Iterated Bernstein polynomial approximations

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1 Ieraed Berse polyomal approxmaos arxv: v3 [mah.ca] 16 Oc 2009 Zhog Gua Deparme of Mahemacal Sceces, Idaa Uversy Souh Bed, 1700 Mshawaka Aveue, P.O. Box 7111 Souh Bed, IN , U.S.A. Absrac Ieraed Berse polyomal approxmaos of degree for couous fuco whch also use he values of he fuco a /, = 0,1,...,, are proposed. The rae of covergece of he classc Berse polyomal approxmaos s sgfcaly mproved by he eraed Berse polyomal approxmaos whou creasg he degree of he polyomals. The close form expresso of he lmg eraed Berse polyomal approxmao of degree whe he umber of he eraos approaches fy s obaed. The same dea apples o he q-berse polyomals ad he Szasz-Mrakya approxmao. The applcao o umercal egral approxmaos whch gves surprsgly good resuls s also dscussed. MSC: 41A10; 41A17; 41A25. Keywords: Berse polyomals; Bézer curves; Covexy preservao; Ieraed Berse polyomals; Numercal egrao; q-berse polyomals; Rae of approxmao; he Szasz-Mrakya operaors. 1 Iroduco The Berse polyomals [1] have bee used for approxmaos of fucos may areas of mahemacs ad oher felds such as smoohg sascs ad cosrucg Bézer curves [see 2, 3, for examples] whch have mpora applcaos compuer graphcs. Oe of he advaages of he Berse polyomal approxmao of a couous fuco f s ha approxmaes f o 1

2 [0, 1] uformly usg oly he values of f a /, = 0, 1,...,. I case whe he evaluao of f s dffcul ad expesve, he Berse polyomal approxmao s preferred. The properes of he Berse polyomal approxmao have bee suded exesvely by may auhors for decades. However he slow opmal rae O(1/) of covergece of he classcal Berse polyomal approxmao makes o so aracve. May auhors have made remedous effors o mprove he performace of he classcal Berse polyomal approxmao. Amog may ohers, Buzer[4] roduces lear combaos of he Berse polyomals ad Phllps[5] proposes he q-berse polyomals whch s a geeralzao of he classcal Berse polyomal approxmao. However, Buzer[4] s approxmao volves o oly he Berse polyomals of degree bu also degree of 2 whch requres more sampled values of he fuco o be approxmaed a he raher ha + 1 uform paro pos of [0, 1]. The q-berse polyomal approxmaes a fuco f oly whe q 1. For q > 1, seems ha f(z) has o be a aalyc complex fuco o dsk {z : z < r}, r > q, so ha he q-berse polyomal approxmao of degree has a beer rae of covergece, O(q ), ha he bes rae of covergece, O( 1 ), of he classcal Berse polyomal approxmao of degree [see 6, 7, for example]. If q > 1, he q-berse polyomal approxmao of degree uses he sampled values of he fuco a + 1 ouform paro pos of [0, 1]. These pos excep = 1 are araced oward = 0 whe q s geg larger so ha he approxmao becomes worse he he eghborhood of he rgh ed-po. Ths s a serous drawback of he q-berse polyomal approxmao whch lms he scope of s applcaos. I hs paper, we propose a smple procedure o geeralze ad mprove he classcal Berse polyomal approxmao by repeaedly approxmag he errors usg he Berse polyomal approxmaos. Ths mehod volves oly he eraes of he Berse operaor appled o he base Berse polyomals of degree ad he sampled values of he fuco beg approxmaed a he same se of +1 uform paro pos of [0, 1]. The mproveme made by he q-berse polyomal approxmao wh properly chose q ca be acheved by he eraed Berse polyomals whou messg up he rgh boudary. 2

