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Beurlings theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations

A. G. Baskakov, N. S. Kaluzhina

Voronezh State University

Abstract: The results of the paper are obtained for functions from homogeneous spaces of functions defined on a locally compact Abelian group. The notion of the 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 $\mathrm{c}$-limit of a linear combination of shifts of the function in question. The notion of a slowly varying function at infinity is introduced, and the properties of such functions are considered. For a parabolic equation with initial function 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.

Keywords: Beurling spectrum of a function, locally compact Abelian group, parabolic equation, continuous unitary character, Banach space, Fourier transform, Banach module, directed set, Stepanov set.

UDC: 517.9

Received: 28.10.2010
Revised: 08.06.2011

DOI: 10.4213/mzm8963

 English version: Mathematical Notes, 2012, 92:5, 587–605

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