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Approximation Properties of Phillips Operators

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Mathematics Without Boundaries

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Abstract

This paper deals with presenting a survey-cum-expository account of some developments concerning the approximation properties of well known Phillips operators. These operators are sometimes called as genuine Szász Durrmeyer operators, because of their property of reproducing constants as well as linear functions. We give the alternate form to present these operators in terms of Hypergeometric functions, which are related to the modified Bessel’s function of first kind of index 1. Also, we observe that the r-th moment can be represented in terms of confluent hypergeometric functions, and further it can be written in terms of generalized Laguerre polynomials. In addition, we will present some known results on such operators, which include simultaneous approximation, linear and iterative combinations, global direct and inverse results, rate of convergence for functions of bounded variation, and q-analogues of these operators.

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References

  1. Agrawal, P.N..: Simultaneous approximation by Micchelli combination of Phillips operators. In: Singh B., Murari K., Gupta U.S., Prasad G. ‚ Sukavanam N.(eds.) Proceedings of Conference ‛'Mathematics and its applications in engineering and industry”, University of Roorkee, India, pp. 569–577 (1997)

    Google Scholar 

  2. Agrawal, P.N., Gupta, V.: On the iterative combination of Phillips operators. Bull. Inst. Math. Acad. Sinica 18(4), 361–368 (1990)

    MathSciNet  MATH  Google Scholar 

  3. Agrawal, P.N., Gupta, V.: On convergence of derivatives of Phillips operators. Demonstr. Math. 27(2), 501–510 (1994)

    MathSciNet  MATH  Google Scholar 

  4. Agrawal, P.N., Gupta, V., Gairola, A. R.: On the derivatives of iterative combinations of Phillips operators. Nonlinear Funct. Anal. Appl. 12(2), 195–203 (2007)

    MathSciNet  MATH  Google Scholar 

  5. Butzer, P.L.: Linear combinations of Bernstein polynomials. Canad. J. Math. 5, 559–567 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  6. De Sole, A., Kac, V.: On integral representations of q-gamma and q-beta functions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 16(1), 11–29 (2005)

    MathSciNet  MATH  Google Scholar 

  7. DeVore, R.A., Lorentz, G.G.: Constructive approximationAQ1. Springer, Berlin (1993)

    Book  MATH  Google Scholar 

  8. Ditzian, Z., May, C.P.: A saturation result for combinations of Bernstein polynomials. Tôhoku Math. J. 28, 363–373 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ditzian, Z., Totik, V.: Moduli of smoothness. Springer, New York, (1987)

    Book  MATH  Google Scholar 

  10. Ditzian, Z., Ivanov, K.G.: Strong converse inequality. J. Anal. Math. 61, 61–111 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  11. Finta, Z.: Direct and converse theorems for integral-type operators. Demonstr. Math. 36, 137–147 (2003)

    MathSciNet  MATH  Google Scholar 

  12. Finta, Z., Gupta, V.: Direct and inverse estimates for Phillips type operators. J. Math. Anal. Appl. 303(2), 627–642 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Govil, N.K., Gupta, V., Noor, M. A.: Simultaneous approximation for the Phillips operators. Int. J. Math. Math. Sci. 2006(7), (2006) (1–9 Art Id. 49094)

    Google Scholar 

  14. Gupta, V.: Rate of approximation by a new sequence of linear positive operators. Comput. Math. Appl. 45, 1895–1904 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gupta, V.: Rate of convergence by Bézier variant of Phillips operators. Taiwanese J. Math. 8(2), 183–190 (2004)

    MathSciNet  MATH  Google Scholar 

  16. Gupta, V.: A note on modified Phillips operators. Southeast Asian Bull Math. 34, 847–851 (2010)

    MathSciNet  MATH  Google Scholar 

  17. Gupta, V., Agrawal, P.N.: Linear combinations of Phillips operators. Indian Acad. Math. 11(2), 106–114 (1989)

    MathSciNet  MATH  Google Scholar 

  18. Gupta, V., Agrawal, P.N.: Lp-approximation by iterative combination of Phillips operators. Publ. Inst. Math. (Beograd) 52(66), 101–109 (1992)

