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 Mathematical Physics and Computer Simulation: Year: Volume: Issue: Page: Find

 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
$$X (n + N) = Bx (n) + y (n), n \in \mathbb {Z}, \tag{1}$$
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

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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}