3 2 Prelmary Resuls Abou he Classcal Berse Polyomal Le f be a fuco o [0, 1]. The classcal Berse polyomal of degree s defed as f() = B (1) f() = f ( ) B (), 0 1, (1) where s called he Berse operaor ad () = ( ) (1 ), = 0,...,, are called he Berse bass polyomals. Noe ha he Berse polyomal of degree, B (1) f, uses oly he sampled values of f a = /, = 0, 1,...,. Noe also ha for = 0,...,, β () ( + 1) (), 0 1, s he desy fuco of bea dsrbuo bea( + 1, + 1 ). Le Y () be a bomal b(, ) radom varable. The E{Y ()} =, var{y ()} = E{Y () } 2 = (1 ), E{Y () } 3 = (1 )(1 2), ad f() = E[f{Y ()/}]. The error of B (1) f s Err{B (1) f}() = B (1) f() f(). (2) Le f be a member of C (r) [0, 1], he se of all couous fucos ha have couous frs r dervaves. C[0, 1] = C (0) [0, 1]. Le he modulus of couy of he rh dervave f (r) be ω r (δ) = max f (r) (s) f (r) (), δ > 0. s <δ Abou he rae of covergece of B (1) f we have he followg well kow resuls [see 8]. Theorem 1. Suppose f C (r) [0, 1], r = 0, 1. For each > 1 Err{B (1) f}() = f() f() C r r/2 ω r ( 1/2 ), where C r s a cosa depedg o r oly. Oe ca choose C 0 = 5/4 ad C 1 = 3/4. The resul accordg o r = 0 s due o Popovcu[9]. The order of approxmao of f C (r) [0, 1] by arbrary polyomals s gve by he heorem of Duham Jackso [10] 3

4 Theorem 2 (Duham Jackso). Suppose f C (r) [0, 1], r 0. For each > r here exss a polyomal P of degree a mos so ha P () f() C r r ω r ( 1 ), where C r s a cosa depedg o r oly. If r = 0, oe ca choose C 0 = 3. The followg s a resul of Voroovskaya [11] abou he asympoc formula of he Berse polyomal approxmao. Theorem 3 (E. Voroovskaya). Suppose ha f has secod dervave f. The Err{B (1) (1 ) f}() = f() f() = 2 f () + 1 ε (), (3) where ε () s a sequece of fucos whch coverge o 0 as. From Theorem 3 follows ha he bes rae of covergece of B (1) f, as, s O( 1 ) eve f f has couous secod or hgher order dervaves [8]. Ths s o as good as he case of arbrary polyomal approxmao whch f f has couous rh dervave he he rae of covergece of a sequece of arbrary polyomals P of degree a mos ca be a leas o( r ) [10]. Berse [12] geeralzes hs asympoc formula o coa erms up o he (2k)h dervave ad proposes a polyomal cosruced based o boh f(/) ad f (/), = 0, 1,...,. Buzer [4] cosders some combaos of Berse polyomals of dffere degrees ad shows ha hey have beer rae of covergece whch s much faser ha O(1/). Cosable e al [13] geeralze he lear combaos of he Berse polyomals proposed by of [4], [14] ad [15]. The q-berse polyomals of [5] has beer rae of covergece. However, f 0 < q < 1, he q-berse polyomals of fuco f do o approxmae f. For q > 1, he q-berse polyomals do approxmae f a a rae of O(q m ) bu f(z) has o be aalyc a complex dsk wh radus greaer ha q. The aalycy of f may be oo resrcve for applcaos. Eve f we are sure ha f s aalyc, we have o deal wh he choce of q. I some cases, he approxmaos are very sesve o he choce of q. 3 The Ieraed Berse Polyomals ad he Rae of Covergece The error Err{B (1) f}() s also a couous fuco o [0, 1] whose values a = /, = 0, 1,...,, deped o f( ), = 0, 1,...,, oly. So we ca 4

5 approxmae hs error fuco by he Berse polyomal B (1) [Err{B (1) f}]() ad he subrac he approxmaed error fuco from B (1) f() o oba he secod order Berse polyomal of degree B (2) f() = B(1) f() B(1) [Err{B(1) f}](). (4) Ths dea s closely relaed o, alhough was o aed by, he proposal of Berse [12] whch he secod dervave raher ha he error of he Berse polyomal s approxmaed. Iducvely, B (k+1) f() = f() { f() f()}, k 1. (5) Ths erao procedure ca be performed furher ul a sasfacory approxmao precso s acheved because he error Err{ f()} = f() f() ca be esmaed by { f() f()} = f() B (k+1) f(). Lemma 4. Geerally he k-h order Berse polyomal of degree ca be wre as k ( ) k f() = ( 1) 1 B f(), k 1, 0 1. (6) =1 Defe B 0 f() = f(). The he error of he k-h Berse polyomal of degree ca be wre as k ( ) k Err{ f()} = B(k) f() f() = ( 1) 1 B f() = (I ) k f(), (7) where I = B 0 s he dey operaor. Proof. B (k+1) f() = f() { k ( k = =1 k ( k = =1 k+1 ( k + 1 = =1 f() f()} ( ) k ) ( 1) 1 B f() k =1 ) k+1 ( 1) 1 B f() + =2 ( k 1 ( 1) 1 B +1 f() + f() ) ( 1) 1 B f() + f() ) ( 1) 1 B f(). (8) By duco, (1) ad (8) assure ha (6) s rue for every posve eger k. Equao (7) s he obvous. 5

