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Optimal Estimates with Moduli of Continuity

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Abstract

In this paper we obtain estimates with optimal constants for the pointwise approximation of functions by linear positive functional, with the aid of a new second order modulus with a parameter, as well as with the aid of the first order modulus of the derivatives of functions.

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References

  1. J. A. Adell and J.de la Cal, Preservation of moduli of continuity by Bernstein-type operators, in Approximation, Probability and Related Fields (A. Anastassiou and S. T. Rachev Eds.), Plenum Press, New York, 1994, 1–18

    Chapter  Google Scholar 

  2. C. Badea, I. Badea and H. H. Gonska, New estimates for simultaneous approximation by Bernstein operators, ”Bulletino U.M.I” Piazza Porta, S. Donata 5, I-40.127, Bologna, Italia

  3. Ju. A. Brudnyi, On a method of approximation of bounded functions defined in an interval (in Russian), in ”Studies in Contemporary Problems. Constructive Theory of Functions”, Proc. Second All-Union Conference, Baku, 1962, I. I. Ibrahimov, ed., Izdat. Akad. Nauk. Azerbaid\({\widetilde z}\)an. SSR, Baku, (1965), 40–45.

  4. E. Censor, Quantitative results for positive linear approximation operators, J. Approx. Theory 4 (1971), 442–450.

    Article  MathSciNet  MATH  Google Scholar 

  5. R. A.De Vore, The approximation of continuous functions by positive linear operators, Berlin-Heidelberg-New York: Springer (1972).

    Google Scholar 

  6. G. Freud, On approximation by positive linear methods II, Studia Sci. Math. Hungar. 3 (1968), 365–370.

    MathSciNet  Google Scholar 

  7. N. G. Gasharov, On estimates for the approximation by linear operators preserving linear functions (Russian) In ”Approximation of Functions by Special Classes of Operators”, Vologda, (1987), 28–35.

  8. I. Gavrea and I. Raşa, Remarks on some quantitative Korovkin type results (to apear).

  9. H. H. Gonska, Quantitative Aussagen zur Approximation durch positive lineare Operatoren, Dissertation, Universität Duisburg, (1979).

  10. H. H. Gonska, On approximation of continuously differentiable functions by positive linear operators, Bull. Austral. Math. Soc. 27 (1983), 73–81.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. H. Gonska, On approximation by linear operators: Improved estimates, Anal. Numér. Théor. Approx. 14 (1985), 7–32.

    MathSciNet  MATH  Google Scholar 

  12. H. H. Gonska and R. K. Kovacheva, The second order modulus revisited: remarks, applications, problems. Confer. Sem. Math. Univ. Bari 257 (1994), 1–32.

    MATH  Google Scholar 

  13. R. G. Mamedov, On the order of the approximation of differentiable functions by linear positive operators, (Russian), Doklady, S.S.S.R., 128 (1959), 674–676.

    MATH  Google Scholar 

  14. J. Meier, Zur Verallgemeinerung eines Satzes von Censor und DeVore, Facta Univ. Nis Ser. Math. Inform. 1 (1986), 45–52.

    MathSciNet  MATH  Google Scholar 

  15. B. Mond and R. Vasudevan, On approximation by linear positive operartors, J. Approx. Theory 30 (1980), 334–336.

    Article  MathSciNet  Google Scholar 

  16. R. Păltănea, General estimates for linear positive operators, that preserve linear functions, Anal. Numer. Theor. Approx. 18 (1989), 147–159.

    MathSciNet  MATH  Google Scholar 

  17. R. Păltănea, On the estimate of the pointwise approximation of functions by linear positive functionals, Studia Univ. ”Babeş-Bolyai”, Mathematica, 35, 1, (1990), 11–24.

    MATH  Google Scholar 

  18. R. Păltănea, Best constants in estimates with second order moduli of continuity, ”Approximation Theory” (Proc. Int. Dortmund Meeting on Approximation Theory 1995, ed by M. W. Müller et.al.), Berlin Akademie Verlag, 1995, 251–275.

  19. J. Peetre, Exact interpolation theorems for Lipschitz continous functions, Ricerche Mat. 18 (1969), 239–259.

    MathSciNet  MATH  Google Scholar 

  20. A. Sahai and G. Prasad, Sharp estimates of approximation by some positive linear operators, Bull. Austral. Math. Soc. 29 (1984), 13–18.

    Article  MathSciNet  MATH  Google Scholar 

  21. F. Schurer and F. W. Steutel, The degree of local approximation of function in C1[0,1] by Bernstein polynomials, J. Approx. Theory 19 (1977), 69–82.

    Article  MathSciNet  MATH  Google Scholar 

  22. O. Shisha and B. Mond, The degree of convergence of linear positive operators, Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 1196–1200.

    Article  MathSciNet  MATH  Google Scholar 

  23. S. P. Sing and O. P. Varshney, On positive linear operators. J. Orissa Math. Soc. 1 (1982), no. 2, 51–56.

    MathSciNet  Google Scholar 

  24. O. P. Varshney and S. P. Sing, On degree of approximation by positive linear operators, Rend. Mat. (7)2 (1982), 219-225.

    Google Scholar 

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  1. Dept. of Mathematics, Transilvania University, 29 Eroilor Str. Braşov, 2200, Romania

    Radu Păltănea

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  1. Radu Păltănea
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Păltănea, R. Optimal Estimates with Moduli of Continuity. Results. Math. 32, 318–331 (1997). https://doi.org/10.1007/BF03322143

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