamma_ 0\), \(\beta_ 0\) are constants, \(0<\beta_ 0 < 2\pi\), and \(\alpha (x)\) is the characteristic function of the positive \(x\)-axis. Finally, the author shows how the given solution method can be used to construct explicitly a spectral representation for a class of integral operators reminiscent of Wiener-Hopf operators: \(Pv(t) = e^{-t} \int^ \infty_{- \infty} c(t - s) e^{-s} v(s)ds\), where the function \(c\) is the Fourier inverse of \(d\) in \((*)\).
Mathematics Subject Classification ID: 39B32
Mathematics Subject Classification ID: 47B35
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 599813
zbMATH Keywords: vibration
zbMATH Keywords: cochlea
zbMATH Keywords: functional equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: Laplace equation
zbMATH Keywords: growth condition
zbMATH Keywords: spectral representation
zbMATH Keywords: integral operators reminiscent of Wiener-Hopf operators
reviewed by: Pavel Talpalaru
MaRDI profile type: Publication
full work available at URL: jfan.1994.1011
OpenAlex ID: W2089880219
DOI: 10.1006/JFAN.1994.1011
description: scientific article; zbMATH DE number 599813