Constructive Approximation

, Volume 2, Issue 1, pp 377–392 | Cite as

Converse theorems for approximation by bernstein polynomials in Lp[0,1] (1<p<∞)

  • K. G. Ivanov
Article

Abstract

The class of all continuous functions possessing n−α(1/p<α≤1) order of approximation by Bernstein polynomials inLp[0, 1] is characterized.

AMS classification

41A25 41A27 41A36 41A40 

Key words and phrases

Bernstein polynomials Order of approximation inLp Saturation classes Inverse theorems 

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References

  1. 1.
    H. Berens, G. G. Lorentz (1972):Inverse theorems for Bernstein polynomials. Indiana Univ. Math. J.,21:693–708.Google Scholar
  2. 2.
    R. DeVore, S. D. Riemenschneider, R. C. Sharpley (1979):Weak interpolation in Banach spaces. J. Fund. Anal.,33:58–94.Google Scholar
  3. 3.
    Z. Ditzian (1980):On interpolation of L p [a,b] and weighted Sobolev spaces. Pacific J. Math.,90:307–324.Google Scholar
  4. 4.
    K. G. Ivanov (1982):On a new characteristic of functions. Serdica,8:262–279.Google Scholar
  5. 5.
    K. G. Ivanov (1982):On Bernstein polynomials. C. R. Acad. Bulgare Sci.,35:893–896.Google Scholar
  6. 6.
    K. G. Ivanov (1983):A constructive characteristic of the best algebraic approximation in L p[−1, 1] (1≤p≤∞). Constructive Function Theory '81, Sofia, pp. 357–367.Google Scholar
  7. 7.
    K. G. Ivanov (1984):Approximation by Bernstein polynomials in L p metric. Constructive Theory of Functions '84, Sofia, pp. 421–429.Google Scholar
  8. 8.
    K. G. Ivanov (1985):On the behavior of two moduli of functions. C. R. Acad. Bulgare Sci.,38:539–542.Google Scholar
  9. 9.
    K. G. Ivanov (1985):On the behavior of two moduli of functions, II. Serdica (to appear).Google Scholar
  10. 10.
    K. G. Ivanov (unpublished):A characterization of weighted Peetre K-functionals.Google Scholar
  11. 11.
    K. de Leeuw (1959):On the degree of approximation by Bernstein polynomials. J. Analyse Math.,7:89–104.Google Scholar
  12. 12.
    V. Maier (1978):L p-approximation by Kantorovic operators. Anal. Math.,4:289–295.Google Scholar
  13. 13.
    S. D. Riemenschneider (1978):The L p-saturation of Bernstein-Kantorovitch polynomials. J. Approx. Theory,23:158–162.Google Scholar
  14. 14.
    V. Totik (1983):L p (p>1)-approximation by Kantorovich polynomials. Analysis,3:79–100.Google Scholar
  15. 15.
    V. Totik (1984):The necessity of a new kind of modulus of smoothness. Anniversary Volume on Approximation Theory and Functional Analysis, ISNM 65, pp. 233–248.Google Scholar

Copyright information

© Springer-Verlag New York Inc. 1986

Authors and Affiliations

  • K. G. Ivanov
    • 1
  1. 1.Institute of MathematicsBulgarian Academy of SciencesSofiaBulgaria

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