6 The lm of B k f(), as k, has bee gve by Kelsky ad Rvl [16]. A shor ad elemeary proof of [16] s resul s gve by [17]. Afer we fshed he frs verso of hs paper, we realzed ha [18] obaed he formula (7) ad vesgaed he properes of f() usg smulao mehod. The cos of f() s oly some smple algebrac calculaos addo o he evaluao of f a /, = 0, 1,...,. Abou he eraes of he Berse operaor we have he followg resul. Lemma 5. For k 1, B k f() = where B 0 () = (), ad Whe k = 1, f ( ) B k 1 ( )(), k 1; 0 1, (9) B k+1 () = {B k }(), k 1. (10) B 1 () = ( B j ) Bj (). (11) j=0 Proof. The heorem ca be easly proved by duco ad he fac ha he Berse operaor s lear. By (6) ad (7) we have Theorem 6. The k-h Berse polyomal approxmao ca be calculaed ducvely as f() = f ( ) k j=1 Clearly, for every k 1, Err{ f()} = ( ) k ( 1) j 1 B j 1 (), k 1, 0 1. (12) j preserves lear fucos. Therefore { ( f } ) k ( ) k f() ( 1) j 1 B j 1 (), k 1; 0 1. j j=1 (13) Expresso (12) ca easly mplemeed compuer laguages usg erave algorhm. Defe dcaor fucos { 1, = I () = ; 0,. (14) 6

7 The () = I () = B (1) I (), = 0, 1,...,, ad, by Theorem 6, (12) ad (13) ca be smplfed as f() = f ( ) B (k) I (), k 1, 0 1. (15) Err{ f()} = { ( f } ) f() B (k) I (), k 1; 0 1. (16) () = B(k) I () = k j=1 ( ) k ( 1) j 1 B j 1 (). (17) j The followg heorem shows ha he eraed Berse polyomals, lke he classcal oes, have o error a he edpos of [0, 1]. Theorem 7. For ay fuco f defed o [0, 1] ad ay eger k 0, f(0) = f(0), B(k) f(1) = f(1). (18) Proof. I s kow ha B 0 () = () = I () for = 0, 1, = 0,...,. Assume ha B k 1 () = I () for = 0, 1, = 0,...,, ad some k 1. By Theorem 9, f = 0, 1, B k () = ( B j ) B k 1 (j )() = j=0 ( B j ) Ij () = () = I (). So by duco, for all oegave egers k, = 0, 1, ad = 0,...,, B k () = I (). By (17), we have, f = 0, 1, Thus by (15), f = 0, 1, () = B(k) I () = k j=1 f() = j=0 ( ) k ( 1) j 1 I () = I (). j f ( ) I () = f(). Clearly, for each k 1, f() ca be wre as where F (k) (k) f() = F () = (f (k) ) 1 (+1) s a ( + 1) row vecor, ad () = {0 (),..., ()} T. 7

