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Abstract

In this note we will present sharpened versions of two theorems of Harold Shapiro on comparison between generalized moduli of continuity. By specialization of the measures defining the moduli of continuity in question our theorem gives a sharp form of the Jackson and Bernstein theorems. In particular our theorem implies the known fact that the order of best approximation by trigonometric polynomials for any continuous and periodic function f satisfies En (f) = O((log n)−1) if and only if the modulus of continuity of f is O(|log t|−1).

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References

  1. [1]
    Björk, J.-E., On Subalgebras of M(R n) generated by Smooth Measures carried by Smooth Submanifolds in R n. Mimeographed, Dept. Math., Univ. Stockholm, 1973.Google Scholar
  2. [2]
    Boman, J. — Shapiro, H.S., Comparison theorems for a Generalized Modulus of Continuity. Ark. Mat. 9 (1971), 91–116.CrossRefGoogle Scholar
  3. [3]
    Lorentz, G.G., Approximation of Functions. Holt, Rinehart and Winston 1966.Google Scholar
  4. [4]
    Shapiro, H.S., A Tauberian theorem related to Approximation Theory. Acta Math. 120 (1968), 279–292.CrossRefGoogle Scholar
  5. [5]
    Shapiro, H.S., Smoothing and Approximation of Functions. Van Nostrand Reinhold, 1969.Google Scholar
  6. [6]
    Shapiro, H.S., Topics in Approximation Theory. Berlin/Heidelberg, Lecture Notes in Mathematics nr 187, Springer-Verlag 1971.Google Scholar
  7. [7]
    Varopoulos, N., Studies in Harmonic Analysis. Proc. Cambridge Philos. Soc. 60 (1964), 465–516.CrossRefGoogle Scholar

Copyright information

© Springer Basel AG 1974

Authors and Affiliations

  • Jan Boman
    • 1
  1. 1.Department of MathematicsUniversity of StockholmStockholmSweden

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