We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein operators

Jia, Jin-Ping

Wenming, Lu

English

We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein operators

WEN-MING LU

JIN-PING JIA

NEUROSURGERY ENTHUSIASTIC WOMEN SOCIETY

Pointwise Weighted Approximation of Functions with Endpoint Singularities by Combinations of Bernstein Operators

Scitech Research Journals

                                                                      Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1039|  SCITECH                                                                         Volume 7, Issue 3  RESEARCH ORGANISATION                  Published online: April 27, 2016| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals     Pointwise Weighted Approximation of Functions with Endpoint Singularities by Combinations of Bernstein Operators WEN-MING LU1, JIN-PING JIA2 1School of Science, Hangzhou Dianzi Unviersity, Hangzhou, People's Republic of China 2School of Mathematics and Statistics, Tianshui Normal University, People's Republic of China   Abstract. We give direct and inverse theorems for the weighted approximation of functions with endpoint singularities by combinations of Bernstein operators. 1991 Mathematics Subject Classification: Primary 41A10, Secondary 41A17. Keywords and phrases: Combinations of Bernstein polynomials; Functions with endpoint singularities; Direct and inverse results.  Introduction The set of all continuous functions, defined on the interval I, is denoted by C(I). For any the corresponding Bernstein operators are defined as follows:  Where  Approximation properties of Bernstein operators have been studied very well (see [2], [3], [5]-[8], [12]-[14], for example). In order to approximate the functions with singularities, Della Vecchia et al. [3] and Yu-Zhao [12] introduced some kinds of modified Bernstein operators. Throughout the paper, C denotes a positive constant independent of n and x, which may be different in different cases. Let  and  The norm in  is defined by  Define                                                                        Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1040|                                                                        Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1041|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1042|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1043|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1044|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1045|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1046|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1047|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1048|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1049|                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1050|                                                                        Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1051|  References [1] P.L. Butzer, Linear combinations of Bernstein polynomials, Canad. J. Math. 5 (1953), pp. 559-567. [2] H. Berens and G. Lorentz, Inverse theorems for Bernstein polynomials, Indiana Univ. Math. J. 21 (1972), pp. 693-708. [3] D. Della Vechhia, G. Mastroianni and J. Szabados, Weighted approximation of functions with endpoint and inner singularities by Bernstein operators, Acta Math. Hungar. 103 (2004), pp. 19-41. [4] Z. Ditzian, A global inverse theorem for combinations of Bernstein polynomials, J. Approx. Theory 26 (1979), pp. 277-292. [5] Z. Ditzian and V. Totik, Moduli of Smoothness, Springer-Verlag, Berlin, New York (1987). [6] S.S. Guo, C.X. Li and X.W. Liu, Pointwise approximation for linear combinations o Bernstein operators, J. Approx. Theory 107 (2000), pp. 109-120. [7] S.S. Guo, H. Tong and G. Zhang, Pointwise weighted approximation by Bernstein operators, Acta Math. Hungar. 101 (2003), pp. 293-311. [8] G.G. Lorentz, Bernstein Polynomial, University of Toronto Press, Toronto (1953).                                                                       Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                               ISSN: 2395-0218   Volume 7, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                1052| [9] L.S. Xie, Pointwise simultaneous approximation by combinations of Bernstein operators, J. Approx. Theory 137 (2005), pp. 1-21. [10] L.S. Xie, The saturation class for linear combinations of Bernstein operators, Arch. Math. 91 (2008), pp. 86-96. [11] D.S. Yu,weighted approximation of functions with singularities by combinations of Bernstein operators, J.Applied Mathematics and Computation. 206(2008),pp.906-918. [12] D.S. Yu and D.J. Zhao, Approximation of functions with singularities by truncated Bernstein operators, Southeast Bull. Math. 30 (2006), pp. 1178-1189. [13] D.X. Zhou, Rate of convergence for Bernstein operators with Jacobi weights, Acta Math. Sinica 35 (1992), pp. 331-338. [14] D.X. Zhou, On smoothness characterized by Bernstein type operators, J. Approx. Theory 81 (1994), pp. 303-315. [15] J.J. Zhang, Z.B. Xu, Direct and inverse approximation theorems with Jacobi weight for combinations and higher derivatives of Baskakov operators(in Chinese), Journal of systems science and mathematical sciences. 2008 28 (1), pp. 30-39.  

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Pointwise Weighted Approximation of Functions with Endpoint Singularities by Combinations of Bernstein Operators

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