Abstract
The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted ℓ p-spaces. In particular, the stability of the finite section method on ℓ 2 implies its stability on weighted ℓ p-spaces. Our approach uses recent results from the theory of Banach algebras of matrices with off-diagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of unstructured non-hermitian matrices as well as to least squares problems.
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K.G. and Z.R. were supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154, T.S. was supported by NSF DMS Grant 0511461.
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Gröchenig, K., Rzeszotnik, Z. & Strohmer, T. Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices. Integr. Equ. Oper. Theory 67, 183–202 (2010). https://doi.org/10.1007/s00020-010-1775-x
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DOI: https://doi.org/10.1007/s00020-010-1775-x