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

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.