Pointwise Approximation Theorems for Combinations of Bernstein Polynomials with Inner Singularities

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1 Appled Mathematcs do:46/am447 Pblshed Ole Apl ( Potse Appomato Theoems o ombatos o Beste Polyomals th Ie Sglates Wemg L L Zhag Depatmet o Mathematcs Hagzho Daz Uvesty Hagzho ha Depatmet o Mathematcs Zheag Uvesty Hagzho ha E-mal: l_emg@6com lyz@zedc godyal@6com Receved Octobe 8 ; evsed Jaay 4 ; accepted Jaay 7 Abstact It s ell-o that Beste polyomals ae vey mpotat stdyg the chaactes o smoothess theoy o appomato A e type o combatos o Beste opeatos ae gve [] I ths pape e gve the Beste-Maov eqaltes th step-eght ctos o combatos o Beste polyomals th e sglates as ell as dect ad vese theoems Keyods: Beste Polyomals Ie Sglates Potse Appomato Beste-Maov Ieqaltes Dect ad Ivese Theoems Itodcto The set o all cotos ctos deed o the teval I s deoted by I Fo ay the coespodg Beste opeatos ae deed as ollos: B : p hee p : Appomato popetes o Beste opeatos have bee stded vey ell (see [-7] o eample) I ode to appomate the ctos th sglates Della Veccha et al [8] todced some ds o moded Beste opeatos Thoghot the pape deotes a postve costat depedet o ad hch may be deet deet cases Dtza ad Tot eteded the method o combatos ad deed the ollog combatos o Beste opeatos: th the codtos: : B B a) b) c) ( ) ( ) o d) Fo ay postve tege e cosde the detemat A : We obta A! Ths thee s a qe solto o the system o ohomogeeos lea eqatos: opyght ScRes

2 9 Let W M LU ET AL aa a aa 4a aa 44a! aa 4a a a a : 4 () th the coecets a a a satsyg () ( ) Fom () e see that ( ) ( ) o Moeove t holds that ad Let ad : H l l : Fthe let 4 ad 4 Set : H F F We have F 5 H 5 H H Obvosly F als o degee ad that s lea epodces polyom- F povded No e ca dee o e combatos o Beste opeatos as ollos: hee B B F B F () satsy the codtos (a)-(d) The Ma Reslts Let : R be a admssble step-eght cto o the Dtza-Tot modls o smoothess that s satses the ollog codtos: ) Fo evey pope sbteval ests a costat ab sch that o ab ) Thee ae to mbes ad o hch as ~ as ab thee (X~Y meas - Y X Y o some ) ombg codtos (I) ad (II) o e ca dedce that hee opyght ScRes

3 W M LU ET AL 9 Let ad \ : lm The om s deed as : sp Dee W : : A W : : A Fo e dee the eghted modls o smoothess by : sp sp W t h ht hee h + h + h h Recetly Felte shoed the ollog to theoems [4]: Theoem A Let = ad let : R be a admssble step-eght cto o the Dtza-Tot modls o smoothess ([]) sch that ad ae cocave The o ad B Theoem B Let = : R ad let be a admssble step-eght cto o the Dtza-Tot modls o smoothess sch that ad ae cocave The o ad B mples t Ot O ma eslts ae the ollog: Theoem Fo ay m e have B () Theoem Fo ay e have W () B Theoem Fo m e have B Lemmas O t O t / () Lemma Fo ay o-egatve eal ad v e have v v p () Lemma I Lemma Fo ay R the p () e have W F () Poo We st pove 5 (The same as the othes) e have F : F I I Obvosly Fo I e have I I F F By [] e have F 5 H So 5 5 opyght ScRes

4 9 W M LU ET AL I H 5 5 : T T By Taylo epaso e have! s sd s! It ollos om (4) ad the detty e have v v l v v (4) H l! l s sds! v v v l l s s d s! hch mples that H l s sd s! Sce l o 5 It ollos om s s s betee ad the H So / s s ds I The the lemma s poved Accodg to methods o Lemma e ca easly get: Lemma 4 I the W g H g g Lemma 5 Fo ay e have Poo By () e have (5) B (6) B B F F p F p F p : I I I No the theoem ca be poved easly Lemma 6 Let m the o N t t t ad e have 8 t t t t d d t (7) Lemma 7 Let A : p / the A o ad Poo I the the statemet s tval Hece assme ( the case ca be teated smlaly) The o a ed the mamm o p s attaed o : By sg Stlg s omla e get e p e e opyght ScRes

