Beurling's theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations
type: Item
instance of: scholarly article
title: Beurling's theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations
zbMATH Open document ID: 1264.43008
DOI: 10.1134/S0001434612110016
author: N. S. Kaluzhina
author: Anatoly G. Baskakov
published in: Mathematical Notes
publication date: 2013-04-15T00:00:00Z
review text: The authors consider functions from homogeneous spaces of functions defined on a locally compact Abelian group. The notion of Beurling spectrum, or essential spectrum, of functions is introduced. If a continuous unitary character is an essential point of the spectrum of a function, then it is the c-limit of a linear combination of shifts of the function in question. The notion of a slowly varying function at infinity is introduced, and properties of such functions are considered. The authors consider the Cauchy problem \[ \frac{\partial x}{\partial t}=\Delta x, \qquad x(0,s)=x_0(s), \;s\in {\mathbb R}^n, \] with initial function \(x_0\) from a homogeneous space. It is proved that the weak solution as a function of the first argument is a slowly varying function at infinity.
Mathematics Subject Classification ID: 43A70
Mathematics Subject Classification ID: 14M17
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 6153790
zbMATH Keywords: Beurling spectrum of a function
zbMATH Keywords: locally compact Abelian group
zbMATH Keywords: parabolic equation
zbMATH Keywords: continuous unitary character
zbMATH Keywords: Banach space
zbMATH Keywords: Fourier transform
zbMATH Keywords: Banach module
zbMATH Keywords: directed set
zbMATH Keywords: Stepanov set
reviewed by: Lubomira Softova
MaRDI profile type: Publication
cites work: Q3049314
cites work: Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations
cites work: Representation theory for Banach algebras, Abelian groups, and semigroups in the spectral analysis of linear operators
cites work: Un théorème sur les fonctions bornees et uniformement continues sur l'axe réel
cites work: On the spectral analysis of bounded functions
cites work: Q5615626
cites work: Théorèmes taubériens et théorie spectrale
cites work: Some Results of Narrow Spectral Analysis.
cites work: Q3249967
cites work: Q5336643
cites work: Sur un mode de croissance régulière. Théorèmes fondamentaux
cites work: Regularly varying functions
cites work: Q3848344
cites work: Q5507899
cites work: Stabilization of the solution of the Cauchy problem for parabolic equations
full work available at URL: s0001434612110016
OpenAlex ID: W2155248901
description: scientific article; zbMATH DE number 6153790