8 If k = 1, f (1) = f ( ) 1, = 1,..., + 1. Defe ( + 1) ( + 1) square marx = (u T ) = (b j ) (+1) (+1) where u = ( 0 T., 1,..., ) Tha s ( b j = B j 1 ), 1,, j = 1,..., + 1. I s easy o see ha s osgular ad have all he egevalues (0, 1] amog hem exacly wo are oes whch correspod o egevecors u ad 1 +1 = (1,...,1) T R +1. We have he followg heorem. Theorem 8. For k 1, k ( ) k F (k) = ( 1) 1 F (1) B 1 =1 = F (1) where B 0 = I +1, he ( + 1)s order u marx. If k 1, F (k+1) Proof. I s easy o show ha B 1 {I +1 (I +1 ) k }, (19) = F (k) {I +1 } + F (1). (20) F (2) = 2F (1) F (1). By duco, (19) ad (20) ca be easly proved. More mporaly, we have Theorem 9. The opmal Berse polyomal approxmao of degree s where Moreover, B ( ) ( ) f() = F ( ) F ( ) () = F (1) = lm k F (k) preserves lear fucos. = F (1) B 1 (), (21) B 1. (22) Proof. Sce all he egevalues of marx are (0, 1] ad exacly wo of hem are oes, all he egevalues of marx I +1 are [0, 1) ad exacly wo of hem are zeros. Thus lm k (I +1 ) k = O, he zero marx. Because preserves lear fucos for ay posve eger k, so does ( ). Ths ca also be proved by he followg facs ha F (1) = F (1) f ad oly f F (1) B 1 = F (1) ad ha F (1) = F (1) s rue provded ha f s lear. 8

9 Numercal examples (see 6) show ha he maxmum absolue approxmao error seems o be mmzed by opmal Berse polyomal approxmao ( ) f() f f s fely dffereable. For osmooh fucos such as f() = 0.5 ad fxed, seems ha he maxmum absolue approxmao error s mmzed by he eraed Berse polyomal approxmao f() for some k. The ex heorem shows ha f k > 1 he f() s deed a beer polyomal approxmao of f ha he classcal Berse polyomal. Theorem 10. Suppose ha f C dkr [0, 1], d kr = 2(k 1) + r ad r = 0, 1. The where C kr Err{ f()} = B(k) s a cosa depedg o r ad k oly. Proof. Ths resul follows easly from Theorems 1 ad 3. f() f() d C kr kr 2 ωdkr ( 1/2 ), (23) Remark 3.1. From hs heorem wh k = 2 ad r = 0, we see ha f f has couous secod dervave he he rae of covergece of he secod Berse polyomal approxmao B (2) f s a leas o( 1 ). Remark 3.2. From Theorem 10 wh k = 2 we see ha f f has couous fourh dervave, he he rae of covergece of B (2) f ca be as fas as O( 2 ). Ths seems he fases rae ha B (2) f ca reach eve f f has couous ffh or hgher dervaves. Remark 3.3. I ca also be proved ha f f has couous (2k)h dervave, he he rae of covergece of f ca be as fas as O( k ). Alhough hese mprovemes upo f() are sll o as good as hose saed Theorem 2, hey are good eough for applcao compuer graphcs ad sascs. Remark 3.4. I s a very eresg projec o vesgae he relaoshp bewee C kr ad k, ad he rae of covergece of B( ) f whch s cojecured o be expoeal. 4 The Dervaves ad Iegrals of f() ad Applcaos 4.1 The Dervaves of f() Theorem 11. For ay posve egers k ad r, d r! r k ( ) k d r B(k) f() = ( 1) j 1 r (B j 1 f) ( ( r)! j ) B r, (), (24) j=1 9

10 where r s he rh forward dfferece operaor wh creme h = 1/, f() = f( + h) f(), r ( ) r r f() = ( 1) f ( ) + r h. Proof. If k = 1, s well kow ha for ay fuco f d d B(1) f() = d d f() = 1 f ( ) B 1, (). (25) Assume ha (24) wh r = 1 s rue for he kh eraed Berse polyomal of ay fuco f. By (5) we have d d B(k+1) f() = d d I follows from (25) ad (9) ha [ B (k) f() { f() f()} ] = d d B(k) f() + d d f() d d { }f(). (26) 1 d d { }f() = f( ) B 1, () 1 = 1 = j=0 k l=1 f ( j ) k l=1 Combg (25), (9), (26), ad (27) we arrve a d d B(k+1) 1 k+1 f() = j=1 ( ) k ( 1) l 1 B l 1 l B j( ) B 1, () ( ) k ( 1) l 1 B l l f( ) B 1, (). (27) ( 1) j 1 ( k + 1 j ) B j 1 f ( ) B 1, (). (28) The proof of (24) wh r = 1 ad k 1 s complee by duco. Smlarly (24) wh r 1 ad k 1 ca be proved usg duco. I s o hard o prove by adopg he mehod of [8] ha Theorem 12. () If f has couous rh dervave f (r) o [0, 1], he for each fxed k, as, dr d r f() coverge o f (r) () uformly o [0, 1]. () If f bouded o [0, 1] ad s rh dervave f (r) () exss a [0, 1], he for each fxed k, as, dr d r f() coverge o f (r) (). 10

