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Applied Mathematics, 2011, 2, 389-397 doi:10.4236/am.2011.24047 Published Online April 2011 (http://www.SciRP.org/journal/am) Copyright © 2011 SciRes. AM Pointwise Approximation Theorems for Combinations of Bernstein Polynomials with Inner Singularities Wenming Lu1, Lin Zhang2 1Department of Mathematics, Hangzhou Dianzi University, Hangzhou, China 2Department of Mathematics, Zhejiang University, Hangzhou, China E-mail: lu_wenming@163.com, linyz@zju.edu.cn, godyalin@163.com Received October 8, 2010; revised January 14, 2011; accepted January 17, 2011 Abstract It is well-known that Bernstein polynomials are very important in studying the characters of smoothness in theory of approximation. A new type of combinations of Bernstein operators are given in [1]. In this paper, we give the Bernstein-Markov inequalities with step-weight functions wx for combinations of Bernstein polynomials with inner singularities as well as direct and inverse theorems. Keywords: Bernstein Polynomials, Inner Singularities, Pointwise Approximation, Bernstein-Markov Inequalities, Direct and Inverse Theorems 1. Introduction The set of all continuous functions, defined on the inter- val I , is denoted by CI . For any 0,1fC, the corresponding Bernstein operators are defined as fol- lows: , 0 ,: n nk k Bpx k fxf n , where ,:C1 nk kk nk n pxx x , 0,1,,kn, 0,1x. Approximation properties of Bernstein operators have been studied very well (see [2-7], for example). In order to approximate the functions with singularities, Della Vecchia et al. [8] 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. Ditzian and Totik extended the method of combinations and defined the following combinations of Bernstein operators: 1 , 0 ,: , i r nri n i Bfx CnBfx , with the conditions: a) 01 1r nn nnCn , b) 1 0() r i iCn C , c) 1 0() 1 r i iCn , d) 1 00 rk ii iCnn , for 1, ,1kr. For any positive integer r, we consider the determi- nant 11 11 212 22341 221 2121 2223441 : 22132242322 41 r rr rr rrr rrrrr A rrr rr We obtain 2 2!0 r rj Aj . Thus, there is a unique solution for the system of nonhomogeneous linear equations: W. M. LU ET AL. Copyright © 2011 SciRes. AM 390 12 21 12 21 12 21 12 21 1, 21 22410, 212 22214410, 21!3 2122410. r r r r aa a rar ara rrarrarra ra rarra (1.1) Let 21 2241 12 21 1, 01, :0, 0, 1, 1. rr r r axaxa xx xx x with the coefficients 12 21 ,,,r aa a satisfying (1.1). From (1.1), we see that (2 ) (,) () r xC , 01x for 01 x . Moreover, it holds that 11 , 00 i , 0,1,, 2i r and 10 i , 1,,2ir. Let 1 1 ,: r ii i H fxf x lx , and 1 1, 1 1, : r j ji j ir j ji ij x x lx x x , 12 i nr i xn , 1,2,,1ir. Further, let 1 2 , nn xn 2, nn xn 3, nn xn 4 2nn xn and 1 1 21 x x x x x , 3 2 43 x x x x x . Set 12 12 ,: 1 1. n F fx Fx fxxx xxHx We have 52 32 11 5232 32 12 22 1232 ,0,,1, 1,,, , ,,, 1,,. rr rr n rr rr fxx xx fxxxHxx xx Ffx Hxx xx Hxxxfxxxx Obviously, , n F fx is linear, reproduces polynomi- als of degree r, and 2 ,0,1 r n Ffx C, provided that 20, 1 r fC. Now, we can define our new com- binations of Bernstein operators as follows: 1 ,, 0 ,, , i r nrnr ninn i BfxBFx CnBFx (1.2) where i Cn satisfy the conditions (a)-(d). 