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Some topics of haar-like spaces of F[a, b]

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1576)

Keywords

  • Analytic Function
  • Differentiable Function
  • Derivative Space
  • Finite Dimensional Subspace
  • Unicity Space

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. E. W. Cheney and D. E. Wulbert, The existence and unicity of best approximations, Math. Scand. 24 (1969), 113–140.

    MathSciNet  MATH  Google Scholar 

  2. G. Gierz and B. Shekhtman, On spaces with large Chebyshev subspaces, J. Approx. Theory 54 (1988), 155–161.

    CrossRef  MathSciNet  MATH  Google Scholar 

  3. A. Haar, Die Minkowskische Geometrie und die Annäherung an stetige Funktionen, Math. Ann. 78(1918), 294–311.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. C. R. Hobby and C. R. Rice, A moment problem in L 1-approximation, Proc. Amer. Math. Soc. 65 (1965), 665–670.

    MathSciNet  MATH  Google Scholar 

  5. D. Jackson, The Theory of Approximation, AMS Vol. XI, Colloq. Publ. Providence, Rhode Island, 1930.

    Google Scholar 

  6. S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics, Wilely-Interscience Publ., New York, 1966.

    MATH  Google Scholar 

  7. K. Kitahara, On Tchebysheff systems, Proc. Amer. Math. Soc. 105(1989), 412–418.

    CrossRef  MathSciNet  MATH  Google Scholar 

  8. K. Kitahara, Best L 1-approximations, 505–512, Progress in Approximation Theory (P. Nevai and A. Pinkus Eds.), Academic Press, New York, 1991.

    Google Scholar 

  9. K. Kitahara, Representations of weak Tchebycheff systems, Num. Func. Anal. Optim. 14(1993), 383–388.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. M. G. Krein, The L-problem in abstract linear normed space, Some Questions in the Theory of Moments (N. Akiezer and M. Krein Eds.), Translations of Mathematical Monographs, Vol. 2, Amer. Math. Soc., Providence, Rhode Island, 1962.

    Google Scholar 

  11. B. R. Kripke and T. J. Rivlin, Approximation in the metric of L 1(X,μ), Trans. Amer. Math. Soc. 119(1965), 101–122.

    MathSciNet  MATH  Google Scholar 

  12. A. Kroó, On an L 1-approximation problem, Proc. Amer. Math. Soc. 94(1985), 406–410.

    CrossRef  MathSciNet  MATH  Google Scholar 

  13. A. Kroó, A general approach to the study of Chebyshev subspaces in L 1 approximation of continuous functions, J. Approx. Theory 51(1987), 98–111.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. A. Kroó, Best L 1-approximation with varying weights, Proc. Amer. Math. Soc. 99(1987), 66–70.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. A. Kroó, D. Schmidt and M. Sommer, On A-spaces and their relation to the Hobby-Rice theorem, J. Approx. Theory 68(1992), 136–154.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. W. Li, Weak Chebyshev subspaces and A-subspaces of C[a, b], Trans. Amer. Math. Soc. 322(1990), 583–591.

    MathSciNet  MATH  Google Scholar 

  17. C. A. Micchelli, Best L 1-approximation by weak Chebyshev systems and the uniqueness of interpolating perfect splines, J. Approx. Theory 19(1977), 1–14.

    CrossRef  MathSciNet  MATH  Google Scholar 

  18. R. R. Phelps, Uniqueness of Hahn-Banach extensions and unique best approximation, Trans. Amer. Math. Soc. 95(1960), 238–255.

    MathSciNet  MATH  Google Scholar 

  19. R. R. Phelps, Cebysev subspaces of finite dimension in L 1, Proc. Amer. Math. Soc. 17(1966), 646–652.

    MathSciNet  MATH  Google Scholar 

  20. A. Pinkus, Unicity subspaces in L 1-approximation, J. Approx. Theory 48 (1986), 226–250.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. A. Pinkus and B. Wajnryb, Necessity conditions for uniqueness in L 1-approximation, J. Approx. Theory 53(1988), 54–66.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. V. Ptak, On approximation of continuous functions in the metric ε b a |x(t)|dt, Czechoslovak Math. J. 8(1958), 267–274.

