Convergence analysis of the finite section method and Banach algebras of matrices (Q989599)

Many applications in science and engineering result in a problem involving linear operator equations. This is usually represented as a system of linear equations of the form \(Ax=b\), where \(A=(a_{kl})_{k,l\in \mathbb{Z}}\) is an infinite matrix and \(b\) belong to some Banach space of sequences. The ``finite section method'' is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework the authors present a convergence analysis of the method for unstructured matrices on weighted \(l^{p}\)-spaces. The finite section of \(A\) is defined to be \(A_{n}: \text{Im} P_{n}AP_{n}\) restricted to \(P_{n}\) i.e. \[ A_{n}:\text{Im} P_{n}\rightarrow \text{Im} P_{n}\subset l^{2}(\mathbb{Z}^{d}). \] The finite system yielding the approximations \(x_{n}\) are given by \[ A_{n}x_{n}=b_{n}, \] here \(b_{n}=P_{n}b\). In the sequel the paper deals with further convergence and stability conditions for the method.