11 Numercal examples show ha he larger he r s, he slower he above covergece s. For ay posve egers k, he secod dervave of he eraed Berse polyomal f s d 2 2 d 2B(k) f() = ( 1) k ( ) k ( 1) j 1 2 (B j 1 f) ( j ) B 2, (). (29) j=1 I s well kow ha f f s covex o [0, 1], he d2 B (1) d 2 f() 0 ad hus B (1) f() s also covex ad B (1) f() f() o [0, 1]. So he classcal Berse polyomals preserve he covexy of he orgal fuco ad has oegave errors. However examples of 6 show ha whe k 2 he eraed Berse polyomal f does o preserve he covexy of he orgal fuco ucodoally. The eraed Berse polyomals sll preserve he mooocy of f f s o oo fla aywhere. Theorem 13. If f s srcly creasg (decreasg) o [0, 1], for ay k 1, f() s also srcly creasg (decreasg) o [0, 1]. Proof. The heorem s rue for k = 1 eve f f s creasg (decreasg), bu o srcly, o [0, 1]. I suffces o prove he heorem whe f s srcly creasg o [0, 1]. Assume ha he heorem s rue for some k 1. Sce f s srcly creasg o [0, 1], B k f() are also srcly creasg o [0, 1] for all k 1. Remark 4.1. If k = 1, he codo of src mooocy s o ecessary. However, f k > 1, he codo of src mooocy ca be relaxed. For example, f(x) = x, f 0 x < 1/3, = 1/3, f 1/3 x < 2/3, ad = x 1/3, f 2/3 x 1. I ca be show ha d d B(2) f(x) < 0 for x a eghborhood of x = 1/ The Iegrals of f() The followg heorem s very useful for mplemeg he erave algorhm compuer laguages. Theorem 14. Suppose f s couous o [0, 1]. For 1 k ad x [0, 1], we have x f()d = f (k) S (x) = F (k) S (x), (30) 0 where S (x) = {S 0 (x),...,s (x)} T ad S (x) = x 0 ()d = x 0 β ()d, S (1) = 1 +1.

12 Corollary 15. If g s couous o [a, b], a < b, he for 1 k, b a g()d where F (k) s calculaed based o f() = 1 g[a + (b a)]. b a f (k) = F (k) 1 +1, (31) Remark 4.2. Noe ha umercal egrao (31) does o volve ay egrals. I coas oly algebrac calculaos. See Example 9 of 6 for some umercal examples. Proof. The heorem follows mmedaely from Theorems 8 ad 9. The followg heorem follows mmedaely from Theorems 10 ad 14. Theorem 16. Uder he codo of Theorem 10, for ay x [0, 1] x x f()d f()d d C kr kr 2 ωdkr ( 1/2 ), (32) where C kr 0 s a cosa depedg o r ad k oly. 0 5 Ieraed Szasz Approxmao ad Ieraed q- Berse Polyomal The dea used o cosruc he eraed Berse polyomal approxmao s smple ad very effecve. The same dea seems also applcable o oher operaors or approxmaos such as he Szasz operaor [19] [or he Szasz-Mrakya (Mrakja) operaor] ad he q-berse polyomal wh q > 1. We wll gve some umercal examples 6 ad he aalogues of resuls of Seco 3 could be be obaed by usg he aalogue resuls abou he rae of covergece of he Szasz- Mrakya approxmao [20]. We hope hese would spre more vesgaos wh rgorous mahemacs. 5.1 Ieraed Szasz Approxmao The so-called Szasz-Mrakya approxmao s defed as S f(x) = f ( ) P (x), x [0, ), (33) 12

