An Infinite Algebraic System in the Irregular Case

L. G. Arabadzhyan

Matematičeskie zametki, Tome 89 (2011) no. 1, pp. 3-11

Résumé

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_j0, \\ 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.

Détail

Keywords: algebraic system, co-conservative system, Toeplitz matrix, Wiener–Hopf equation.