2. The Main Results Let :0,1R , 0 be an admissible step-weight function of the Ditzian-Totik modulus of smoothness, that is, satisfies the following conditions: 1) For every proper subinterval ,0,1ab there exists a constant 1,0CCab such that 1 1 1 CxC for , x ab. 2) There are two numbers 00 and 10 for which 0 1 ,as 0, 1,as 1. xx x xx ~ (X~Y means C-1Y ≤ X ≤ CY for some C). Combining conditions (I) and (II) on , we can de- duce that 1 22 Cx xCx , 0, 1x, where 1 0 21xxx . W. M. LU ET AL. Copyright © 2011 SciRes. AM 391 Let wx x ,01 ,0 and 0,1\: lim0 wx CfC wfx . The norm in w C is defined as 01 :sup w wC x f wfwfx . Define 1 :: 0,1, rr rr w WfCf ACwf , 1 ,:: 0,1, rr rr w WfCfAC wf . For w fC, we define the weighted modulus of smoo- thness by 001 ,:supsup rr hx whtx Wftwx fx , where 0 1+ 2 rk rk hr k r f xCfxkhx , 0 1C + rk rk hr k f xfxrkh . Recently Felten showed the following two theorems in [4]: Theorem A. Let =1 x xx and let :0,1 R , 0 be an admissible step-weight func- tion of the Ditzian-Totik modulus of smoothness ([3]) such that 2 and 22 are concave. Then, for 0, 1fC and 02 , 212 ,, n x Bfx fxCfn x . Theorem B. Let =1 x xx and let :0,1 R , 0 be an admissible step-weight func- tion of the Ditzian-Totik modulus of smoothness such that 2 and 22 are concave. Then, for 0,1fC and 02 , 12 , n x Bfx fxCnx implies 2, f tOt . Our main results are the following: Theorem 2.1. For any 0 , 1 min0 ,1, 2w f C , we have 2 ,1 , r r r nr w xx BfxCnwf . (2.1) Theorem 2.2. For any 0 , r f W , we have ,1 , rr rr nr wxxBfxw fC . (2.2) Theorem 2.3. For w f C , 0 , 1 min0,12 , 00, r , 1 nxx n , we have 0 0 ,1 1/2 , , nr nr w wxf xBfx x OftOt nx (2.3) 3. Lemmas Lemma 3.1. For any non-negative real u and v, we have 1 , 1 1 uv nv u nk k nn pxCx x knk . (3.1) Lemma 3.2. If R , then 2 , 0 n nk k pxknx Cnx . (3.2) Lemma 3.3. For any r f W , 0 , we have rr rr n wFCw f . (3.3) Proof. We first prove 5232 , rr xxx (The same as the others), we have 12 : rr rr n r r n wx xFxwx xfx wxxf xF I Ix Obviously 1 r r Cw fI For 2 I , we have 0 2 2 r r n i rri r n i wxxf xFx wxxnfxFx I By [3], we have 52 32 52 3252 32 , 22 ,, . rr rr rr ri n xx ri i r xx xx fxFx Cnf Hnf So W. M. LU ET AL. Copyright © 2011 SciRes. AM 392 52 32 52 32 1 , 2 2 , 2 : rr rr r r xx r r xx Cn w xxfH Cw xxTf I T By Taylor expansion, we have 1 0 1 ! 1d. 1! i u ru i i u xrr i x xx fxf x u x sfss r (3.4) It follows from (3.4) and the identity 1 rvv ii i x lx Cx , 0,1, ,vr. we have 10 1 1 10 1 1 1 ,! 1 d 1! 