Harmonic analysis of functions in homogeneous spaces and harmonic distributions that are periodic or almost periodic at infinity

Abstract

Vector-valued functions in homogeneous spaces and harmonic distributions that are periodic or almost periodic at infinity are investigated. The concept of the Fourier series of a function (distribution), periodic or almost periodic at infinity, with coefficients that are functions (distributions) slowly varying at infinity, is introduced. The properties of the Fourier series are investigated and an analogue of Wiener’s theorem on absolutely convergent Fourier series is obtained for functions periodic at infinity. Special attention is given to criteria ensuring that solutions of differential or difference equations are periodic or almost periodic at infinity. The central results involve theorems on the asymptotic behaviour of a bounded operator semigroup whose generator has no limit points on the imaginary axis. In addition, the concept of an asymptotically finite-dimensional operator semigroup is introduced and a theorem on the structure of such a semigroup is proved. Bibliography: 39 titles.

Method of similar operators in the problem of bi-invariant subspaces

“…The space of F -almost-periodic vectors from X will be denoted by AP F X and the space of Fperiodic vectors of period ω -by P F ω (X ). We note that periodic and almost periodic at infinity functions that were considered, e. g., in [35,36,38] are examples of X c -(almost-)periodic functions in the module (X , T ), where X = L ∞ (R, X ) and T = M as in Example 2.4.…”

“…It is not hard to see that if F is a closed submodule of M, then the spaces AP F M and P F ω (M), ω > 0, are also A-harmonious spaces and Banach algebras (see Definition 2.20). An operator V ∈ M = M 0 = End L 2 from Example 3.3 is M c -almost-periodic if the function v is almost periodic at infinity [35,36,38].…”

“…Spectral invariance results such as the one above have been used extensively in various areas of analysis and applications. For example, in differential equations, such results lead to existence of solutions in a certain class [17,25,35, and references therein], whereas in frame theory, spectral invariance is the key tool for studying localization of dual frames [2, 3, 4, 5, 55, 81, and references therein].…”