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Linear Combinations of Bernstein Polynomials

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Bernstein Operators and Their Properties
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Abstract

For complex functions we do not find such a good variety of results as the ones presented in previous chapters. For instance, there are no converse results.

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References

  1. P.N. Agrawal, Direct, inverse and saturation theorems in ordinary and simultaneous approximation by iterative combinations of exponential type operators, Thesis, Kanpur, 1979

    Google Scholar 

  2. P.N. Agrawal, H.S. Kasana, On the iterative combinations of Bernstein polynomials. Demonst. Math. 17, 777–783 (1984)

    MathSciNet  MATH  Google Scholar 

  3. P.L. Butzer, Linear combinations of Bernstein polynomials. Can. J. Math. 15, 559–567 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  4. L. Chen, L. Xie, Equivalent theorems on simultaneous approximation by combination of Bernstein operators. Anal. Theory Appl. 22 (3), 246–253 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Della Vecchia, G. Mastroianni, J. Szabados, Weighted approximation of functions with endpoint and inner singularities by Bernstein operators. Acta Math. Hungar. 103, 19–41 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  6. C. Ding, F. Cao, K-functionals and multivariate Bernstein polynomials. J. Approx. Theory 155, 125–135 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Z. Ditzian, A global inverse theorem for combinations of Bernstein polynomials. J. Approx. Theory 26, 277–292 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  8. Z. Ditzian, G. Freud, Linear approximating processes with limited oscillation. J. Approx. Theory 12, 23–31 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  9. Z. Ditzian, C.P. May, A saturation result for combinations of Bernstein polynomials. Tôhoku Math. J. 28, 363–372 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  10. Z. Ditzian, V. Totik, Moduli of Smoothness (Springer, New York, 1987)

    Book  MATH  Google Scholar 

  11. B.R. Draganov, Upper estimates of the approximation rate of combinations of iterates of the Bernstein operator. Annuaire Univ. Sofia Fac. Math. Inform. 101, 95–104 (2013)

    MathSciNet  Google Scholar 

  12. B.R. Draganov, On simultaneous approximation by iterated Boolean sums of Bernstein operators. Result. Math. 66, 21–41 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  13. B.R. Draganov, Strong estimates of the weighted simultaneous approximation by the Bernstein and Kantorovich operators and their iterated Boolean sums. J. Approx. Theory 200, 92–135 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  14. G. Felbecker, Linearkombinationen von iterierten Bernsteinoperatoren. Manuscrip. Math. 29, 229–248 (1979) (in German)

    Article  MathSciNet  MATH  Google Scholar 

  15. M. Frentiu, Linear combinations of Bernsteln polynomials and of Mirakjan operators. Studia Univ. Babes-Bolyai Ser. Math. Mech. 15, 63–68 (1970) (in Romanian)

    MathSciNet  Google Scholar 

  16. W. Gawronski, U. Stadtmüller, Linear combinations of iterated generalized Bernstein functions with an application to density estimation. Acta Sci. Math. (Szeged) 47, 205–221 (1984)

    MathSciNet  MATH  Google Scholar 

  17. H.H. Gonska, X.-L. Zhou, Approximation theorems for the iterated Boolean sums of Bernstein operators. J. Comput. Appl. Math. 53, 21–31 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  18. S. Guo, C. Li, Q. Qi, Approximation by Bernstein operators. J. Math. Res. Res. Expos. 25 (4), 749–756 (2005) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  19. S. Guo, C. Li, X. Liu, Z. Song, Pointwise approximation for linear combination of Bersntein operators. J. Approx. Theory 107, 109–120 (2000)

    Article  MathSciNet  Google Scholar 

  20. S. Guo, H. Tong, G. Zhang, Pointwise weighted approximation by Bernstein operators. Acta Math. Hungar. 101 (4), 293–311 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Guo, S. Yue, C. Li, G. Yang, Y. Sun, A pointwise approximation theorem for linear combinations of Bernstein polynomials. Abstract Appl. Anal. 1 (4), 397–406 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  22. X. Huo, C. Li, Q. Yao, Pointwise approximation for the iterated Boolean sums of Bernstein operators. Chin.Quart. J. Math. 28 (2), l29–l38 (2013)

    Google Scholar 

  23. H. Jiang, L. Xie, Direct and inverse theorems on simultaneous approximation by combinations of Bernstein operators. Chinese J. Eng. Math. 20 (3), 99–104 (2003) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  24. W. Lu, Direct and inverse estimates for combinations of Bernstein polynomials with endpoint singularities. J. Prog. Res. Math. 5 (4), 582–593 (2015)

    Google Scholar 

  25. G. Mastroianni, M.R. Occorsio, Una generalizzazione dell’operatore di Bernstein. Rend. Accad. Sci. Fis. Mat. Napoli 44 (4), 151–169 (1977)

    MathSciNet  MATH  Google Scholar 

  26. O.P. May, 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 

  27. C.A. Michelli, Saturation classes and iterates of positive operators, Thesis, Stanford University (1969)

    Google Scholar 

  28. C. Micchelli, The saturation class and iterates of the Bernstein polynomials. J. Approx. Theory 8, 1–18 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  29. D.A. Naiko, Approximation of the derivatives of some combinations of operators of class B. Ukrain. Math. J. 39 (5), 473–477 (1987)

