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Extensions of Bernstein’s Lethargy Theorem

  • Asuman Güven Aksoy
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 264)

Abstract

In this paper, we examine the aptly-named “Lethargy Theorem” of Bernstein and survey its recent extensions. We show that one of these extensions shrinks the interval for best approximation by half while the other gives a surprising connection to the space of bounded linear operators between two Banach spaces.

Keywords

Best approximation Bernstein’s Lethargy theorem Approximation numbers 

Mathematics Subject Classification (2000)

41A25 41A50 46B20 

References

  1. 1.
    Aksoy, A.G., Al-Ansari, M., Case, C., Peng, Q.: Subspace condition for Bernstein’s lethargy theorem. Turk. J. Math. 41(5), 1101–1107 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aksoy, A.G., Peng, Q.: Constructing an element of a Banach space with given deviation from its nested subspaces. Khayyam J. Math. 4(1), 59–76 (2018)MathSciNetGoogle Scholar
  3. 3.
    Aksoy, A.G., Almira, J.: On Shapiro’s lethargy theorem and some applications. Jaén J. Approx. 6(1), 87–116 (2014)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Aksoy, A.G., Lewicki, G.: Bernstein’s lethargy theorem in Fréchet spaces. J. Approx. Theory 209, 58–77 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Aksoy, A.G., Lewicki, G.: Diagonal operators, \(s\)-numbers and Bernstein pairs. Note Mat. 17, 209–216 (1999)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Albinus, G.: Remarks on a theorem of S. N. Bernstein. Stud. Math. 38, 227–234 (1970)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Almira, J.M., Luther, U.: Compactness and generalized approximation spaces. Numer. Funct. Anal. Optim. 23(1–2), 1–38 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Almira, J.M., Luther, U.: Generalized approximation spaces and applications. Math. Nachr. 263(264), 3–35 (2004)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Almira, J.M., del Toro, N.: Some remarks on negative results in approximation theory. In: Proceedings of the Fourth International Conference on Functional Analysis and Approximation Theory, vol. I (Potenza, 2000). Rend. Circ. Mat. Palermo (2) Suppl. No. 68, Part I, pp. 245–256 (2002)Google Scholar
  10. 10.
    Almira, J.M., Oikhberg, T.: Approximation schemes satisfying Shapiro’s theorem. J. Approx. Theory 164(5), 534–571 (2012)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bernstein, S.N.: On the inverse problem in the theory of best approximation of continuous functions, Collected works (in Russian), Izd. Akad. Nauk, USSR, vol. II, pp. 292–294 (1954)Google Scholar
  12. 12.
    Borodin, P.A.: On the existence of an element with given deviations from an expanding system of subspaces. Math. Notes 80(5), 621–630 (2006). (translated from Mat. Zametki 80(5), 657–667)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Carl, B., Stephani, I.: Entropy, Compactness and the Approximation of Operators. Cambridge University Press, Cambridge (1990)CrossRefGoogle Scholar
  14. 14.
    Deutsch, F., Hundal, H.: A generalization of Tyuriemskih’s lethargy theorem and some applications. Numer. Func. Anal. Optim. 34(9), 1033–1040 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Diestel, J.: Sequences and Series in Banach Spaces. Springer, New York (1984)CrossRefGoogle Scholar
  16. 16.
    Enflo, P.: A counterexample to the approximation property. Acta Math. 13, 308–317 (1973)Google Scholar
  17. 17.
    Hutton, C., Morrell, J.S., Retherford, J.R.: Diagonal operators, approximation numbers and Kolmogorow diameters. J. Approx. Theory 16, 48–80 (1976)CrossRefGoogle Scholar
  18. 18.
    Imomkulov, S.A., Ibragimov, Z.Sh: Uniqueness property for Gonchar quasianalytic functions of several variables. Topics in Several Complex Variables. Contemporary Mathematics, vol. 662, pp. 121–129. American Mathematical Society, Providence (2016)CrossRefGoogle Scholar
  19. 19.
