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Approximation mit splinefunktionen und quadraturformeln

Part of the Lecture Notes in Mathematics book series (LNM,volume 501)

Abstract

This paper is concerned with the problem of approximating functions in the L1-norm by spline functions with fixed and free knots and its applications to the approximation of linear functionals. For this best L1-approximation characterizations are given which involve perfect splines. In addition, one-sided approximation is studied in more detail. The results are used to give another proof of the existence of a monospline with maximal number of zeros.

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Literatur

  1. Barrodale I.: On computing best L1 approximations. In: Approximation theory (edit. by A. Talbot), London-New York, Academic Press 1970.

    Google Scholar 

  2. Bojanic R. and R. DeVore: On polynomials of best one-sided approximation, L'Enseignement Math. 12 (1966), 139–144.

    MathSciNet  MATH  Google Scholar 

  3. De Boor C.: A remark concerning perfect splines, Bull. Amer. Math. Soc. 80 (1974), 724–727.

    CrossRef  MathSciNet  MATH  Google Scholar 

  4. Braess D.: Chebyshev Approximation by Spline Functions with free knots, Num. Math. 17 (1971), 357–366.

    CrossRef  MathSciNet  MATH  Google Scholar 

  5. Brosowski B.: Nichtlineare Approximation in normierten Vektorräumen, In “Abstract Spaces and Approximation”, ISNM 10, 140–159, Birkhäuser-Verlag, 1970.

    Google Scholar 

  6. Cavaretta A. S., Jr.: On Cardinal Perfect Splines of Least Sup-Norm on the Real Axis, J. Approx. Theory 8 (1973), 285–303.

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. Curry K. B. and I. J. Schoenberg: On spline distributions and their limits: The Polya distribution functions, Bull. Amer. Math. Soc. 53 (1947), 1114.

    Google Scholar 

  8. DeVore R.: One-sided approximation of functions, J. Approx. Theory 1 (1968), 11–25.

    CrossRef  MathSciNet  MATH  Google Scholar 

  9. Elsner L.: Ein Optimierungsproblem der Analysis, Bericht 018 des Instituts für Angewandte Mathematik I, Erlangen.

    Google Scholar 

  10. Favard J.: Sur l'interpolation, J. Math. Pures Appl. (9) 19 (1940), 281–306.

    MathSciNet  MATH  Google Scholar 

  11. Glaeser G.: Prolongement extrémal de fonctions différentiables d'une variable, J. Approx. Theory 8 (1973), 249–261.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. Handscomb D. C.: Characterization of best spline approximations with free knots, In: Approximation Theory (edit. by A. Talbot), London-New York, Academic Press 1970.

    Google Scholar 

  13. Hobby C. R. and J. R. Rice: A moment problem in L1 approximation, Proceed. Amer. Math. Soc., 16 (1965), 665–670.

    MathSciNet  MATH  Google Scholar 

  14. Jetter K.: Splines und optimale Quadraturformeln, Dissertation Tübingen, 1973.

    Google Scholar 

  15. Karlin S.: Total Positivity, Stanford University Press, Stanford, California, 1968

    MATH  Google Scholar 

  16. -: Best quadrature formulas and interpolation by splines satisfying boundary conditions. In: Approximations with special emphasis on spline functions, ed. by I.J. Schoenberg, 447–466. Academic Press, New York, 1969.

    Google Scholar 

  17. -: The fundamental theorem of algebra for monosplines satisfying certain boundary conditions and applications to optimal quadrature formulas. In: Approximations with special emphasis on spline functions, ed. by I.J. Schoenberg, 467–484. Academic Press, New York, 1969.

    Google Scholar 

  18. -: Best quadrature formulas and splines, J. Approx. Theory 4 (1971), 59–90.

    CrossRef  MathSciNet  MATH  Google Scholar 

  19. -: Some variational problems on certain Sobolev spaces and perfect splines, Bull. Amer. Math. Soc., 79 (1973), 124–128.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. Karlin S. and C. Micchelli: The fundamental theorem of algebra for monosplines satisfying boundary conditions, Israel J. Math. 11 (1972), 405–451.

    CrossRef  MathSciNet  MATH  Google Scholar 

  21. Karlin S. and L. Schumaker: The fundamental theorem of algebra for Tchebyscheffian monosplines, J. Analyse Math. 20 (1967), 233–270.

