Abstract
This paper deals with a fundamental property of an approximation sequence which is responsible for the uniformity of certain limiting processes: its fractality. Roughly speaking, a sequence is fractal if the knowledge of any of its infinite subsequences allows to reconstruct the whole sequence up to a sequence tending to zero in the norm. Typical features of a fractal sequence (A n) are the existence of the limit of the condition numbers of the matrices A n and the existence of the limit in the sense of the Hausdorff metric of the spectra, the pseudospectra, and the numerical ranges of the A n. It will be moreover shown that every approximation sequence possesses a fractal subsequence.
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References
W. Arveson, C*-algebras and numerical linear algebra. J. Funct. Anal. 122 (1994), 333–360.
A. Böttcher, Pseudospectra and singular values of large convolution operators. J. Integral Equations Appl. 6 (1994), 3, 267–301.
A. Böttcher, B. Silbermann, Introduction to Large Truncated Toeplitz Matrices. Springer-Verlag, New York, Berlin, Heidelberg 1999.
F. F. Bonsall, J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras. London Math. Soc. Lecture Note Series 2, Cambridge University Press, Cambridge 1971.
F. Chatelin, Spectral Approximation of Linear Operators. Academic Press, New York 1983.
K. E. Gustafson, D. K. M. Rao, Numerical Range. The Field of Values of Linear Operators and Matrices. Springer Verlag, New York 1996.
R. Hagen, S. Roch, B. Silbermann, Spectral Theory of Approximation Methods for Convolution Equations. Birkhäuser Verlag, Basel, Boston, Berlin 1995.
F. Hausdorff, Set Theory. Chelsea Publ. Comp., New York 1957.
H. Landau, The notion of approximate eigenvalues applied to an integral equation of laser theory. Q. Appl. Math., April 1977, 165–171.
B. V. Limaye, Spectral Perturbation and Approximation with Numerical Experiments. Proc. of the Centre for Math. Analysis Vol. 13, Australian National University 1986.
L. Reichel, L. N. Trefethen, Eigenvalues and pseudo-eigenvalues of Toeplitz matrices. Linear Algebra Appl. 162 (1992), 153–185.
S. Roch, Numerical ranges of large Toeplitz matrices. Linear Algebra Appl. 282 (1998), 185–198.
S. Roch, Spectral approximation of Wiener-Hopf operators with almost periodic generating function. Submitted to: Numer. Funct. Anal. Optimization.
S. Roch, Pseudospectra of operator polynomials. Submitted to: Proceedings of the IWOTA 1998, Groningen.
S. Roch, B. Silbermann, Limiting sets of eigenvalues and singular values of Toeplitz matrices. Asymptotic Anal. 8 (1994), 293–309.
S. Roch, B. Silbermann, C*-algebra techniques in numerical analysis. J. Oper. Theory 35 (1996), 2, 241–280.
L. N. Trefethen, Pseudospectra of matrices. — In: D. F. Griffiths, G. A. Watson (Eds.), Numerical Analysis 1991, Longman 1992, 234–266.
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Dedicated to the memory of Siegfried Prössdorf
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Roch, S. (2001). Algebras of approximation sequences: Fractality. In: Elschner, J., Gohberg, I., Silbermann, B. (eds) Problems and Methods in Mathematical Physics. Operator Theory: Advances and Applications, vol 121. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8276-7_25
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DOI: https://doi.org/10.1007/978-3-0348-8276-7_25
Publisher Name: Birkhäuser, Basel
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