We give direct and converse results for the weighted approximation of functions with inner singularities by a new type of Bernstein operators

Jia, Jin-ping

Lu, Wen-ming

English

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Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     561  SCITECH                                                                           Volume 5, Issue 3  RESEARCH ORGANISATION|                Published online: October 3, 2015| Journal of Progressive Research in Mathematics www.scitecresearch.com/journals     Another weighted approximation of functions with singularities by combinations of Bernstein operators Wen-ming Lu1, Jin-ping Jia2 1 School of Science, Hangzhou Dianzi Unviersity, Hangzhou, 310018 P.R. China. 2 School of Mathematics and Statistics, Tianshui Normal University.  Abstract We give direct and converse results for the weighted approximation of functions with inner singularities by a new type of Bernstein operators. Keywords: Combinations of modified Bernstein polynomials; Functions with singularities; Weighted approximation; Direct and inverse results.  1. Introduction The present work continues to study modified Bernstein operators following [11]. Here, the notation is referred to [11], for convenience, these notations will be listed. 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  Let  and   The norm  is defined as  and Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     562  For we define the weighted modulus of smoothness by  where  and  The weighted K-function is given by  It was shown in [5] that   Della Vecchia et al. firstly introduced  and  in [3], where the properties of                   and                   are studied. Among others, they prove that  Where  In [11], for any                                                          there hold  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     563  and for any   The main purpose of the present paper is to give another new type of combinations of Bernstein operators so as to obtain higher approximation order. Throughout the paper, C denotes a positive constant independent of  and which may be different in different cases. 2. Main Results For any positive integer r, we consider the determinant  We obtain  Thus, there is a unique solution for the system of nonhomogeneous linear equations:  Let  with the coefficients       satisfying (2.1). From (2.1), we see that   for                            Moreover, it holds that                  and  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     564 Let  Where   and  Where   Further, let  Set    is a linear is a linear polynomial of degree r, and                                                    provided that  We define our combinations of Bernstein operators as follows:  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     565 where   satisfy the conditions (a)-(d).  We have our main results as follows: Theorem. For any  we have                                                                                           and for                    3. Lemmas Lemma 1. ([13]) For any non-negative real u and v, we have                             Lemma 2. For any positive real  and we have                             Proof. It follows from Kolmogolov's inequality that                                      Moreover,                   Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     566 When   then   Morever we haveTherefore  Which, together with (3.3), gives that  Similarly, we can prove that the above inequality also holds when Therefore we obtain that  Now, the result follows from (3.4) when  and thus the result can be deduced from (3.4) by induction when  Lemma 3.  we have                                             The proof is similar to the lemma 3 of [?],we don't give a proof here. Lemma 4. For any   and  we have                               The proof is similar to the lemma 4 of [11], we don't give a proof  here. Lemma 5. ([3]) Let                                                                      Then                                       for  and  Lemma 6. For we have Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     567                     Proof. By (3.1) and the lemma 5, we have  4 Proof of Theorem 1 4.1 Proof of (2.2) We first prove   (The same as  now   We have Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     568  Similarly, we can get and  When  according to [5], we have    Where  is a linear function. If                  when    we have  then and  By (3.8), then  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     569 By a simple calculation, we have  By (3.1), then  We note  If    we have   If   then  we have  It follows from combining the above inequalities (4.1) and (4.2) that the theorem is proved. 4.2 Proof of (2.3) (1) When  we discuss it as follows: Case 1. If by (2.2), we have  Case2. If  we have                        Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     570  Where  We can easily get                                                                                        By bringing these facts (4.3) and (4.4) together, the theorem is proved. 2) When  we have                   If  we have                          If  we have                              Similarly  By (4.5) and (4.6), we have Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     571  By (4.6), we have  Where  is a linear function. If   when                 we have                                        by (3.1) and (3.7), then  Similarly, we can get   Soand by (4.7), we have Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     572  Analogously,  then the theorem is proved. Corollary 1. If and  we have  Corollary 2. If  and  we have  4.3 Proof of (2.4)  Analogously, the theorem can be proved easily.   4.4. Proof of (2.5) We assume  then Recall that [?], then                                    Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     573 Case 1.  By using Taylor expansion, we have  By (3.2), (3.7) and (4.9), we have for  then  By (4.8) and (4.10) , we have  If  is between  and     , we have  Then   We have Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     574  Analogously, we can get  We note that  max   is a linear function. If   then    So we have             If  by  we have                 For  we have  Similarly, we have  By bringing these facts together, we have  Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     575 Case 2.  (Similarly as    By using Taylor expansion, we have   Analogously, we can get  So, we have  Then Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     576  If  there exists                      by (2.4)  and the first inequality of (2.5), then  4.5. Proof of (2.6) From the proof of (2.5), we actually have   Therefore,  implies  By (2.3) and (2.4), we may choose g properly such that  and  Hence, by [5], we obtain the converse inequality. 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. Journal of Progressive Research in Mathematics(JPRM)                                                                                                                                                            Volume 5, Issue 3 available at www.scitecresearch.com/journals/index.php/jprm                                                     577 [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 of 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). [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. 

Another weighted approximation of functions with singularities by combinations of Bernstein operators

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Another weighted approximation of functions with singularities by combinations of Bernstein operators

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