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Uniform approximation of vector-valued functions

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Summary

Uniform approximation of vector-valued functions is defined, together with analogs of extreme points and H-sets. Characterizations of best approximations are given in terms of these, and some applications are presented.

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Bibliography

  1. 1.

    Bacopoulos, A. C.: Approximation with vector-valued norms in linear spaces. Thesis, University of Wisconsin 1966.

  2. 2.

    Eggleston, H. G.: Convexity. Cambridge: Cambridge University Press 1958.

  3. 3.

    Garabedian, H. L. (ed.): Approximation of functions. Amsterdam: Elsevier Publishing Co. 1965.

  4. 4.

    Johnson, L. W.: Unicity in approximation of a function and its derivatives. Mathematics of Computation22, 873–875 (1968).

  5. 5.

    Kammerer, J. W.: Optimal approximation of functions: One-sided approximations and extrema preserving approximations. Thesis, University of Wisconsin 1959.

  6. 6.

    Kolmogoroff, A. N.: A remark concerning the polynomials of P. L. Tchebycheff which deviate least from a given function. [Russian.] Uspekhi Math. Nauk3, 216–221 (1948).

  7. 7.

    Laurent, P. J.: Theoremes de caracterisation d'une meilleure approximation dans un espace norme et generalization de l'algorithme de Remes. Numerische Mathematik10, 190–208 (1967)

  8. 8.

    Lawson, C. L.: Contributions to the theory of linear least maximum approximation. Thesis, UCLA 1961.

  9. 9.

    Meinardus, G.: Approximation of functions: Theory and numerical methods. Berlin-Heidelberg-New York: Springer 1967

  10. 10.

    Moursund, D. G.: Chebyschev approximation using a generalized weight function. SIAM Journal of Numerical Analysis3, 435–450 (1966).

  11. 11.

    Moursund, D. G.: Optimal starting values for the Newton-Raphson calculation of\(\sqrt x\). Comm. ACM10, 430–432 (1967)

  12. 12.

    —— andG. D. Taylor: Optimal starting values for the Newton-Raphson calculation of the inverses of certain functions. SIAM Journal of Numerical Analysis5, 138–150 (1968).

  13. 13.

    —— Chebyshev approximation of a function and its derivatives. Mathematics of Computation18, 382–389 (1964).

  14. 14.

    Taylor, G. D.: On approximation by polynomials having restricted ranges. SIAM Journal of Numerical Analysis5, 258–268 (1968).

  15. 15.

    Zuhovickii, S. I., andS. B. Steckin: On the approximation of abstract functions. [English translation.] American Mathematical Society Translations, (Series 2)16, 401–406 (1960).

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Most of the work for this paper was done at Michigan State University, East Lansing; as a Ph. D. Thesis directed by D. G. MOURSUND. The author wishes to thank Prof. MOURSUND for his kind assistance.

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Johnson, L.W. Uniform approximation of vector-valued functions. Numer. Math. 13, 238–244 (1969). https://doi.org/10.1007/BF02167554

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Keywords

  • Mathematical Method
  • Extreme Point
  • Uniform Approximation