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ASYMPTOTIC PROPERTIES OF KOLMOGOROV WIDTHS

Part of: Approximations and expansions Normed linear spaces and Banach spaces; Banach lattices

Published online by Cambridge University Press:  25 March 2010

MIKHAIL I. OSTROVSKII
MIKHAIL I. OSTROVSKII*
Affiliation:
Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, NY 11439, USA (email: [email protected])
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Abstract

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We consider two problems concerning Kolmogorov widths of compacts in Banach spaces. The first problem is devoted to relations between the asymptotic behavior of the sequence of n-widths of a compact and of its projections onto a subspace of codimension one. The second problem is devoted to comparison of the sequence of n-widths of a compact in a Banach space 𝒴 and of the sequence of n-widths of the same compact in other Banach spaces containing 𝒴 as a subspace.

Keywords

absolute widthsBanach spaceKolmogorov widths

MSC classification

Secondary: 46B20: Geometry and structure of normed linear spaces 41A65: Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 82 , Issue 1 , August 2010 , pp. 10 - 17
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

References

[1]Hutton, C. V., ‘On the approximation numbers of an operator and its adjoint’, Math. Ann. 210 (1974), 277–280.CrossRefGoogle Scholar
[2]Ismagilov, R. S., ‘Diameters of sets in normed linear spaces, and the approximation of functions by trigonometric polynomials’, Uspekhi Mat. Nauk 29(3(177)) (1974), 161–178 (in Russian); Russian Math. Surveys 29(3) (1974), 169–186 (English translation).Google Scholar
[3]Johnson, W. B. and Lindenstrauss, J., ‘Basic concepts in the geometry of Banach spaces’, in: Handbook of the Geometry of Banach Spaces, Vol. 1 (eds. Johnson, W. B. and Lindenstrauss, J.) (Elsevier, Amsterdam, 2001), pp. 1–84.CrossRefGoogle Scholar
[4]Kashin, B. S., ‘The widths of certain finite-dimensional sets and classes of smooth functions’, Izv. Akad. Nauk SSSR Ser. Mat. 41(2) (1977), 334–351 (in Russian); Math. USSR-Izv. 11(2) (1977), 317–333 (1978) (English translation).Google Scholar
[5]Kochurov, A. S., ‘Absolute widths and cowidths’, Mat. Zametki 48(1) (1990), 38–46 (in Russian); Math. Notes 48 (1990), 647–652 (English translation).Google Scholar
[6]Kolmogoroff, A., ‘Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse’, Ann. of Math. (2) 37 (1936), 107–111.CrossRefGoogle Scholar
[7]Lorentz, G. G., Golitschek, M. v. and Makovoz, Y., Constructive Approximation. Advanced Problems, Grundlehren der mathematischen Wissenschaften, 304 (Springer, Berlin, 1996).CrossRefGoogle Scholar
[8]Oikhberg, T., ‘Absolute widths of some embeddings’, J. Approx. Theory 81(1) (1995), 120–126.CrossRefGoogle Scholar
[9]Ostrovskii, M. I. and Shulman, V. S., ‘Weak operator topology, operator ranges and operator equations via Kolmogorov widths’, Integral Equations Operator Theory 65 (2009), 551–572.CrossRefGoogle Scholar
[10]Pełczyński, A., ‘Projections in certain Banach spaces’, Studia Math. 19 (1960), 209–228.CrossRefGoogle Scholar
[11]Pinkus, A., n-Widths in the Approximation Theory (Springer, Berlin, 1985).CrossRefGoogle Scholar
[12]Pisier, G., The Volume of Convex Bodies and Banach Space Geometry, Cambridge Tracts in Mathematics, 94 (Cambridge University Press, Cambridge, 1989).CrossRefGoogle Scholar
[13]Szarek, S. J., ‘On Kashin’s almost Euclidean orthogonal decomposition of ℓ1n’, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26(8) (1978), 691–694.Google Scholar
[14]Tikhomirov, V. M., ‘Diameters of sets in functional spaces and the theory of best approximations’, Uspekhi Mat. Nauk 15(3) (1960), 81–120 (in Russian); Russian Math. Surveys, 15(3) (1960), 75–111 (English translation).Google Scholar