Journal of Data Analysis and Information Processing, 2013, 1, 85-89 Published Online November 2013 (http://www.scirp.org/journal/jdaip) http://dx.doi.org/10.4236/jdaip.2013.14009 Open Access JDAIP The Average Errors for Linear Combinations of Bernstein Operators on the Wiener Space * Yanjie Jiang#, Ziqing Zhang Department of Mathematics and Physics, North China Electric Power University, Baoding, China Email: #jiangyj@126.com Received September 18, 2013; revised October 20, 2013; accepted November 4, 2013 Copyright © 2013 Yanjie Jiang, Ziqing Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. ABSTRACT In this paper, we discuss the average errors of function approximation by linear combinations of Bernstein operators. The strongly asymptotic orders for the average errors of the combinations of Bernstein operators sequence are deter- mined on the Wiener space. Keywords: Linear Combinations; Bernstein Operators; Weighted L-Norm; Average Error; Wiener Space 1. Introduction Let be a real separable Banach space equipped with a probability measure on the Borel sets of . Let be another normed space such that is continu- ously embedded in . By X we denote the norm in . Any such that :TFX X fTf is a measurable mapping is called an approximation operator. The p-ave rage error of is defined as T 1 ,,, :d. p pXX F eTFfTf f Let 0:0,10FfC f 0. For every 0 F set 01 :max . Ct ft Then 0,C F becomes a separable Banach space. Denote by 0 B the Borel class of 0, FC and by 0 the Wiener measure on 0 B (see [1]). From [1, p. 70] we know 00d 1 min,,,0,1 . 2 Ffsft f sts ts tst (1) The Bernstein operato r on 0, 1C defined by , 0 ,:n nn k k Bfxfp x n k ,. , where ,1,0,1, nk k nk n pxxx kn k This operator turned out to be a very interesting ope- rator, easy to deal with and having many applications in approximation theory and practice. Since Bernstein operators cannot be used in the inves- tigation of higher orders of smoothness, Butzer [2] in- troduced combinations of Bernstein operators. Ditzian and Totik [3, p. 116] extended this method and defined the combinations as 1 ,0 ,: , i m nmi n i Lfx CnBfx , (2) where and i n i Cn satisfy the following conditions: 01 1 0 1 0 1 0 () ; () ; () 1; ( )0,1,2,,1. m m i i m i i m ii i annn Cn bCnC cCn dCnn m (3) *Supported by National Natural Science Foundation of China (Project no 10871132 and 11271263) and by a grant from Hebei province higher school science and technology research (Z2010160). #Corres ondin autho . Throughout this paper, denotes a positive constant C
Y. J. JIANG, Z. Q. ZHANG 86 independent of and n , which may be a different constant in different cases. For 10,1 0,1Lp, , the weighted L-norm of 0,1fC is de fined by 1 0 :d. p 1 , p ftt t 2. Main Result Recently G. Q. Xu [4] studied the average errors of Bern- stein operators approximation on the Wiener space. Mo- tivated by [4], we considered the average errors of func- tion approximation by linear combinations of Bernstein operators. The strongly asymptotic orders for the average errors of the linear combinations of Bernstein operators sequence are determined on the Wiener space. We ob- tain: Theorem 1. Let , be given by (2), 1p ,, nm Lfx 0x 1 and 0, 1L , is continuous on . Then we have (0,1) 1 4 ,0 0 , 1 2 11 00 1 14 0 ,, , 2 2 1d , pnm p j i mm ji ij ij iij pp p eL F n n nn CnCn nnn xx xxn where 2 2 1ed 2 x p p. x Here and in the following the notation nn ab for sequences and means that n a n b lim 0 nnn ab . 3. Proof of Theorem 1 To prove Theorem 1 we need the following two lemmas. Lemma 1([5, p. 15]). If 1 02 , then , k n k nk xn pxCn for each , the constant depending only on 0kC and . k Lemma 2 ([5, p. 15]). Fo r fix ed 1 01, 3 x , the asymptotic relation 1 2 , 2 , 1 21 exp21 nk k nk nk n pxx x k nk xxn x xxn Px holds uniformly for all values of satisfying the ine- quality k . k n n (4) In other words, ,, lim 1 nk nk npxPx uniformly for all satisfying (4). k Proof of Theorem 1. From [1, p.107] we have 2 0 ,0 0 , 12 ,0 0. ,, , ,d d p pn mp pnm F LF fxLfxfx x p e v (5) By (2), 0 0 0 0 2 ,0 1 20, 00 11 000 ,, 0 00 123 ,d d2 d d 2. i i j i ij nm F n m ink Fik mm ij Fij i n n nkn sF ks ij fx Lfxf fxfCn p x k fxffCnCn n ks pxpxf ff nn AxAx Ax (6) On using (1), we obtain 0 2 10 d F. xfxf x (7) Note that ,, 00 222 , 0 1, , 1, nn nk nk kk n nk k px kpxnx kpxnx nxx (8) by (1), we have Open Access JDAIP