5 W M LU ET AL 9 No om the eqaltes e ad e We have that the secod eqalty s vald To pove the st oe e cosde the cto Hee e e e hece Hece o p ep ep Ths e A e A easy calclato shos that hee the mamm s attaed he ad the lemma ollos Lemma 8 Fo e have (8) p Poo By () ad the lemma 7 e have p ( ) p e have Lemma 9 Fo ay Poo We st pove o W B (9) (The same as B! F p! F p F p F p F p : H H H We have H F p Smlaly e ca get H ad H Whe accodg to [] e have B B F Q A F p Q H p : hee A: A he H s a lea cto I e have also Q Q ad By () the opyght ScRes

6 94 W M LU ET AL : F p p I I By a smple calclato e have I By () the I p We ote that H ma H H 4: Ha 4 a So 4 a H a e have the I the 4 a by (8) e have ah a p It ollos om combg the above eqaltes that the lemma s poved 4 Poo o Theoems 4 Poo o Theoem Whe m e dscss t as ollos: ase I by (9) e have ( ) B (4) B ase I e have B B F Q hee So F p Q ad Q B F p F A p H p : hee A: e ca easly get (4) ad By bgg these acts togethe the theoem s poved 4 Poo o Theoem Whe W by [] e have B F F p (4) opyght ScRes

7 W M LU ET AL 95 I e have F F d (44) I e have Smlaly d F F (45) F F d (46) By (4) e have B F p F p F p F p hch combg th (44)-(46) gve B (47) ombg th the theoem ad theoem e ca obta oollay Fo ay { B e have ma W 4 Poo o Theoem 4 The Dect Theoem We o! F t F F t t t F d (48) (49) B (4) Accodg to the deto o W o ay e have B g B G g ad t thee o R G t t G d G B G g W G B G B R G t t t G B d t t t G B d also B t t d e have t t t t t (4) d d By () () ad (4) e have G B G G B t G By () (5) ad (4) he g W (4) (4) the g B g g G g G g B g g H g δ δ G g (44) Fo e choose pope g W by (6) ad (44) the B g B g g B g δ 4 The Ivese Theoem The eghted K-cto s gve by opyght ScRes

8 96 g K t g t g : g W By [] e have t K t t (45) Poo Let by (45) e choose pope g so Obvosly W M LU ET AL that g g (46) t t Fo N t ad e have 8 B B g B g h h h h h h h h B g d d h h h h h h By (9) ad (46) e have B g d d : J J J (47) J / (48) d d δ (49) h h h h J g h By the st eqalty o (48) ad (46) e let the By (7) ad (46) e have h h h h J g d d h δ (4) d d δ δ h h (4) h h J g h No by (47)-(4) thee ests a costat M so that h m / h h h M / h Whe e have hoosg pope δ N so that Theeoe hch mples h h t h opyght ScRes

9 W M LU ET AL 97 So by Bees-Loetz lemma [] e get 5 Reeeces t t [] D S Y Weghted Appomato o Fctos th Sglates by ombatos o Beste Opeatos Joal o Appled Mathematcs ad omptato Vol 6 No 8 pp [] Z Dtza A Global Ivese Theoem o ombatos o Beste Polyomals Joal o Appomato Theoy Vol 6 No 979 pp 77-9 do:6/-945(79)965- [] Z Dtza ad V Tot Modl o Smoothess Spge- Velag Bel 987 [4] M Felte Dect ad Ivese Estmates o Beste Polyomals ostctve Appomato Vol 4 No 989 pp do:7/s [5] S S Go X L ad X W L Potse Appomato o Lea ombatos o Beste Opeatos Joal o Appomato Theoy Vol 7 No pp 9- do:6/ath54 [6] G G Loetz Beste Polyomal Uvesty o Tooto Pess Tooto 95 [7] J J Zhag ad Z B X Dect ad Ivese Appomato Theoems th Jacob Weght o ombatos ad Hghe Devatves o Basaov Opeatos Joal o Systems Scece ad Mathematcal Sceces I hese Vol 8 No 8 pp -9 [8] D D Vechha G Mastoa ad J Szabados Weghted Appomato o Fctos th Edpot ad Ie Sglates by Beste Opeatos Acta Mathematca Hgaca Vol No - 4 pp 9-4 do:/b:hu e opyght ScRes

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