Harmonic analysis of periodic sequences at infinity

Let be a complex Banach space and End𝑋 be a Banach algebra. By 𝑙∞ = 𝑙∞(𝑍,𝑋) we denote the Banach space of two-sided sequences of vectors in X with the norm ‖𝑥‖∞ = sup 𝑛∈Z ‖𝑥(𝑛)‖, : Z → 𝑋, ∈ 𝑙∞. By 𝑐0 we denote the (closed) subspace of sequences of 𝑙∞, decreasing at infinity, i.e. lim 𝑛→∞‖𝑥(𝑛)‖ = 0. In the space 𝑙∞, let us consider the group of operators 𝑆(𝑛) : 𝑙∞ → 𝑙∞, ∈ Z where (𝑆(𝑛)𝑥)(𝑘) = 𝑥(𝑘 + 𝑛), ∈ Z, ∈ 𝑙∞. The sequence ∈ 𝑙∞ is called slowly varying at infinity if S(1)x-x ∈ c0, i.e. N→∞‖x(n + 1) - x(n)‖ = 0. The sequence x of l∞ is called periodic at infinity period ≥ 1, ∈ N, if S(𝑁)x - x ∈ c0. An example of a sequence slowly varying at infinity is sequence 𝑥(𝑛) = = sin(ln( + 𝑛)), ∈ Z, where > 0. The set of slowly varying at infinity sequences form a closed subspace of 𝑙∞ which is denoted by 𝑙∞𝑠𝑙,∞. The set of periodic at infinity period form a closed subspace of 𝑙∞, which is denoted by 𝑙∞𝑁,∞. Note that 𝑐0 ⊂ 𝑙∞𝑠𝑙,∞ ⊂ 𝑙∞𝑁,∞ for any ≥ 1. Suppose that = 𝑖2 𝑁, 0 ≤ ≤ - 1, - the roots of unity. Note that they form a group, denoted further by 𝐺𝑁. One of the main results is Theorem 1. Each periodic at infinity sequence ∈ 𝑙∞ period ≥ 1 representation of the form 𝑥(𝑛) = 𝑁-1 Σ︁𝑘=0 𝑥𝑘(𝑛) 𝑛𝑘, where ∈ 𝑙∞𝑠𝑙,∞, 0 ≤ ≤ - 1. In a Banach space 𝑙∞(Z,𝑋), where - finite-dimensional Banach space, consider the difference equation 𝑋(𝑛 + 𝑁) = 𝐵𝑥(𝑛) + 𝑦(𝑛), ∈ Z, (1) where ∈ 𝑐0(Z,𝑋),𝐵 ∈ End𝑋 with the property Σ0 = (𝐵) ∩ T = { 1, 2..., 𝑚} - set of simple eigenvalues, where T = { ∈ C : | | = 1} and (𝐵) denotes the spectrum of the operator 𝐵. Theorem 2. Each bounded solution : Z → of the equation (1) is a periodic sequence at infinity, which is a representation of the form 𝑋(𝑛) = Σ︁𝑘=1 𝑥𝑘(𝑛) 𝑛𝑘, where ∈ 𝑙∞𝑠𝑙,∞, ∈ T, 0 ≤ ≤ N-1.