H-sets and minimal conditions can be used as criteria for best uniform approximations. In this paper both concepts are compared and examples are given that show where these criteria are applicable and where not. For the minimal conditions the dependence of the extremal values on the parameters is investigated and a simple extension of the conditions is pointed out.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Brannigan: H-sets in linear approximation. J. Approximation Theory 20 (1977), 153–161.CrossRefGoogle Scholar
  2. [2]
    L. Collatz: Approximation von Funktionen bei einer und bei mehreren unabhängigen Veränderlichen. Z. Angew. Math. Mech. 36 (1956), 198–211.CrossRefGoogle Scholar
  3. [3]
    L. Collatz: Tschebyscheffsche Annäherung mit rationalen Funktionen. Abh. Math. Sem. Univ. Hamburg 24 (1960), 70–78.CrossRefGoogle Scholar
  4. [4]
    L. Collatz: Inclusion theorems for the minimal distance in rational Tschebyscheff approximation with several variables. In: Approximation of Functions, Proc. Sympos. General Motors Res. Lab., (1964), 43–56. Elsevier Publ. Co., Amsterdam, 1965.Google Scholar
  5. [5]
    L. Collatz: Rationale trigonometrische Tschebyscheff-Approximation in zwei Variablen. Publ. Inst. Math. (Beograd) (N. S. ) 6 (20) 1966, 57–63.Google Scholar
  6. [6]
    L. Collatz: The determination of H-sets for the inclusion theorem in nonlinear Tschebyscheff approximation. Approximation Theory (Proc. Sympos., Lancaster, 1969), 179–189. Academic Press, London, 1970.Google Scholar
  7. [7]
    L. Collatz, W. Krabs: Approximationstheorie. 208 p. B. G. Teubner, Stuttgart, 1973.Google Scholar
  8. [8]
    C. Dierieck: Some remarks on H-sets in linear approximation theory. J. Approximation Theory 21 (1977), 188–204.CrossRefGoogle Scholar
  9. [9]
    R. Hettich: Kriterien zweiter Ordnung für lokal beste Approximationen. Numer. Math. 22 (1974), 409–417.CrossRefGoogle Scholar
  10. [10]
    G. D. Taylor: On minimal H-sets. J. Approximation Theory 5 (1972), 113–117.CrossRefGoogle Scholar
  11. [11]
    W. Wetterling: Definitheitsbedingungen für relative Extrema bei Optimierungs- und Approximationsaufgaben. Numer. Math. 15 (1970), 122–136.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1979

Authors and Affiliations

  • Wolfgang Wetterling

There are no affiliations available

Personalised recommendations