Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices

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 ℓp-spaces. In particular, the stability of the finite section method on ℓ2 implies its stability on weighted ℓp-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.

[1]  J. Sjöstrand,et al.  Wiener type algebras of pseudodifferential operators , 1995 .

[2]  A. I. Perov.  Estimates for the elements of inverse matrices under the conditions of regularity criteria , 1999 .

[3]  Qiyu Sun,et al.  WIENER’S LEMMA FOR INFINITE MATRICES , 2007 .

[4]  Steffen Roch,et al.  C* - Algebras and Numerical Analysis , 2000 .

[5]  Nathanial P. Brown.  Quasi-diagonality and the finite section method , 2007, Math. Comput..

[6]  B. Silbermann,et al.  Algebras of Approximation Sequences: Finite Sections of Band-Dominated Operators , 2001 .

[7]  K. Grōchenig,et al.  Banach algebras of pseudodifferential operators and their almost diagonalization , 2007, 0710.1989.

[8]  Thomas Strohmer,et al.  QUANTITATIVE ESTIMATES FOR THE FINITE SECTION METHOD , 2006 .

[9]  Thomas Strohmer.  Four short stories about Toeplitz matrix calculations , 2000 .

[10]  William F. Moss,et al.  Decay rates for inverses of band matrices , 1984 .

[11]  Stéphane Jaffard.  Propriétés des matrices « bien localisées » près de leur diagonale et quelques applications , 1990 .

[12]  T. Strohmer.  Approximation of Dual Gabor Frames, Window Decay, and Wireless Communications , 2000, math/0010244.

[13]  I. M. Gelfand,et al.  Commutative Normed Rings , 1968 .

[14]  A. Böttcher,et al.  Introduction to Large Truncated Toeplitz Matrices , 1998 .

[15]  A. Baskakov,et al.  Estimates for the entries of inverse matrices and the spectral analysis of linear operators , 1997 .

[16]  I. Gohberg,et al.  Convolution Equations and Projection Methods for Their Solution , 1974 .

[17]  Israel Gohberg,et al.  Basic Classes of Linear Operators , 2004 .

[18]  Marko Lindner,et al.  Infinite Matrices and their Finite Sections: An Introduction to the Limit Operator Method , 2006 .

[19]  Karlheinz Gröchenig,et al.  Time-Frequency Analysis of Sjöstrand's Class , 2004 .

[20]  Siegfried Prössdorf,et al.  Some classes of singular equations , 1978 .

[21]  William Arveson.  C*-Algebras and Numerical Linear Algebra , 1992 .

[22]  B. Silbermann,et al.  Fredholm theory and finite section method for band-dominated operators , 1998 .

[23]  A. Baskakov,et al.  Wiener's theorem and the asymptotic estimates of the elements of inverse matrices , 1990 .

[24]  Thomas Strohmer,et al.  Pseudodifferential operators and Banach algebras in mobile communications , 2006 .

[25]  Bernd Silbermann,et al.  Analysis of Toeplitz Operators , 1991 .

[26]  Анатолий Григорьевич Баскаков,et al.  Оценки элементов обратных матриц и спектральный анализ линейных операторов@@@Estimates for the entries of inverse matrices and the spectral analysis of linear operators , 1997 .

[27]  Steffen Roch,et al.  Limit Operators And Their Applications In Operator Theory , 2004 .

[28]  Karlheinz Gröchenig,et al.  Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices , 2006 .

[29]  Israel Gohberg,et al.  The band method for positive and strictly contractive extension problems: An alternative version and new applications , 1989 .

[30]  U. G. Kurbatov.  Functional Differential Operators and Equations , 1999 .

[31]  Bernd Silbermann,et al.  The finite section method for TOEPLITZ operators on the quarter‐plane with piecewise continuous symbols , 1983 .

[32]  Thomas Strohmer.  Rates of convergence for the approximation of dual shift-invariant systems in ℓ2(ℤ) , 1999 .

[33]  Symmetry of Matrix Algebras and Symbolic Calculus for Infinite Matrices , 2022 .

[34]  K. Gröchenig,et al.  Wiener's lemma for twisted convolution and Gabor frames , 2003 .

[35]  Karlheinz Gröchenig,et al.  Time-Frequency Analysis of Sj\"ostrand's Class , 2004 .