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aequationes mathematicae

, Volume 54, Issue 1–2, pp 56–73 | Cite as

A modulus of smoothness based on an algebraic addition

  • M. FeltenEmail author
Research Papers

Summary

The purpose of this paper is to present a new approach to smoothness of nonperiodic functions. We consider the space of continuous functions on [−1, 1] as well as the weighted Lp-space and introduce a modulus of smoothness that is based on an algebraic addition ⊕ defined on [−1, 1]. The present paper is mainly concerned with general properties and groundwork, whereas a second paper [4] is devoted to more complex properties, in particular to an equivalent K-functional and to the characterization of best algebraic approximation. Moreover the equivalence with the Butzer-Stens modulus will be shown there.

AMS classification

41A17 

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References

  1. [1]
    Butzer, P. L. andStens, R. L.,Chebyshev transform methods in the theory of best algebraic approximation. Abh. Math. Sem. Hamburg45 (1976), 165–190.zbMATHMathSciNetGoogle Scholar
  2. [2]
    Ditzian, Z. andTotik, V. Moduli of smoothness. Springer-Verlag, New York, 1987.zbMATHGoogle Scholar
  3. [3]
    Felten, M.,Ein neuer Glättemodul für nichtperiodische Funktionen. Dissertation, Universität Dortmund, 1992.Google Scholar
  4. [4]
    Felten, M.,Characterization of best algebraic approximation by an algebraic modulus of smoothness. Journal of Approximation Theory89 (1997), 1–25.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    Nikolskil, K.,On the best approximation of functions satisfying a Lipschitz’s condition by polynomials. Izv. Akad. Nauk SSSR10 (1946), 295–322.Google Scholar

Copyright information

© Birkhäuser Verlag 1997

Authors and Affiliations

  1. 1.Lehrstuhl VIII für MathematikUniversität DortmundDortmundGermany

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