Convergence analysis of the finite section method and banach algebras of matrices

Integral Equations and Operator Theory

Volumn 67, Issue 2, 2010, Pages 183-202

  Gröchenig, Karlheinz ^a   Rzeszotnik, Ziemowit ^b   Strohmer, Thomas ^c  

^a UNIVERSITY OF VIENNA   (Austria)

^b UNIVERSITY OF WROC AW   (Poland)

^c UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Finite section method; Inverse closed Banach algebras; Spectral invariance; Stability

Indexed keywords

EID: 77953021941     PISSN: 0378620X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00020-010-1775-x     Document Type: Article

References (31)

1. *-algebras and numerical linear algebra. J. Funct. Anal. 122(2), 333-360 (1994).
2. Baskakov A. G.: Wiener's theorem and asymptotic estimates for elements of inverse matrices. Funktsional. Anal. i Prilozhen. 24(3), 64-65 (1990).
3. Baskakov A. G.: Estimates for the elements of inverse matrices, and the spectral analysis of linear operators. Izv. Ross. Akad. Nauk Ser. Mat. 61(6), 3-26 (1997).
4. Böttcher A., Silbermann B.: The finite section method for Toeplitz operators on the quarter-plane with piecewise continuous symbols. Math. Nachr. 110, 279-291 (1983).
5. Böttcher A., Silbermann B.: Analysis of Toeplitz Operators. Springer, Berlin (1990).
6. Böttcher A., Silbermann B.: Introduction to Large Truncated Toeplitz Matrices. Universitext. Springer, New York (1999).
7. Brown, N. P.: Quasi-diagonality and the finite section method. Math. Comp. 76(257), 339-360 (electronic) (2007).
8. Gelfand I., Raikov D., Shilov G.: Commutative Normed Rings (Translated from the Russia). Chelsea Publishing Co., New York (1964).
9. Gohberg, I., Fel'dman, I.: Convolution Equations and Projection Methods for Their Solution, Vol. 41. American Mathematical Society, Providence, R. I., Translated from the Russian by F. M. Goldware, Translations of Mathematical Monographs (1974).
10. Gohberg I., Goldberg S., Kaashoek M.: Basic Classes of Linear Operators. Birkhäuser, Basel (2003).
11. Gohberg I., Kaashoek M. A., Woerdeman H. J.: The band method for positive and strictly contractive extension problems: an alternative version and new applications. Integr. Equ. Oper. Theory 12(3), 343-382 (1989).
12. Gröchenig K.: Time-frequency analysis of Sjöstrand's class. Revista Mat. Iberoam. 22(2), 703-724 (2006).
13. Gröchenig K., Leinert M.: Wiener's lemma for twisted convolution and Gabor frames. J. Am. Math. Soc. 17, 1-18 (2004).
14. Gröchenig K., Leinert M.: Symmetry of matrix algebras and symbolic calculus for infinite matrices. Trans. Am. Math. Soc. 358, 2695-2711 (2006).
15. Gröchenig K., Rzeszotnik Z.: Banach algebras of pseudodifferential operators and their almost diagonalization. Ann. Inst. Fourier (Grenoble) 58(7), 2279-2314 (2008).
16. Gröchenig, K., Rzeszotnik, Z., Strohmer, T.: Quantitative estimates for the finite section method. http://arxiv.org/pdf/math.FA/0610588.
17. *-Algebras and Numerical Analysis, Volume 236 of Monographs and Textbooks in Pure and Applied Mathematics. Marcel Dekker Inc., New York (2001).
18. Jaffard S.: Propriétés des matrices "bien localisées" près de leur diagonale et quelques applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(5), 461-476 (1990).
19. Kurbatov V.: Functional Differential Operators and Equations, volume 473 of Mathematics and Its Applications. Kluwer, Dotrecht (1999).
20. Lindner M.: Infinite Matrices and Their Finite Sections: An Introduction to the Limit Operator Method (Frontiers in Mathematics). Birkhäuser, Basel (2006).
21. Prössdorf, S.: Some Classes of Singular Equations. North Holland, Amsterdam, German original: Birkhäuser, Basel and Stuttgart 1974 (1977).
22. Rabinovich V., Roch S., Silbermann B.: Limit Operators and Their Applications in Operator Theory, Volume 150 of Operator Theory: Advances and Applications. Birkhäuser, Basel (2004).
23. Rabinovich V. S., Roch S., Silbermann B.: Fredholm theory and finite section method for band-dominated operators. Integr. Equ. Oper. Theory 30(4), 452-495 (1998).
24. Rabinovich, V. S., Roch, S., Silbermann, B.: Algebras of approximation sequences: finite sections of band-dominated operators. Acta Appl. Math. 65(1-3), 315-332 (2001) (Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday).
25. Sjöstrand, J.: Wiener type algebras of pseudodifferential operators. In: Séminaire sur les Équations aux Dérivées Partielles, 1994-1995, pages Exp. No. IV, 21. École Polytech., Palaiseau (1995).
26. 2. J. Four. Anal. Appl. 5(6), 599-615 (2000).
27. Strohmer, T.: Four short stories about Toeplitz matrix calculations. Linear Algebra Appl. 343/344, 321-344 (2002) (Special issue on structured and infinite systems of linear equations).
28. Strohmer T.: Approximation of dual Gabor frames, window decay, and wireless communications. Appl. Comput. Harmonic Anal. 11(2), 243-262 (2001).
29. Strohmer T.: Pseudodifferential operators and Banach algebras in mobile communications. Appl. Comput. Harmonic Anal. 20(2), 237-249 (2006).
30. Sun, Q.: Wiener's lemma for infinite matrices. Trans. Am. Math. Soc. 359(7), 3099-3123 (electronic) (2007).
31. Tessera, R.: The Schur algebra is not spectral. Monatsh. Math. (to appear).