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Handbook of Functional Equations pp 247–271Cite as

On Approximation Properties of Szász–Mirakyan Operators

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Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 95))

Abstract

In the present chapter, we present approximation properties of the well-known Szász-Mirakyan operators. These operators were introduced in the middle of last century and because of their important properties, researchers continued to work on such operators and their different modifications. Although there are several modifications of the Szász-Mirakyan operators available in the literature viz. integral modifications due to Kantorovich, Durrmeyer and mixed operators, but here we discuss only the discrete modifications of these operators which were proposed by several researchers in last 60 years. In the recent years, overconvergence properties were studied by considering the complex version of Szász-Mirakyan operators. In the last section, we consider complex Szász-Stancu operators and establish upper bound and a Voronovskaja type result with quantitative estimates for these operators attached to analytic functions of exponential growth on compact disks.

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References

  1. Abel, U., Ivan, M., Zeng, X.M.: Asymptotic expansion for Szász-Mirakyan operators. AIP Conf. Proc. 936, 779–782 (2007)

    Article  Google Scholar 

  2. Agarwal, R.P., Gupta, V.: On q-analogue of a complex summation-integral type operators in compact disks. J. Inequal. Appl. 2012, Article 111 (2012). doi:10.1186/1029-242X-2012-111

    Google Scholar 

  3. Aral, A.: A generalization of Szász-Mirakyan operators based on q-integers. Math. Comput. Model. 47 (9–10), 1052–1062 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Aral, A., Gupta, V.: q-derivatives and applications to the q-Szász Mirakyan operators. CALCOLO 43(3), 151–170 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Aral, A., Gupta, V., Agarwal, R.P.: Applications of q Calculus in Operator Theory, 265 p. Springer, Berlin (2013) (ISBN 978-1-4614-6945-2)

    Google Scholar 

  6. Cheng, F.: On the rate of convergence of the Szász-Mirakyan operators for functions of bounded variation. J. Approx. Theory 40, 226–241 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  7. Comtet, L.: Advanced Combinatorics. Reidel, Dordrecht (1974) (ISBN 90-0022-247X)

    Book  MATH  Google Scholar 

  8. Favard, J.: Sur les multiplicateurs d’interpolation. J. Math. Pures Appl. 23(9), 219–244 (1944)

    MATH  MathSciNet  Google Scholar 

  9. Gal, S.G.: Approximation of analytic functions without exponential growth conditions by complex Favard-Szász-Mirakjan operators. Rend. Circ. Mat. Palermo 59, 367–376 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  10. Gal, S.G.: Approximation by Complex Bernstein and Convolution Type Operators. World Scientific, Singapore (2009)

    MATH  Google Scholar 

  11. Gal, S.G., Gupta, V.: Quantative estimates for a new complex Durrmeyer operator in compact disks. Appl. Math. Comput. 218(6), 2944–2951 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  12. Gal, S.G., Gupta, V.: Approximation by a Durrmeyer-type operator in compact disk. Ann. Univ. Ferrara 57, 261–274 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  13. Gal, S.G., Gupta, V.: Approximation be certain integrated Bernstein type operators in compact disks. Lobachevskii J. Math. 33(1), 39–46 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gupta, V., Verma, D.K.: Approximation by complex Favard-Szász-Mirakjan-Stancu operators in compact disks. Math. Sci. 6, Article 25 (2012). doi:10.1186/2251-7456-6-25

    Google Scholar 

  15. Hermann, T.: Approximation of unbounded functions on unbounded intervals. Acta Math. Acad. Sci. Hung. 29(3–4), 393–398 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  16. Jain, G.C.: Approximation of functions by a new class of linear operators. J. Austral. Math. Soc. 13(3), 271–276 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  17. Kasana, H.S., Agrawal, P.N.: Approximation by linear combination of Szász-Mirakian operators. Colloq. Math. 80(1), 123–130 (1999)

    MATH  MathSciNet  Google Scholar 

  18. Lorentz, G.G.: Bernstein Polynomials, 2nd edn. Chelsea, New York (1986)

    MATH  Google Scholar 

  19. Lupas, A.: Some properties of the linear positive operators, II. Mathematica (Cluj) 9(32), 295–298 (1967)

    MathSciNet  Google Scholar 

  20. Mahmudov, N.I., Gupta, V.: Approximation by genuine Durrmeyer-Stancu polynomials in compact disks. Math. Comput. Model. 55, 278–285 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. May, C.P.: Saturation and inverse theorems for combinations of a class of exponential type operators. Can. J. Math. XXVIII, 1224–1250 (1976)

    Article  Google Scholar 

  22. Mirakjan, G.M.: Approximation des fonctions continues au moyen de polynômes de la forme \(e^{-nx}\sum k={0}^{m_n} C_{k,n}x^k \) [Approximation of continuous functions with the aid of polynomials of the form \(e^{-nx}\sum k={0}^{m_n} C_{k,n}x^k \)] (in French). C. R. Acad. Sci. URSS 31, 201–205 (1941)

    Google Scholar 

  23. Ren, M.Y., Zeng, X.M.: Approximation by a kind of complex modified q-Durrmeyer type operators in compact disks. J. Inequal. Appl. 2012, Article 212 (2012)

    Google Scholar 

  24. Singh, S.P.: On the degree of approximation by Szász operators. Bull. Austral. Math. Soc. 24, 221–225 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  25. Stancu, D.D.: Use of probabilistic methods in the theory of uniform approximation of continuous functions. Rev. Roum. Math. Pures Appl. 14, 673–691 (1969)

    MATH  MathSciNet  Google Scholar 

  26. Stypińki, Z.: Theorem of Voronovskaya for Szász-Chlodovsky operators. Funct. Approx. Comment. Math. 1, 133–137 (1974)

    Google Scholar 

  27. Sun, X.H.: On the simultaneous approximation of functions and their derivatives by the Szász-Mirakian operators. J. Approx. Theory 55, 279–288 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  28. Szász, O.: Generalizations of S. Bernstein’s polynomial to the infinite interval. J. Res. Nat. Bur. Stand. 45, 239–245 (1950)

    Article  Google Scholar 

  29. Totik, V.: Uniform approximation by Szász-Mirakyan type operators. Acta Math. Hung. 41 (3–4), 291–307 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  30. Walczak, Z.: On certain linear positive operators in polynomial weighted spaces. Acta. Math. Hung. 101(3), 179–191 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Walczak, Z., Gupta, V.: Uniform convergence with certain linear operators. Indian J. Pure. Appl. Math. 38(4), 259–269 (2007)

    MATH  MathSciNet  Google Scholar 

  32. Zeng, X.M.: On the rate of convergence of the generalized Szász type operators for functions of bounded variation. J. Math. Anal. Appl. 226, 309–325 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  33. Zeng, X.M., Piriou, A.: Rate of pointwise approximation for locally bounded functions by Szász operators. J. Math. Anal. Appl. 307, 433–443 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Netaji Subhas Institute of Technology, Sector 3 Dwarka, New Delhi, 110078, India

    Vijay Gupta

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  1. Vijay Gupta
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Correspondence to Vijay Gupta .

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Editors and Affiliations

  1. Department of Mathematics, National Technical University of Athens, Athens, Greece

    Themistocles M. Rassias

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Gupta, V. (2014). On Approximation Properties of Szász–Mirakyan Operators. In: Rassias, T. (eds) Handbook of Functional Equations. Springer Optimization and Its Applications, vol 95. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1246-9_11

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