1 d, 1! i i u rr u i i iu rxrr ii x i ru r uv uvv uii uv i rxrr ii x i xx Hfxf xlx u lxxsfs s r f xC fxCxxlx lxxsfs s r which implies that 1 1 , 1d, 1! i r rxrr r ii x i wxxf xHfx wxxl xxsfss r Since i lx C, for 5232 , rr xxx , 1, ,ir. It follows from 11rr ii x sxx ws wx , s between i x and x , then 1 1 /2 , d . i r rxrr r i x i r r r wxxf xHfx Cw xxxsfss C nwf So 2 r r Cw fI . Then, the lemma is proved. According to methods of Lemma 3.3, we can easily get: Lemma 3.4. If r f W , 0 , then , r r nr x wxgxH gxCwg nx (3.5) Lemma 3.5. For any 0 , w fC, we have ,1nr wBfC wf . (3.6) Proof. By (1.2), we have ,1 ,1 1 1 , 01 1 ,0 0 1 , 0 123 ,, . 0 1 : i i i ii nrnr n n r in nk ik i r inn i r innn i wxBfxwxBF x k wxCn Fpx n wxC n Fpx wxC n Fp II x I Now, the theorem can be proved easily. Lemma 3.6. Let 1 min0 ,12 , then for rN , 1 08 tr and 1 22 rt rt x , we have 22 1 1 22 dd tt r rrr kr tt k x uu uCtx (3.7) Lemma 3.7. Let , : nnk knn A xwxpx , then /2 n Ax Cn for 01 and 0 . Proof. If 3 xn , then the statement is trivial. Hence assume 3 0xn ( the case 31x n can be treated similarly). Then for a fixed x the maxi- mum of ,nk px is attained for : n kk nn . By using Stirling’s formula, we get , 1n n nnn n nk k nk knk nn nn nnx x e pxC knk knk ee 11 . nn knk nn nn knxknx C knk n W. M. LU ET AL. Copyright © 2011 SciRes. AM 393 Now from the inequalities 1 1, 2 n knx nnnxnx nnx and 2 1 2 1uu ue , 1u ue, 0u. We have that the second inequality is valid. To prove the first one we consider the function 2 1 21 uu ueu . Here 00 , 2 1 2 11 uu uue , 00 , 2 1 2 20 uu uuue , whence 0u for 0u. Hence 2 , 1 exp 2 n nn nk nn nn knxknx C px kknx kk n 2 2 exp 2 Cn x n n knx Ce k n . Thus 2 Cn x n AxCxe . An easy calcula- tion shows that here the maximum is attained when C xn and the lemma follows. Lemma 3.8. For 01 , , 0 , we have 2 ,nk knn wxpx knxCnx (3.8) Proof. By (3.2) and the lemma 3.7, we have 21 2 1 2, 1 2 22 ,() . n n nnk knn n n nk kn n wxwxp x pxknxCn x Lemma 3.9. For any 0 , r f W ,we have ,1 rr nr wBfCn wf . (3.9) Proof. We first prove 1 0,xn (The same as 1 1,1xn , now ,1 2 1, 00 2 , 000 2 ,0 00 2 00 , ! ! i i i i i i r nr nr rr i innrk ik ii n nr rr rj irn nrk ikj i rr rj irnnr ij i rr rj i irn ij i wxB fx nk wxC nFpx nr n kr j Cw xnCFpx n rj Cw xnCFpx n nj Cw xnCFn , 1 2 , 010 123 :. ii i i nrnr nr rr rj irn nrk ikji px kr j Cw xnCFpx n HHH We have 2 1,0 00 2 00 1 . i i rr r in nr ij i rr nr i r ij i r rj HCwx nFpx n nx Cn wfx rjn Cn wf Similarly, we can get 2 r H Cn wf, and 3 r H Cn wf. When 11 ,1xnn , according to [3], we have 23 ,1 ,1 2 2 00 , 2 2 00 ,12 ,, , , : i i i i rr nrnr n rr rj jiii ij j nnk kn Aii rr rj jiii ij j nk xknx ii wxBfxwxBF x wxxQxn Cnn kk xFpx nn wxxQxn Cnn kk xHpx nn where 23 :0, ,1Axx , H is a linear function. If i k A n , when 2 1ii i wx Cnknx wkn , we have 2 i i n kn , also 2 ,1 rj ji i Qxnnx x , and 22 2 , rj rj jii i xQxnnCnx . By (3.2), then W. M. LU ET AL. Copyright © 2011 SciRes. AM 394 2 2 1 , 2 000 2 2 2 ,12 2 000 1:. i i i i rj j n rr i nnk