    MathSciNet  Google Scholar 

  19. Gupta, V., Sahai, A.: On linear combination of Phillips operators. Soochow J. Math. 19(3), 313–323 (1993)

    MathSciNet  MATH  Google Scholar 

  20. Gupta, V., Srivastava, G.S.: On the rate of convergence of Phillips operators for functions of bounded variation. Comment. Math. 36, 123–130 (1996)

    MathSciNet  MATH  Google Scholar 

  21. Heilmann, M., Tachev, G.: Commutativity, direct and strong converse results for Phillips operators. East J. Approx. Theory 17(3), 299–317 (2011)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  23. Jackson, D.: The theory of approximation, Vol. 11. American Mathematical Society Call Publications, New York (1930)

    Google Scholar 

  24. López-Moreno, A.J.: Weighted simultaneous approximation with Baskakov type operators, Acta Math. Hungar. 104(1–2), 143–151 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Lupas, A. A q-analogue of the Bernstein operator, In: Seminar on numerical and statistical calculus (Cluj-Napoca, 1987), 85–92, Preprint, 87-9 University, Babes-Bolyai, Cluj.

    Google Scholar 

  26. Mahmoodov, N., Gupta, V., Kaffaoglu, H.: On certain q-Phillips operators, Rocky Mountain J. Math. 42(4), 1291–1312 (2012)

    Article  MathSciNet  Google Scholar 

  27. Mahmudov, N.I.: On q-parametric Szász-Mirakjan operators. Mediterranen J. Math. 7(3), 297–311 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  28. May, C.P.: Saturation and inverse theorems for combinations of a class of exponential type operators. Canad. J. Math. 28, 1224–1250 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  29. May, C.P.: On Phillips operators. J. Approx. Theory. 20, 315–322 (1977)

    Article  MATH  Google Scholar 

  30. Mazhar, S.M., Totik, V.: Approximation by modified Szász operators. Acta. Sci. Math. 49, 257–269 (1985)

    MathSciNet  MATH  Google Scholar 

  31. Micchelli, C.A.: The saturation class and iterates of the Bernstien polynomials. J. Appro. Theory. 8, 1–18 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  32. Natanson, I.P..: Constructive function theory, vol. 1, Frederic Unger, New York (1964)

    MATH  Google Scholar 

  33. Phillips, R.S.: An inversion formula and semi-groups of linear operators. Ann. Math. 59, 325–356 (1954)

    Article  MATH  Google Scholar 

  34. Phillips, G.M.: Bernstein polynomials based on the q-integers. Ann. Numer. Math. 4, 511–518 (1997)

    MathSciNet  MATH  Google Scholar 

  35. Rathore, R.K.: Linear combinations of linear positive operators and generating relations in special functions, Ph. D. Thesis submitted to IIT Delhi (1973)

    Google Scholar 

  36. Srivastava, H.M., Gupta, V.: A certain family of summation-integral type operators, Math. Comput. Model 19, 1307–1315 (2003)

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  38. Zygmund, A.: Trigonometric series. Dover Pub, Inc., New York (1995)

    Google Scholar 

  39. Yuksel, I.: Approximation by q-Phillips operators. Hacettepe J. Math. Stat. 40(2), 191–205 (2011)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, 36849-5108, Auburn, AL, USA

    N. K. Govil

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

    Vijay Gupta

Authors
  1. N. K. Govil
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  2. Vijay Gupta
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Correspondence to N. K. Govil .

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Editors and Affiliations

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

    Themistocles M. Rassias

  2. Dept. Industrial & Systems Engineering, University of Florida, Gainesville, Florida, USA

    Panos M. Pardalos

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Govil, N., Gupta, V. (2014). Approximation Properties of Phillips Operators. In: Rassias, T., Pardalos, P. (eds) Mathematics Without Boundaries. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-1106-6_8

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