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This article is cited in 3 scientific papers (total in 3 papers)
An Infinite Algebraic System in the Irregular Case
L. G. Arabadzhyanab a Institute of Mathematics, National Academy of Sciences of Armenia
b Armenian State Teachers' Training University named after Khachatur Abovian
Abstract:
We obtain sufficient conditions for the nontrivial solvability of systems of the form
$$
\varphi_i=b_i+\lambda_i\sum^\infty_{j=0} a_{i-j}\varphi_j,\qquad
i\in\mathbb Z_+\overset{\text{def}}{=}\{0,1,2,\dots,n,\ldots\},
$$
and of the corresponding homogeneous systems. It is assumed that the sequences $b=(b_0,b_1,b_2,\ldots)$ and $\lambda=(\lambda_0,\lambda_1,\lambda_2,\ldots)$ and the Toeplitz matrix $A=(a_{i-j})$ satisfy the conditions
\begin{gather*}
a_j\ge 0,\quad j\in {\mathbb Z},\qquad \sum^\infty_{j=-\infty}a_j=1,\qquad \sum^\infty_{j=-\infty}|j|a_j<\infty,\qquad \sum^\infty_{j=-\infty}ja_j<0,
\\
b_j\ge 0,\quad j\in {\mathbb Z}_+,\qquad \sum^\infty_{j=0}b_j<\infty,\qquad 1\le\lambda_i\le
\biggl(\,\sum^i_{j=-\infty}a_j\biggr)^{-1},\quad i\in {\mathbb Z_+}.
\end{gather*}
Under these conditions, we construct bounded solutions of homogeneous and inhomogeneous systems of the form indicated above.
Keywords:
algebraic system, co-conservative system, Toeplitz matrix, Wiener–Hopf equation.
Received: 12.11.2008
Citation:
L. G. Arabadzhyan, “An Infinite Algebraic System in the Irregular Case”, Mat. Zametki, 89:1 (2011), 3–11; Math. Notes, 89:1 (2011), 1–10
Linking options:
https://www.mathnet.ru/eng/mzm6578https://doi.org/10.4213/mzm6578 https://www.mathnet.ru/eng/mzm/v89/i1/p3
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