    Kaiser, R., Retherford, J.: Eigenvalue distribution of nuclear operators: a survey. Vector Measures and Integral Representation of Operators, pp. 245–287. Essen University Press, Essen (1983)zbMATHGoogle Scholar
  20. 20.
    Kalton, N.J., Peck, N.T., Roberts, J.W.: An F-space Sampler. London Mathematical Society Lecture Notes Series. Cambridge University Press, Cambridge (1984)CrossRefGoogle Scholar
  21. 21.
    König, H.: A formula for the eigenvalues of a compact operator. Stud. Math. 65, 141–146 (1979)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Konyagin, S.V.: Deviation of elements of a Banach space from a system of subspaces. Proc. Steklov Inst. Math. 284(1), 204–207 (2014)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Lewicki, G.: Bernstein’s “lethargy” theorem in metrizable topological linear spaces. Monatsh. Math. 113, 213–226 (1992)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lewicki, G.: A theorem of Bernstein’s type for linear projections, Univ. lagel. Acta Math. 27, 23–27 (1988)zbMATHGoogle Scholar
  25. 25.
    Marcus, A.S.: Some criteria for the completeness of a system of root vectors of a linear operator in a Banach space. Trans. Am. Math. Soc. 85, 325–349 (1969)Google Scholar
  26. 26.
    Micherda, B.: Bernstein’s lethargy theorem in SF-spaces. Z. Anal. Anwendungen 22(1), 3–16 (2003)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Oikhberg, T.: Rate of decay of s-numbers. J. Approx. Theory 163, 311–327 (2011)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Pietch, A.: Eigenvalues and s-Numbers. Cambridge University Press, Cambridge (1987)Google Scholar
  29. 29.
    Pietch, A.: Operator Ideals. North Holland, Amsterdam (1980)Google Scholar
  30. 30.
    Pietch, A.: History of Banach Spaces and Linear Operators. Boston, Birkhauser (2007)Google Scholar
  31. 31.
    Pinkus, A.: Weierstrass and approximation theory. J. Approx. Theory 107(1), 1–66 (2000)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Pleśniak, W.: On a theorem of S. N. Bernstein in \(F\)-spaces. Zeszyty Naukowe Uniwersytetu Jagiellonskiego, Prace Mat. 20, 7–16 (1979)MathSciNetzbMATHGoogle Scholar
  33. 33.
    Pleśniak, W.: Quasianalytic functions in the sense of Bernstein. Diss. Math. 147, 1–70 (1977)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Pleśniak, W.: Characterization of quasi-analytic functions of several variables by means of rational approximation. Ann. Pol. Math. 27, 149–157 (1973)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Rolewicz, S.: Metric Linear Spaces. PWN, Warszawa (1982)zbMATHGoogle Scholar
  36. 36.
    Shapiro, H.S.: Some negative theorems of approximation theory. Mich. Math. J. 11, 211–217 (1964)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer, Berlin (1970)CrossRefGoogle Scholar
  38. 38.
    Timan, A.F.: Theory of Approximation of Functions of a Real Variable. Dover publications, New York (1994)Google Scholar
  39. 39.
    Tyuriemskih, I.S.: On one problem of S. N. Bernstein. In: Scientific Proceedings of Kaliningrad State Pedagogical Institute, vol. 52, pp. 123–129 (1967)Google Scholar
  40. 40.
    Tyuriemskih, I.S.: The \(B\)-property of hilbert spaces. Uch. Zap. Kalinin. Gos. Pedagog. Inst. 39, 53–64 (1964)MathSciNetGoogle Scholar
  41. 41.
    Vasil’ev, A.I.: The inverse problem in the theory of best approximation in \(F\)-spaces, (In Russian). Dokl. Ross. Akad. Nauk. 365(5), 583–585 (1999)zbMATHGoogle Scholar
  42. 42.
    Weyl, H.: Inequalities between two kinds of eigenvalues of a linear trasformation. Proc. Natl. Acad. Sci. USA 35, 408–411 (1949)CrossRefGoogle Scholar
  43. 43.
    Wojtaszczyk, P.: Banach Spaces for Analysts. Studies in Advanced Mathematics, vol. 25. Cambridge University Press, Cambridge (1991)CrossRefGoogle Scholar

Copyright information

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Authors and Affiliations

  1. 1.Department of Mathematical SciencesClaremont McKenna CollegeClaremontUSA

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