ijk ii rj j n rr i ii nk ij ki nkk Cw xxFpx nn x nk Cwfnk nxxpxII n x By a simple calculation, we have 1 r I Cn wf. By (3.2), then 2 2 2 2 , 2 00 0 i i rj n rr jjr i iink ij k n I CwfnknxpxCnwf x . We note that 14 max ,: i k H Hx HxHa n . if 14 , x xx ,we have wx wa.So, if 14 , x xx , then 2 rr Cn waHaCnwf . If 14 , x xx , then 2 i wa n , by (3.8), we have 23 2 2 2 22 00 ,. i i rj rr i i ij j r nk xknx i n Cw aHawxnx k xpxCnwf n It follows from combining the above inequalities that the lemma is proved. 4. Proof of Theorems 4.1. Proof of Theorem 2.1 When w fC, min0,11 2 , we discuss it as follows: Case 1. If 1 0xn , by (3.9), we have () ,1 2 ,1 , , rr nr rr r r nr r wx xBfx x CxwxB fxCnwf x (4.1) Case 2. If 1 xn , we have ,1 ,1 2 2 00 ,, ,. rr nrnr n rr r j j ii i ij BfxBFx x QxnCnn , 0 i i j n nnk kii kk x Fpx nn , where 2 ,1 rj ji i Qxnnx x , and 22 2 , rj rj jii i xQxnnCnx So ,1 2 2 , 2 000 2 2 2 00 2 2 ,2 00 , i i i i r rr nr rj j n rr i nnk ijk ii rj j rr ri n ijknAii rj rr ri nk ij wx xBfxCwxx nkk xFpx nn x nkk Cw xxxF nn x n pxCwx xx k xn 23 ,12 :. i i j nk xknx ii k Hpx n (4.2) where 23 :0, ,1Axx , we can easily get 2 1 r Cn wf , and 2 2 r Cn wf . By bringing these facts together, the theorem is proved. 4.2. Proof of Theorem 2.2 When r f W , by [3], we have 2 ,11 , 00 , i i i nr r rrr nrniinnrk ik i n k BFxCnnFp x n . (4.3) W. M. LU ET AL. Copyright © 2011 SciRes. AM 395 If 0i knr, we have 1 10d i i r r rr n nin ii n kk F CnFu u nn . (4.4) If 0k, we have 1 10 0d i i r r rr n nn n F CuF uu . (4.5) Similarly 1 12 111d. i i rr rr ir ni n n i n nr F CnuFu u n (4.6) By (4.3), we have ,1 2 1, 00 1 2 1, 01 2 1,0 0 2 1, 0 , 0 1. i i i i i i i i ii i r r nr nr r rrr innrk ik i n nr r rrr innrk ik i n r rrr innr in r rrr innrnr in wx xBfx k Cw xxnFpx n k Cw xxnFpx n Cw xxnFpx Cw xxnFpx (4.7) which combining with (4.4)-(4.6) give ,1 , rr rr nr wxxBfxCwf . Combining with the theorem 2.1 and theorem 2.2, we can obtain Corollary For any 0 , 01 , we have ,1 12 1 2 , , max ,,, ,. { r r nr rr r w r rr w wxxBfx Cnnx wff C Cwff W (4.8) 4.3. Proof of Theorem 2.3 4.3.1. The Direct Theorem We know 1 1 d 1! nnn trr n x FtFxFt x tu F uu r (4.9) ,1 ,0 k nr Bxx , 1, 2,,1kr (4.10) According to the definition of r W , for any r g W , we have ,1 ,1 ,, rr nrnr n BgxBGgx , and ,1 ,1 ,,,, nnrn nrrn w xGxBGxwxBR Gtxx , there of 1 ,, d trr rn n x RGtxtuGuu , we have ,1 1 ,1 1 12 ,1 2 1 12 ,1 2 , , d, d,. nnrn r t r r nnr r x r t r r nnr r x r t nr x wx GxBGx tu Cw GwxBdux wu u tu CwGwx Bux u tu Bux wu (4.11) also 11 2222 d, d rrrr tt rr xx tutx tutx uC uC uxwuwx . (4.12) By (3.2), (3.3) and (4.12), we have ,1 ,1 , , nnrn r r rr nnr r r nr n wx GxBGx Cw GxBtxx x CwG nx . (4.13) By (3.3), (3.5) and (4.13), when r g W , then ,1 ,1 ,, ,, , δδ . nr n nnr rr rr nn rr n wx gxBgxwx gxGgx wxGgx Bgxwxgx Hgx xx CwGCwg nx nx (4.14) For , w f C we choose proper r g W , by (3.6) and (4.14), then ,1 ,1 ,1 , ,, δ ,. nr nr nr n r w wxf xBfxwxfxgx wx Bfgxwx gxBgx x Cf nx 4.3.2. The Inverse Theorem The weighted K-function is given by W. M. LU ET AL. Copyright © 2011 SciRes. AM 396 ,,inf : r rrrr rwg KftwfgtwggW . By [3], we have 1 , ,, , rrr r ww w CftKftCft . (4.15) Proof. Let 0 , by (4.15), we choose proper g so that , r w wf gCf , , r rrr w wg Cf . (4.16) For rN , 1 08 tr and 1 22 rt rt x , we have 0 ,1 ,1,1 1 22 2 ,1 1 0 1 22 ,,, 2,dd 2 rrr r hhnrh nrh nr hx hx n r r r j rnrk r hx hx j k h wxfxwxfxBfxwxBf gxwxBgx r xjhx CnwxBfgxu uu r xjhx wx 22 ,11123 1 22 ,dd: hx hxr r nrk r hx x k BgxuuuJJJ (4.17) Obviously 0 11/2 nx JC nx . (4.18) By (3.9) and (4.16), we have 22 21 22 dd ,δ hx hx rrrrr r hx hxw JCnwfgu uCnh xf (4.19) By the first inequality of (4.8) and (4.16), we let 1 , then 22 2 2 2 1 1 22 dd ,δ. hx hx r r r rrrrr kr hx hxw k JCnwfgxuu uCnhxxf (4.20) By (3.7) and (4.16), we have 1 22 3 1 11 22 dd δ,δ hx hxrr r rrrrr kkr hx hxw kk JCwgwxwxuxuuuChf (4.21) Now, by (4.17)-(4.21), there exists a constant 0M so that 0 0 2 1/2 1 2 1/2 min ,,, ,,. r r n rrrrrrrr hrww r nn rrr rrr ww xx wxf xCnnxhfhf nxx xx ChMnfhf x nx When 2n, we have 11 1 22 21 12 nnn nxnx nx , Choosing proper x, δ, nN, so that 111 22 1 nn x x nn x x , Therefore 0, rrrr hw wxf xChf which implies 0 ,, rrrr ww ft Chf . W. M. LU ET AL. Copyright © 2011 SciRes. AM 397 So, by Berens-Lorentz lemma in [3], we get 0 , r w f tCt . 5. References [1] D. S. Yu, “Weighted Approximation of Functions with Singularities by Combinations of Bernstein Operators,” Journal of Applied Mathematics and Computation, Vol. 206, No. 2, 2008, pp. 906-918. [2] Z. Ditzian, “A Global Inverse Theorem for Combinations of Bernstein Polynomials,” Journal of Approximation Theory, Vol. 26, No. 3, 1979, pp. 277-292. doi:10.1016/0021-9045(79)90065-0 [3] Z. Ditzian and V. Totik, “Moduli of Smoothness,” Springer- Verlag, Berlin, 1987. [4] M. Felten, “Direct and Inverse Estimates for Bernstein Polynomials,” Constructive Approximation, Vol. 14, No. 3, 1989, pp. 459-468. doi:10.1007/s003659900084 [5] S. S. Guo, C. X. Li and X. W. Liu, “Pointwise Approxi- mation for Linear Combinations of Bernstein Operators,” Journal of Approximation Theory, Vol. 107, No. 1, 2000, pp. 109-120. doi:10.1006/jath.2000.3504 [6] G. G. Lorentz, “Bernstein Polynomial,” University of Toronto Press, Toronto, 1953. [7] J. J. Zhang and Z. B. Xu, “Direct and Inverse Approxi- mation Theorems with Jacobi Weight for Combinations and Higher Derivatives of Baskakov Operators,” Journal of Systems Science and Mathematical Sciences, In Chi- nese, Vol. 28, No. 1, 2008, pp. 30-39. [8] D. D. Vechhia, G. Mastroianni and J. Szabados, “Wei- ghted Approximation of Functions with Endpoint and Inner Singularities by Bernstein Operators,” Acta Mathe- matica Hungarica, Vol. 103, No. 1-2, 2004, pp. 19-41. doi:10.1023/B:AMHU.0000028234.44474.fe |