13 where f s defed o [0, ) ad P (x) = e x (x) /!. Noe ha, for x > 0, P (x) s he probably ha V (x) = where V (x) s he Posso radom varable wh mea x. Sce he bomal probably () ca be approxmaed by P () for large, he Szasz-Mrakya approxmao ca be vewed as a exeso of he Berse polyomal approxmao. The error of S f as a approxmao of f s Err(S f)(x) = S f(x) f(x) = f ( ) P (x) f(x), x [0, ). (34) Applyg he Szasz-Mrakya operaor o Err(S f)(x), we have S {Err(S f)}(x) = S 2 f(x) S f(x) = f ( ) S P (x) S f(x), x [0, ). So we ca defe he secod Szasz-Mrakya approxmao as Theorem 17. S (k) f(x) = f ( (35) S (2) f(x) = S f(x) S {Err(S f)}(x), x [0, ). (36) ) k j=1 Clearly, for every k 1, S (k) Err{S (k) f(x)} = { ( f } ) k f(x) ( ) k ( 1) j 1 S j 1 P (x), k 1, x [0, ). (37) j preserves lear fucos ad herefore ( k j j=1 ) ( 1) j 1 S j 1 P (), k 1; x [0, ). Fgure 4 gves a example of he eraed Szasz approxmaos. 5.2 Ieraed q-berse Polyomal Le x be a real umber. For ay q > 0, defe he q-umber [x] q = { 1 q x, f q 1; 1 q x, f q = 1. If x s eger, he [x] q s called a q-eger. For q 1, he q-bomal coeffce (Gaussa bomal) s defed by ( ) = r q 1, r = 0; (1 q )(1 q 1 ) (1 q r+1 ) (1 q r )(1 q r 1 ) (1 q), 1 r ; 0, r >. 13 (38)

14 So ( ) = r q r 1 [ ], 0 r, q > 0, r q r where empy produc s defed o be 1. Thus he ordary bomal coeffce ( ) r s he specal case whe q = 1. G. M. Phllps [5] roduced he q-berse polyomal of order for ay couous fuco f() o he erval [0, 1] where (q) Q q f() = ( f = [] q [] q, Q () = (q) ( ) q ) Q (), = 1, 2,..., (1 q j 1 ), = 0, 1,...,. j=1 Clearly, f() = Q 1 f() whch s he classcal Berse polyomal of order. I has bee proved ha f 0 < q < 1 he Q q f() does o approxmae f ad ha f q > 1 ad f(z) s aalyc complex fuco o dsk {z : z < r}, r > q, he Q q f() has beer rae of covergece, O(q ), ha he bes rae of covergece, O( 1 ), of f() [see 6, 7, for example]. Noe ha f q > 1 he pos (q) = [] q /[] q are o loger uform paro pos of he erval [0, 1]. For fxed, lm q (q) = 0, <. So all excep (q) = 1 are araced oward 0 as q geg large. However, eresgly, he larger he q s a cera rage, he closer he q-berse polyomal approxmao Q q f() o f(). For a gve, f q s oo large, he q-berse polyomal approxmao Q q f() becomes worse he eghborhood of he rgh ed-po. Smlarly we have he eraed q-berse polyomals Q (k) q f() = f ( (q) ) k j=1 ( k j ) ( 1) j 1 Qq j 1 Q (), k 1, [0, 1]. (39) See Fgure 5 for a example of he eraed q-berse polyomals. Comparg Fgures 1 ad 5 we see ha creasg q from 1 o 1.1 does mprove he approxmao o [0, 1] excep a pos he eghborhood of he rgh ed-po. The approxmao ear he rgh ed-po could be worse by applyg he eraed q-berse polyomals. The mproveme ca be acheved by he eraed Berse polyomals whou messg up he rgh boudary. 6 Numercal Examples I hs seco some umercal examples are gve wh he hope of more vesgaos o he proposed mehods wh rgorous mahemacs. 14