    CrossRef  MathSciNet  MATH  Google Scholar 

  22. Kripke B. R. and T. J. Rivlin: Approximation in the metric L1 (X, μ), Trans. Amer. Math. Soc. 119 (1965), 101–122.

    MathSciNet  MATH  Google Scholar 

  23. Meinardus G.: Approximation von Funktionen und ihre numerische Behandlung, Berlin-Heidelberg-New York, 1964.

    Google Scholar 

  24. Micchelli C.A.: Some minimum problem for spline functions to quadrature formulas. In: Approximation Theory (edit. by G. Lorentz) London-New York, Academic Press, 1973.

    Google Scholar 

  25. -: Best quadrature formulas at equally spaced nodes, J. Math. Anal. Applic., 47 (1974) 232–249.

    CrossRef  MathSciNet  MATH  Google Scholar 

  26. Micchelli C.A. and T.J. Rivlin: Quadrature Formulae and Hermite-Birkhoff Interpolation, Advances in Math. 11 (1973), 93–112.

    CrossRef  MathSciNet  MATH  Google Scholar 

  27. Nörlund N. E.: “Vorlesungen über Differenzenrechnung”, Springer-Verlag, Berlin 1924.

    CrossRef  MATH  Google Scholar 

  28. Rice J. R.: “The Approximation of Functions”, Addison-Wesley, vol. I 1964, vol. II 1969.

    Google Scholar 

  29. Schmeisser, G.: Optimale Quadraturformeln mit semidefiniten Kernen, Num. Math. 20 (1973), 32–53.

    CrossRef  MathSciNet  MATH  Google Scholar 

  30. Schoenberg, I.J.: On best approximations of linear operators, Nederl. Akad. Wetensch. Indag. Math. 26 (1964), 155–163.

    CrossRef  MathSciNet  MATH  Google Scholar 

  31. On monosplines of least deviation and best quadrature formulae, J. Siam Numer. Anal. Ser. B 2 (1965), 144–170.

    MathSciNet  MATH  Google Scholar 

  32. On monosplines of least square deviation and best quadrature formulae II, J. Siam Numer. Anal. 3 (1966), 321–328.

    CrossRef  MathSciNet  MATH  Google Scholar 

  33. Monosplines and quadrature formulae. In: Theory and applications of spline functions, ed. by T.N.E. Greville, New York, Academ. Press 1969, 157–207.

    Google Scholar 

  34. A second look at approximate quadrature formulae and spline interpolation, Advances in Math. 4 (1970), 277–300.

    CrossRef  MathSciNet  MATH  Google Scholar 

  35. The perfect B-splines and a time optimal control problem, Israel J. Math. 8 (1971) 261–275.

    CrossRef  MathSciNet  MATH  Google Scholar 

  36. Schumaker, L.: Uniform Approximation by Tchebycheffian spline functions, J. Math. Mech. 18 (1968) 369–378.

    MathSciNet  MATH  Google Scholar 

  37. Uniform approximation by chebyshev spline functions II: free knots, Siam J. Numer. Anal. 5 (1968), 647–656.

    CrossRef  MathSciNet  MATH  Google Scholar 

  38. Strauss H.: L1-Approximation mit Splinefunktionen, ISNM 26, Birkhäuser-Verlag, 1975.

    Google Scholar 

  39. Strauss H.: Eindeutigkeit bei der gleichmäßigen Approximation mit Tschebyscheffschen Splinefunktionen, erscheint in Journ. Approx. Theory.

    Google Scholar 

  40. Strauss, H.: Nichtlineare L1-Approximation, Bericht 020 des Instituts für Angewandte Mathematik I, Erlangen, 1973.

    Google Scholar 

  41. Strauss H.: Approximation mit Splinefunktionen und Anwendungen auf die Approximation linearer Funktionale, Bericht 023 des Instituts für Angewandte Mathematik I, Erlangen, 1975.

    Google Scholar 

  42. Micchelli C.A. and A. Pinkus: Moment theory for weak chebyshev systems with applications to monosplines, quadratur formulae and best one-sided L1-approximation by spline functions with fixed knots.

    Google Scholar 

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© 1976 Springer-Verlag

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Strauß, H. (1976). Approximation mit splinefunktionen und quadraturformeln. In: Böhmer, K., Meinardus, G., Schempp, W. (eds) Spline Functions. Lecture Notes in Mathematics, vol 501. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0079756

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  • DOI: https://doi.org/10.1007/BFb0079756

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  • Print ISBN: 978-3-540-07543-1

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