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 Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2017, Issue 1(38), Pages 22–32 (Mi vvgum160)  Mathematics

Harmonic analysis of periodic sequences at infinity

A. A. Ryzhkova

Voronezh State University

Abstract: Let $X$ be a complex Banach space and $\mathrm{End} X$ be a Banach algebra. By $l ^ {\infty} = l ^ {\infty} (Z, X)$ we denote the Banach space of two-sided sequences of vectors in X with the norm
$\|x\|_ {\infty} =\sup \limits_ {n\in \mathbb{Z}} \|x(n)\|$, $X: \mathbb{Z} \rightarrow X$, $x \in l ^ {\infty}$.

By $c_0$ we denote the (closed) subspace of sequences of $l ^ {\infty}$, decreasing at infinity, i.e. $\lim \limits_ {n \rightarrow \infty} \|x (n)\| = 0$.
In the space $l ^ {\infty}$, let us consider the group of operators $S (n): l ^ {\infty } \rightarrow l ^ {\infty}$, $n \in \mathbb{Z}$ where $(S (n) x) (k) = x (k + n)$, $k \in \mathbb{Z}, x \in l ^ {\infty}$.
The sequence $x \in l^{\infty}$ is called slowly varying at infinity if $S(1) x - x \in c_0$, i.e.

$$\lim_{N \rightarrow \infty}\|x (n+1)-x (n)\|= 0.$$

The sequence ${x}$ of ${l} ^ {\infty}$ is called periodic at infinity period $N \geq 1$, $N \in \mathbb{N}$, if ${S} (N) {x} - {x} \in {c}_0$.
An example of a sequence slowly varying at infinity is sequence $x (n) = \sin (\ln (\alpha + n))$, $n \in \mathbb{Z}$, where $\alpha > 0$.
The set of slowly varying at infinity sequences form a closed subspace of $l ^ {\infty}$ which is denoted by $l_ {sl, \infty} ^ {\infty}$.
The set of periodic at infinity period $N$ form a closed subspace of $l ^ {\infty}$, which is denoted by $l_ {N, \infty} ^ {\infty}$. Note that $c_0 \subset {l_ {sl, \infty} ^ {\infty} }\subset l_{N, \infty}^{\infty }$ for any $N \geq 1$.
Suppose that $\gamma_k = e^{\frac {i2 \pi k}{N}}$, $0 \leq k \leq N-1$,—the roots of unity. Note that they form a group, denoted further by $G_N$.
One of the main results is
Theorem 1. Each periodic at infinity sequence $x \in l ^{ \infty}$ period $N \geq 1$ representation of the form

\begin{equation*} x(n)=\sum\limits_{k=0}^{N-1} x_k(n)\gamma_k ^n, \end{equation*}
where $x_k \in l_ {sl, \infty} ^{ \infty}, 0 \leq k \leq N-1$.
In a Banach space $l ^ {\infty} (\mathbb {Z}, X)$, where $X$—finite-dimensional Banach space, consider the difference equation
\begin{equation} X (n + N) = Bx (n) + y (n), n \in \mathbb {Z}, \tag{1} \end{equation}
where $y \in c_0 (\mathbb {Z}, X), B\in \mathrm{End} X$ with the property $\Sigma_0 = \sigma (B) \cap \mathbb {T} =$ $\{\gamma_1, \gamma_2 ..., \gamma_m\}$—set of simple eigenvalues, where $\mathbb{T} = \{\lambda \in \mathbb {C}: |\lambda| = 1\}$ and $\sigma (B)$ denotes the spectrum of the operator $B$.
Theorem 2. Each bounded solution $x: \mathbb {Z} \rightarrow X$ of the equation (1) is a periodic sequence at infinity, which is a representation of the form
$$X (n) = \sum \limits_ {k = 1} ^ {N} x_k (n) \gamma ^ n_k,$$
where $x_k \in l ^ {\infty} _ {sl, \infty}$, $\gamma_k \in \mathbb {T}$, $0 \leq k \leq$ N-1.

Keywords: periodic sequences at infinity, difference equations, eigenvalues, spectral decomposition, projectors.

DOI: https://doi.org/10.15688/jvolsu1.2017.1.3  Full text: PDF file (365 kB) References: PDF file   HTML file

UDC: 517.9
BBK: 22.161

Citation: A. A. Ryzhkova, “Harmonic analysis of periodic sequences at infinity”, Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica, 2017, no. 1(38), 22–32 Citation in format AMSBIB
\Bibitem{Ryz17} \by A.~A.~Ryzhkova \paper Harmonic analysis of periodic sequences at infinity \jour Vestnik Volgogradskogo gosudarstvennogo universiteta. Seriya 1. Mathematica. Physica \yr 2017 \issue 1(38) \pages 22--32 \mathnet{http://mi.mathnet.ru/vvgum160} \crossref{https://doi.org/10.15688/jvolsu1.2017.1.3} 

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