15 Example 1. Fgure 1 shows he frs hree eraed Berse polyomals of f() = s(2π) ad he errors where = 30. The opmal Berse polyomal approxmao s also ploed whch seems o have almos o error. Example 2. Fgure 2 shows he frs hree eraed Berse polyomals of f() = sg( 0.5)( 0.5) 2 (a dffereable bu o wce dffereable fuco) ad he errors where = 30. The opmal Berse polyomal approxmao s o ploed whch becomes very bad ear he wo edpos. Example 3. Fgure 3 shows he frs hree eraed Berse polyomals of f() = 0.5 ad he errors where = 30. The opmal Berse polyomal approxmao s o ploed whch becomes very bad ear he wo edpos. Example 4. Fgure 4 shows he frs hree eraed Szasz approxmao of f(x) = 0.25xe x/2, x 0, ad he errors where = 10. Example 5. Fgure 5 shows he frs hree eraed q-berse polyomals of f(x) = s(πx) ad he errors where = 30, q = 1.1. The performace of he approxmao ear = 1 s very sesve o q. Example 6. Fgure 6 shows he frs hree eraed Berse polyomals of he followg fuco f() = 0.5 ad her dervaves where = 30. Example 7. Fgure 7 shows he frs hree eraed Berse polyomals of he followg fuco f() ad her dervaves where = 30, f() = { ( 1), 0 < 0.5; ( 0.5)3/2, Ths a covex fuco whch has couous frs dervave bu does o have a couous secod dervave. Example 8. Deoe δ = 2 δ where δ s a small posve umber. 3 f() = { f0 (), 0 δ ; p k (), δ < 1, where f 0 () = v r 2 ( u) 2 s poro of a crcle wh radus r (a larger posve umber) ad ceered a (u, v), u, v > 0, p k () s a polyomal of degree k = 3, k p k () = a k = a kk k + a k,k 1 k a k1 + a k0. 15

16 Table 1: Some resuls of umercal egrals ( = 5) k 1 5 Exac value 1 π s(πx)dx ex dx ϕ(x)dx Table 2: Some resuls of umercal egrals ( = 10) k 1 5 Exac value 1 π s(πx)dx ex dx ϕ(x)dx If we choose v = 30 δ ± δ 40(252 δ r2 ), u = r 20 2 v 2 he f(0) = f 0 (0) = 0, f( δ ) = f 0 ( δ ) = 3 δ. We also have f 0 () = u f r2 ( u) 2, 0 () = r 2 {r 2 ( u) 2 } 3/2. Choose he coeffces of p k so ha f(1) = k a k = 0 ad he jh (j = 0, 1,..., k 1) dervave a δ sasfy f (j) ( δ ) = k =j! a ( j)! k j δ = f (j) 0 (1 j δ ). If r s large eough, say r = 70, δ = 0.05, he f() s srcly covex ad has couous posve secod dervave f, bu B (2) f s sll o covex because s secod dervave s egave a some pos ear = 0.4 (see Fgure 8). Example 9. I he followg Tables 1 ad 2 we summarze some he resuls of umercal egrals o [0, 1] usg our proposed mehod gve Corollary 15 for fucos f(x) = π s(πx), f(x) = e x, ad f(x) = ϕ(x) = (1/ 2π) exp( x 2 /2). From hese examples ad he fgures we see ha he error s reduced sgfcaly by usg he eraed Berse polyomal approxmao whou creasg he degree of he polyomal. For o-smooh fuco, he maxmum 16

17 error s reduced more ha 50% by he hrd Berse polyomal. I s also see from Fgure 3 ha ulke he classcal Berse polyomal approxmao he eraed Berse polyomal approxmao f seems o o preserve he covexy of f for k > 1 hs case whe f s o smooh. So s ecessary for f o preserve he covexy of f ha f s smooh ad f s o oo close o zero. For applcaos umercal egrals ad compuer graphcs, somemes s eve much more expesve o evaluae he fuco f ha he smple algebrac calculaos. So s sgfca o apply he eraed or he opmal, f f s fely dffereable, Berse polyomal approxmao. Refereces [1] S. N. Beršeĭ. Démosrao du héorème de Weersrass fodée sur le calcul des probables. Comm. Soc. Mah. Kharkov, 13:1 2, [2] Sor G. Gal. Shape-Preservg Approxmao by Real ad Complex Polyomals. Brkhäuser, Boso, Basel, Berl, [3] Jua Mauel Peña. Shape preservg represeaos compuer-aded geomerc desg, volume Volume 385. Nova Scece Publshers. Ic., [4] P. L. Buzer. Lear combaos of Berse polyomals. Caada J. Mah., 5: , [5] George M. Phllps. O geeralzed Berse polyomals. I Approxmao ad opmzao, Vol. I (Cluj-Napoca, 1996), pages Traslvaa, Cluj-Napoca, [6] Sofya Osrovska. q-berse polyomals ad her eraes. J. Approx. Theory, 123(2): , [7] Hepg Wag ad XueZh Wu. Saurao of covergece for q-berse polyomals he case q 1. J. Mah. Aal. Appl., 337(1): , [8] G. G. Lorez. Berse polyomals. Chelsea Publshg Co., New York, secod edo, [9] T. Popovcu. Sur l approxmao des focos covexes d ordre supéreur. Mahemaca (Cluj), 10:49 54, [10] Duham Jackso. The heorey of approxmao, volume 11. Amer. Mah. Soc. Coll. Publ.,

