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Results in Mathematics

, Volume 63, Issue 3–4, pp 837–863 | Cite as

On a Generalization of Szász–Mirakjan–Kantorovich Operators

  • Francesco AltomareEmail author
  • Mirella Cappelletti Montano
  • Vita Leonessa
Article

Abstract

In this paper we introduce and study a sequence of positive linear operators acting on suitable spaces of measurable functions on [0,+∞[, including L p ([0,+∞[) spaces, 1 ≤ p < +∞, as well as continuous function spaces with polynomial weights. These operators generalize the Szász–Mirakjan–Kantorovich operators and they allow to approximate (or to reconstruct) suitable measurable functions by knowing their mean values on a sequence of subintervals of [0,+∞[ that do not constitute a subdivision of it. We also give some estimates of the rates of convergence by means of suitable moduli of smoothness.

Mathematics Subject Classification (2000)

41A10 41A25 41A36 

Keywords

Szász–Mirakjan–Kantorovich operator positive approximation process weighted space modulus of smoothness 

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References

  1. 1.
    Altomare F.: Korovkin-type theorems and approximation by positive linear operators. Surv. Approx. Theory 5, 92–164 (2010) free available online at http://www.math.techmion.ac.il/sat/papers/13/
  2. 2.
    Altomare F., Campiti M.: Korovkin-type approximation theory and its applications, de Gruyter Studies in Mathematics 17. Walter de Gruyter & Co., Berlin (1994)CrossRefGoogle Scholar
  3. 3.
    Altomare F., Cappelletti Montano M., Leonessa V.: On a generalization of Kantorovich operators on simplices and hypercubes. Adv. Pure Appl. Math. 1(3), 359–385 (2010)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Altomare F., Leonessa V.: On a sequence of positive linear operators associated with a continuous selection of Borel measures. Mediterr. J. Math. 3, 363–382 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bauer H.: Probability theory, de Gruyter Studies in Mathematics 23. Walter de Gruyter & Co., Berlin (1996)Google Scholar
  6. 6.
    Becker M.: Global approximation theorems for Szász–Mirakjan and Baskakov operators in polynomial weight spaces. Indiana Univ. Math. J. 27(1), 127–142 (1978)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Bustamante J., Morales de la Cruz L.: Korovkin type theorems for weighted approximation. Int. J. Math. Anal. 26(1), 1273–1283 (2007)MathSciNetGoogle Scholar
  8. 8.
    Butzer P.L.: On the extensions of Bernstein polynomials to the infinite interval. Proc. Amer. Math. Soc. 5, 547–553 (1954)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Cheney E.W., Sharma A.: Bernstein power series. Canadian J. of Math. 16(2), 241–264 (1964)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    DeVore R.A., Lorentz G.G.: Constructive Approximation, Grundlehren der mathematischen Wissenschaften 303. Springer, Berlin (1993)CrossRefGoogle Scholar
  11. 11.
    Ditzian Z., Totik V.: Moduli of smoothness, Springer Series in Computational Mathematics 9. Springer, New-York (1987)Google Scholar
  12. 12.
    Duman O., Della Vecchia M.A., Della Vecchia B.: Modified Szász–Mirakjan–Kantorovich operators preserving linear functions. Turk. J. Math. 33, 151–158 (2009)zbMATHGoogle Scholar
  13. 13.
    Favard J.: Sur les multiplicateurs d’interpolations. J. Math. Pures Appl. 23(9), 219–247 (1944)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gonska H.: Positive operators and approximation of functions: selected topics. Conf. Semin. Mat. Univ. Bari 288, 28 (2002)MathSciNetGoogle Scholar
  15. 15.
    Holhoş A.: The rate of approximation of functions in an infinite interval by positive linear operators. Stud. Univ. Babeş–Bolyai Math. 55(2), 133–142 (2010)zbMATHGoogle Scholar
  16. 16.
    Mirakjan G.M.: Approximation of continuous functions with the aid of polynomials. (Russian), Dokl. Akad. Nauk SSSR 31, 201–205 (1941)zbMATHGoogle Scholar
  17. 17.
    Păltănea R.: Approximation theory using positive linear operators. Birkhäuser, Boston (2004)zbMATHGoogle Scholar
  18. 18.
    Swetits J.J., Wood B.: Quantitative estimates fo L p approximation with positive linear operators. J. Approx. Theory 38, 81–89 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Szász O.: Generalization of Bernstein’s polynomials to the infinite interval. J. Res. Nat. Bur. Stds. 45, 239–245 (1950)CrossRefGoogle Scholar
  20. 20.
    Totik V.: Approximation by Szász–Mirakjan type operators. Acta Math. Hungarica 41, 291–307 (1983)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Totik V.: Approximation by Szász–Mirakjan–Kantorovich operators in L p(p > 1). Analysis Math. 9, 147–167 (1983)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Francesco Altomare
    • 1
    Email author
  • Mirella Cappelletti Montano
    • 1
  • Vita Leonessa
    • 2
  1. 1.Dipartimento di MatematicaUniversità degli Studi di BariBariItaly
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi della BasilicataPotenzaItaly

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