    MathSciNet  MATH  Google Scholar 

  23. M. A. Rutman, Integral representation of functions forming a Markov series, Soviet Math. Dokl. 164(1965), 1340–1343.

    MathSciNet  MATH  Google Scholar 

  24. D. Schmidt, A theorem on weighted L 1-approximation, Proc. Amer. Math. Soc. 101(1987), 81–84.

    MathSciNet  MATH  Google Scholar 

  25. F. Schwenker, Integral representation of normalized weak Markov systems, J. Approx. Theory 68(1992), 1–24.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. M. Sommer, Approximation in Theorie und Praxis (G. Meinardus, Ed.), Bibliographishes Institut, Mannheim, 1979.

    Google Scholar 

  27. M. Sommer, Weak Chebyshev spaces and best L 1-approximation, J. Approx. Theory 39(1983), 54–71.

    CrossRef  MathSciNet  MATH  Google Scholar 

  28. M. Sommer, Uniqueness of best L 1-approximation of continuous functions, Delay Equations, Approximation and Applications (G. Meinardus and G. Nürenberger, Eds.), Birkhäuser, Basel, 1985.

    Google Scholar 

  29. M. Sommer, Examples of unicity subspaces in L 1-approximation, Num. Func. Anal. Optim. 9(1987), 131–146.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. M. Sommer, Properties of unicity subspaces in L 1-approximation, J. Approx. Theory 52(1988), 269–283.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. H. Strauss, Best L 1-approximation, J. Approx. Theory 41(1984), 297–308.

    CrossRef  MathSciNet  MATH  Google Scholar 

  32. J. W. Young, General theory of approximation by functions involving a given number of arbitrary parameters, Trans. Amer. Math. Soc. 8(1907), 331–344.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. R. A. Zalik, Existence of Tchebycheff extensions, J. Math. Anal. Appl. 51 (1975), 68–75.

    CrossRef  MathSciNet  MATH  Google Scholar 

  34. R. A. Zalik, On transforming a Tchebycheff system into a complete Tchebycheff system, J. Approx. Theory 20(1977), 220–222.

    CrossRef  MathSciNet  MATH  Google Scholar 

  35. R. A. Zalik, Integral representation of Tchebycheff systems, Pacific. J. Math. 68(1977), 553–568.

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. R. A. Zalik, Integral representation and embedding of weak Markov systems, J. Approx. Theory 58(1989), 1–11.

    CrossRef  MathSciNet  MATH  Google Scholar 

  37. R. A. Zalik, Integral representation of Markov systems and the existence of adjoined functions for Haar spaces, J. Approx. Theory 65(1991), 22–31.

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. R. A. Zalik, Nondegeneracy and integral representation of weak Markov systems, J. Approx. Theory 68(1992), 30–42.

    CrossRef  MathSciNet  MATH  Google Scholar 

  39. R. Zielke, On transforming a Tchebyshev-system into a Markov-system, J. Approx. Theory 9(1973), 357–366.

    CrossRef  MathSciNet  MATH  Google Scholar 

  40. R. Zielke, Alternation properties of Tchebyshev-systems and the existence of adjoined functions, J. Approx. Theory 10(1974), 172–184.

    CrossRef  MathSciNet  MATH  Google Scholar 

  41. R. Zielke, Discontinuous Čebyšev Systems, Lecture Notes in Mathematics 707, Springer-Verlag, Berlin-Heidelberg-New York, 1979.

    CrossRef  MATH  Google Scholar 

  42. R. Zielke, Relative differentiability and integral representation of a class of weak Markov systems, J. Approx. Theory 44(1985), 30–42.

    CrossRef  MathSciNet  MATH  Google Scholar 

  43. D. Zwick, Characterization of WT-spaces whose derivatives form a WT-space, J. Approx. Theory 38(1983), 188–191.

    CrossRef  MathSciNet  MATH  Google Scholar 

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© 1994 Springer-Verlag

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Kitahara, K. (1994). Some topics of haar-like spaces of F[a, b]. In: Spaces of Approximating Functions with Haar-like Conditions. Lecture Notes in Mathematics, vol 1576. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0091388

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  • DOI: https://doi.org/10.1007/BFb0091388

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