18 [11] E. Voroovskaya. Déermao de la forme asympoque d approxmao des focos par les polyômes de M. Berse,. Doklady Akadem Nauk SSSR, pages 79 85, [12] S. N. Beršeĭ. Compléeme à l arcle de E. Voroowskaja. C. R. Acad. Sc. U.R.S.S., pages 86 92, [13] F. Cosable, M. I. Gualer, ad S. Serra. Asympoc expaso ad exrapolao for Berse polyomals wh applcaos. BIT, 36(4): , [14] M. Freu. Lear combaos of Beršeĭ polyomals ad of Mrakja operaors. Suda Uv. Babeş-Bolya Ser. Mah.-Mech., 15(1):63 68, [15] C. P. May. Saurao ad verse heorems for combaos of a class of expoeal-ype operaors. Caad. J. Mah., 28(6): , [16] R. P. Kelsky ad T. J. Rvl. Ieraes of Berse polyomals. Pacfc J. Mah., 21: , [17] Ulrch Abel ad Mrcea Iva. Over-eraes of Berse s operaors: A shor ad elemeary proof. Amerca Mahemacal Mohly, 116(6): , [18] Ashok Saha. A erave algorhm for mproved approxmao by Berse s operaor usg sascal perspecve. Appl. Mah. Compu., 149(2): , [19] Oo Szasz. Geeralzao of S. Berse s polyomals o fe erval. Joural of Research of he Naoal Bureau of Sadards, 45(3): , Sepember [20] Vlmos Tok. Approxmao by Berse polyomals. Amer. J. Mah., 116(4): ,

19 f() f() (1) (f) (2) (f) (3) (f) ( ) (f) Error Err{B (1) (f)} Err{B (2) (f)} Err{B (3) (f)} Err{B ( ) (f)} Fgure 1: The eraed Berse polyomals ad errors whe f() = s(2π). The error s mmzed by B ( ) f. 19

20 f() f() (1) f() (2) f() (3) f() Error Err{ (1) f()} Err{ (2) f()} Err{ (3) f()} Fgure 2: The eraed Berse polyomals ad errors whe f() = sg( 0.5)( 0.5) 2 whch s dffereable o [0, 1] bu o wce dffereable a =

21 f() f() (1) f() (2) f() (3) f() Error Err{ (1) f()} Err{ (2) f()} Err{ (3) f()} Fgure 3: The eraed Berse polyomals ad errors whe f() = 0.5 whch s o dffereable a =

22 f(x) f S (1) (f) S (2) (f) S (3) (f) Error Err{S (1) (f)} Err{S (2) (f)} Err{S (3) (f)} x x Fgure 4: The eraed Szasz approxmaos ad errors whe f(x) = 0.25xe x/2, x 0 wh =

23 f() f() (1) (f) (2) (f) (3) (f) Q q Q q Q q Error Err{Q (1) q (f)} Err{Q (2) q (f)} Err{Q (3) q (f)} Fgure 5: The eraed q-berse polyomals ad errors whe f() = s(2π) wh = 30, q =

24 f() f() (1) f() (2) (f) (3) f() f () f (), ½ d 2 d 2 d 2 d 2 (1) f() (2) f() Fgure 6: The eraed Berse polyomals ad her dervaves whe f() = 0.5 whch s covex bu o dffereable a =

25 f() f() (1) f() (2) (f) (3) f() f () f (), ½ d 2 d 2 d 2 d 2 (1) f() (2) f() Fgure 7: The eraed Berse polyomals of f() as Example 7 ad her dervaves where f() s covex, dffereable o [0, 1] bu o wce dffereable a =

26 f() f() (1) f() (2) f() (3) f() f () f () d 2 d 2 d 2 d 2 (1) f() (2) f() Fgure 8: The eraed Berse polyomals of f as Example 8 ad her dervaves. The fuco f s srcly covex bu B (2) f s o covex. 26

27 f() f() (1) f() (2) f() (3) f() f () f (), 2/3 d 2 d 2 d 2 d 2 (1) f() (2) f()