    Article  MathSciNet  Google Scholar 

  30. I. Raşa, Iterated Boolean sums of Bernstein and related operators. Rev. Anal. Numér. Approx. 35 (1), 111–115 (2006)

    MathSciNet  MATH  Google Scholar 

  31. R.K.S. Rathore, Linear combinations of linear positive operators and generating relations in special functions, Thesis, Delhi (1973)

    Google Scholar 

  32. R.K.S. Rathore, P.N. Agrawal, Inverse and saturation theorems for derivatives of exponential type operators. Indian J. Pure Appl. Math. 13 (4), 476–490 (1982)

    MathSciNet  MATH  Google Scholar 

  33. A.P. Roslyakov, Approximations of functions and their derivatives by combinations of Bernstein polynomials. Izv. Vyssh. Uchebn. Zaved. Mat. 6, 34–38 (1993) (in Russian)

    MathSciNet  MATH  Google Scholar 

  34. J.C. Sevy, Convergence of iterated Boolean sums of simultaneous approximants. Calcolo 30 (1), 41–68 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  35. K.S. Singh, Global inverse theorems of general order in simultaneous approximation by linear combinations of certain linear positive operators, Thesis, Kanpur (1990)

    Google Scholar 

  36. R. Song, Equivalence theorems for simultaneous approximation by linear combination of Bernstein operators. J. Zhejiang Univ. Sci. Ed. 27 (1), 35–41 (2000) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  37. V.V. Tihomirov, The approximation of functions with a given rate of convergence. Izv. Vyss. Ucebn. Zaved. Matematika 10 (173), 113–115 (1976) (in Russian)

    MathSciNet  Google Scholar 

  38. L. Xie, Uniform approximation by combinations of Bernstein polynomials. Approx. Theory Appl. 11 (3), 36–51 (1995)

    MathSciNet  MATH  Google Scholar 

  39. L. Xie, On the linear combination of Bernstein polynomials. J. Math. Res. Expos. 15 (4), 577–582 (1995) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  40. L. Xie, Approximation order for combinations of Bernstein polynomials. J. Zhejiang Normal Univ. 18 (1), 75–77 (1995) (in Chinese)

    Google Scholar 

  41. L. Xie, Pointwise direct estimate for combinations of some exponential-type operators. Approx. Theory Its Appl. 16 (2), 71–79 (2000)

    MathSciNet  MATH  Google Scholar 

  42. L. Xie, Direct and inverse results for combination and derivatives of Bersntein operators. Adv. Math. 29 (2), 173–175 (2000)

    Google Scholar 

  43. L. Xie, Pointwise approximation theorems for combinations and derivatives of Bernstein polynomials. Acta Math. Sinica 21 (5), 1241–1248 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  44. L. Xie, Pointwise simultaneous approximation by combinations of Bernstein operators. J. Approx. Theory 137, 1–21 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  45. L. Xie, The saturation class for linear combinations of Bernstein operators. Arch. Math. 91, 86–96 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  46. L. Xie, Strong type of Steckin inequality for the linear combination of Bernstein operators. J. Math. Anal. Appl. 408, 615–622 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  47. L. Xie, T. Xie, A pointwise inverse theorem for combinations of Bernstein polynomials. Chin. Ann. Math. Ser. 20 (4), 467–478 (1999) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  48. L. Xie, X.P. Zhang, Pointwise estimate on simultaneous approximation by linear combination of Bernstein operators. Appl. Math. J. Chinese Univ. Ser. A 17 (4), 479–484 (2002) (in Chinese)

    Article  MathSciNet  MATH  Google Scholar 

  49. D. Yu, Weighted approximation of functions with singularities by combinations of Bernstein operators. Appl. Math. Comput. 206, 906–918 (2008)

    MathSciNet  MATH  Google Scholar 

  50. D.S. Yu, D.J. Zhao, Approximation of functions with singularities by truncated Bernstein operators. Southeast Bull. Math. 30, 1178–1189 (2006)

    MathSciNet  MATH  Google Scholar 

  51. L. Zhang, Theorems for pointwise approximation for linear combination of Bernstein polynomial. J. Lishui Teach. Cole. 16 (2), 3–5 (1994) (in Chinese)

    Google Scholar 

  52. D. Zhao, Rate of convergence of linear combination of Bernstein operators with Jacobi weights. J. Shaoxing Coll. Arts Sci. Nat. Sci. 20 (5), 17–22 (2000) (in Chinese)

    MATH  Google Scholar 

  53. D. Zhao, B. Bao, D. Yu, Linear combinations of Bernstein operator and approximation with Jacobi weights. Appl. Math. J. Chinese Univ. Ser. A 22 (1), 91–102 (2007) (in Chinese)

    MathSciNet  Google Scholar 

  54. D.-X. Zhou, Rate of convergence of Bernstein operators with Jacobi weights. Acta Math. Sin. 35, 331–338 (1992) (in Chinese)

    MathSciNet  MATH  Google Scholar 

  55. D.-X. Zhou, On smoothness characterized by Bernstein type operators. J. Approx. Theory 81, 303–315 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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Bustamante, J. (2017). Linear Combinations of Bernstein Polynomials. In: Bernstein Operators and Their Properties. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-55402-0_9

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