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Transfinite Interpolation by Blending Functions, Best One-Sided L1-Approximation, and Cubature Formulae

  • Dimiter P. Dryanov
Conference paper
Part of the International Series of Numerical Mathematics book series (ISNM, volume 145)

Abstract

The main purpose of this paper is to present results on blending—type interpolastion, best multivariate L1—approximation, and cubature formulae for approximate integration of multiple integrals. Since the classical concept of the Haar—Chebyshev condition is essentially restricted to the univariate case we try to give a new treatment of the notion for a HaarChebyshev system in the multivariate case.

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References

  1. 1.
    N. I. Achieser: Theory of Approximation, Frederic Ungar, New York 1956.zbMATHGoogle Scholar
  2. 2.
    D. H. Armitage, S. J. Gardiner: Best one-sided L 1approximation by harmonic and subharmonic functions, In: Advances in Multivariate Approximation,Mathematical Research Vol.107 43–56, Wiley-VCH, Berlin 1999.Google Scholar
  3. 3.
    D. H. Armitage, S. J. Gardiner, W. Haussmann, L. Rogge: Characterization of best harmonic and superharmonic L 1 approximants, J. Reine Angew. Math. 478 (1996), 1–15.zbMATHMathSciNetGoogle Scholar
  4. 4.
    R. E. Barnhill, R. F. Riesenfeld: Computer Aided Geometric Design, Proceedings of a Conference held at the University of Utah, Academic Press, New York 1974.zbMATHGoogle Scholar
  5. 5.
    G. Birkhoff, R. E. Lynch: Numerical Solution of Elliptic Problems, SIAM Studies in Applied Mathematics Vol. 6, Society for Industrial and Applied Mathematics (SIAM), Philadelphia 1984.CrossRefGoogle Scholar
  6. 6.
    B. D. Bojanov, H. A. Hakopian, A. A. Sahakian: Spline Functions and Multivariate Interpolations, Mathematics and its Applications Vol. 248, Kluwer, Dordrecht 1993.CrossRefGoogle Scholar
  7. 7.
    B. D. Bojanov, D. P. Dryanov, W. Haussmann, G. P. Nikolov: Best one-sided L’-approximation by blending functions, In: Advances in Multivariate Approximation, Mathematical Research Vol. 107 85–106, Wiley-VCH, Berlin 1999.Google Scholar
  8. 8.
    C. de Boor: A Practical Guide to Splines, Applied Mathematical Sciences Vol. 27 Springer, New York-Berlin 1978.CrossRefGoogle Scholar
  9. 9.
    P. J. Davis: Interpolation and Approximation, Blaisdell Publ. Co. Ginn and Co., New York 1963.Google Scholar
  10. 10.
    R. A. DeVore: One-sided approximation of functions, J. Approx. Theory 1 (1968), 11–25.CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    R. A. DeVore, G. G. Lorentz: Constructive Approximation, Springer, Berlin 1993.CrossRefzbMATHGoogle Scholar
  12. 12.
    D. Dryanov, W. Haussmann, P. Petrov: Best one-sided L’ -approximation of bivariate functions by sums of univariate ones, Arch. Math. (Basel) 75 (2000), 125–131.CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    D. Dryanov, W. Haussmann, P. Petrov: Best one-sided L’ -approximation by B2,1-blending functions, In: Recent Progress in Multivariate Approximation, International Series of Numerical Mathematics Vol. 137, 115–134, Birkhäuser, Basel 2001.CrossRefGoogle Scholar
  14. 14.
    D. Dryanov, W. Haussmann, P. Petrov: Best one-sided L’ -approximation by blending functions and approximate cubature, In: Approximation Theory X, Innovations in Applied Mathematics Vol. X, 145–152, Vanderbilt Univ. Press, Nashville 2002.Google Scholar
  15. 15.
    D. Dryanov, O. Kounchev: Polyharmonically exact formula of Euler-Maclaurin, multivariate Bernoulli functions, and Poisson type formula, C. R. Acad. Sci. Paris Sr. I Math. 327 (1998), 515–520.CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    D. Dryanov, O. Kounchev: Multivariate Bernoulli functions and polyharmonically exact cubature of Euler-Maclaurin, Math. Nachr. 226 (2001), 65–83.CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    D. Dryanov, P. Petrov: Best one-sided L1-approximation by blending functions of order (2, 2), J. Approx. Theory 115 (2002), 72–99.CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    L. Flatto: The approximation of certain functions of several variables by sums of functions of fewer variables, Amer. Math. Monthly 73 (1966), 131–132.CrossRefzbMATHMathSciNetGoogle Scholar
  19. 19.
    M. Goldstein, W. Haussmann, K. Jetter: Best harmonic L’ -approximation to subharmonic functions, J. London Math. Soc. 30 (1984), 257–264.CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    M. Goldstein, W. Haussmann, L. Rogge: On the mean value property of harmonic functions and best harmonic L1-approximation, Trans. Amer. Math. Soc. 305 (1988), 505–515.zbMATHMathSciNetGoogle Scholar
  21. 21.
    W. J. Gordon, C. A. Hall: Transfinite element methods: blending-function interpolation over arbitrary curved element domains, Nu-mer. Math. 21 (1973/74), 109–129. CrossRefzbMATHMathSciNetGoogle Scholar
  22. B. Gustafsson, M. Sakai, H. S. Shapiro: On domains in which harmonic functions satisfy generalized mean value properties, Potential Anal. 7 (1997), 467–484.CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    W. Haussmann, K. Zeller: Blending interpolation and best L l approximation, Arch. Math. (Basel) 40 (1983), 545–552.CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    W. Haussmann, K. Zeller: Canonical point sets in multivariate constructive function theory, In: Proceedings of the Ashkelon Workshop on Complex Function Theory (1996), Israel Math. Conf. Proc. 11, 91–104, Bar-Ilan University, Ramat Gan 1997.Google Scholar
  25. 25.
    S. Y. Khavinson: Best Approximation by Linear Superpositions, Amer. Math. Soc., Providence 1997.Google Scholar
  26. 26.
    J. C. Mairhuber: On Haar’s Theorem concerning Chebyshev approximation problems having unique solution,Proc. Amer. Math. Soc. 7 (1956), 609–615.zbMATHMathSciNetGoogle Scholar
  27. 27.
    V. M. Mordashev: Approximation of functions of several variables by a sum of functions of fewer variables, Doklad. Akad. Nauk SSSR 183 (1968), 778–779; English transi., Soviet Math. Dokl. 9 (1968), 1462–1463.zbMATHGoogle Scholar
  28. 28.
    A. Pinkus: On L 1 - Approximation, Cambridge Univ. Press, Cambridge 1989.Google Scholar
  29. 29.
    T. J. Rivlin, R. J. Sibner: The degree of approximation of certain functions of two variables by a sum of functions of one variable, Amer. Math. Monthly 72 (1965), 1101–1103.CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2003

Authors and Affiliations

  • Dimiter P. Dryanov
    • 1
  1. 1.Départment de Mathématiques et de StatistiqueUniversité de MontréalMontréalCanada

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