Y. J. JIANG, Z. Q. ZHANG 87 0 1 2, 00 1 , 00 1 , 00 () d 1 2 1. 2 i i i i i i n m ink F ik i n m ink ik ii n m ink ik i k0 x Cnpxfxff n kk Cnpxx x nn k x Cnpxx n (9) From [4,(3.24)], we know 1 2 , 0 21 . i i n nk i kii xx k pxxn nn Combining(3) and (9) we get 1 2 1 2 1 20 1 0 21 1 2 1. 2 m ii ii mi ii xx Ax xCnn n xx Cn xn n (10) Now, we estimate the term 3 x. From (2) and (8), 0 11 3, 000 0 0 11 ,, 000 0 11 , 000 , 0 d 1 2 1 2 j i ij j i ij i i j j n n mm ijnkns ijk s Fij n n mm ijnkns ijk s ijij n mm ijnk ijk n ns s , xCnCnpxp ks ff f nn Cn Cnpxpx ksks nn nn xCnCnpx k px x . ij s nn (11) Using Lemma 1 and (3), we have 512 2 ,. j j j ns sxn n pxCn Note that 0, ij ks nn 1 , we get 512 2 ,. j j j ns ij sxn n ks px Cn nn (12) By (8) and (12), we obtain 512 512512 512512 23 12 512 ,, 00 ,, 0 ,, ,, ,, j i ij i ij sj nj ij ks i nj n ij ij ks i nj n ij ij ks i nn ij n n nkn s ks ij n nkn s kij xn nkn sij xn xn nkn sij xn xn nkn s xn ks pxpx nn ks px px nn ks px px nn ks px px nn npxpx 512 . jij xn ks nn (13) For 512 i i kxn n and 512 j j sxn n , by Lemma 2, ,, 2 2 11 21 exp . 21 21 ij nkn s ij ij ij j i ij ks pxpx nn ks nn xxnn n nks xx xx nxx n (14) Set 22 121212 ,: exp 21 21 j in n. uuuuuu xx xx For 12 11 11 ,, , ii jj kk ss vv nn nn by the differential mean value theorem we have 2 2 12 22 12 112 2 exp 21 21 exp 21 21 ,, i ij j j j i ij n ks xx x nn xxn ns xxvxv xx n n nxv xv xx xx ks Fv Fv nn , i k (15) Open Access JDAIP
Y. J. JIANG, Z. Q. ZHANG Open Access JDAIP 88 By a simple computation we know where 12 ,, ,. CF C 11 1 ,, , ii j kk ss xx xx nn nn 1 j . From (15), 2 2 11 22 12 121 2 exp 2121 exp . 21 21 j i iji j j i n n ksk s xx nnx xnx xn n n xv xvxvxvn xx xx (16) Integrating two side of (16) about in 12 ,vv 11 ,, ii jj kk ss nn nn we get 2 2 111 122 12 1121 22 exp 21 21 dexp d 21 21 j i ij j i iji j s knj ni ijks nn n n ksk s xx nnx xnx xn n n nnvx vx vx vx vvn xx xx . (17) From (14)-(17), we have 512512 512512 512512 ,, ,, 1 1 1 22 1212 11 d 21 exp 21 21 ij ij ij ij ij ij j i ij ij ij nkn sij ks xn xn nn nkn s ij ks xn xn nn s k ij n n ks nn ks xn xn nn j i ks px px nn ks pxpx nn nn v xx n n xv xvxvxv xx xx 23 12 512512 512512 1 512512 2 1 122 1121 2 1 1 112 d dexp 2 12121 exp 21 j i ij ij ij ijnj i ij s k ij nj ni ks nn ks xn xn nn xn xn ij ni xn xn vn nn n n vxvxvxv xvv xxxx xx n nn n dvxvxv xx 2 d 22 1 122 d. 21 21 j n xvxvv n xx xx (18) Let 112 , 21 21 j in n wxvw xx xx 2 xv , by (18), we get
Y. J. JIANG, Z. Q. ZHANG 89 512512 112 112 11212112 12 ,, 21 2122 1 12 112 21 21 21 dexpd ij ij ij j i ii jj nkn sij ks xn xn nn n n xx xx nnnn ij xx xx ks px px nn xx ww www nn 2 . wn (19) By (3), suppose that 22 , ij ij nn cc nn , from (19) and the convergence of the improper integral 22 112 12 dexp ij wcwcw www 2 d, we have 1 2 512512 1 22 12 ,,1 122 22 12 21 11221 2 1 21 dexpd 21 21 dexpd2d exp ij ij ij j i nkn sij kij s xn xn nn nw ij ji n xx ww ks pxpxwwww n nn nn xx xx ww ww wwwww nn nn ww 1 2 1 2 11 22 2 22 22 12 12 d 11 21 21 ee dd 21 21 11 . ji ij ji jj i i ijijj i ji ij wn nn ww nn xx xx wwn nn xx xx nn nn nn nnnnn n nn nn (20) Combining (11) and (20), we obtain 1 2 11 3,, 00 00 11 00 1 2 21 . 2 j i ij n n mm ij nkns ij ksij mm ij ji ij ij ij ks Ax xCnCnpxpxnn nnn n xx xCnCn nn n (21) From (5)-(7), (10), and (21), we complete the proof of Theorem 1. REFERENCES [1] K. Ritter, “Average-Case Analysis of Numerical Prob- lems,” Springer-Verlag, Berlin, 2000. [2] P. L. Butzer, “Linear Combinations of Bernstein Polyno- mials,” Canadian Journal of Mathematics, Vol. 5, 1953, pp. 559-567. http://dx.doi.org/10.4153/CJM-1953-063-7 [3] Z. Ditzian and V. Totik, “Moduli of Smoothness,” Sprin- ger-Verlag, Berlin, 1987. http://dx.doi.org/10.1007/978-1-4612-4778-4 [4] G. Q. Xu, “The Simultaneous Approximation Average Errors for Bernstein Operators on the R-Fold Integrated Wiener Space,” Numerical Mathematics Theory Methods and Applications, Vol. 5, No. 3, 2012, pp. 403-422. [5] G. G. Lornetz, “Bernstein Polynomials,” University of Toronto, Toronto, 1953. Open Access JDAIP
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