Estimates for solutions of bi-infinite systems of linear equations

We consider the problem of estimating in uniform norm solutions of nonhomogeneous difference equations. Difference equations are considered as bi-infinite systems of linear algebraic equations. We give the estimate for Laurent’s matrices with diagonal dominance. Based on this result and an idea of decomposition of the matrix into a product of matrices associated with factorization of the characteristic polynomial, we propose estimates for any nonsingular banded Laurent matrix. The established estimates are attainable.