On the convergence of derivatives of Bernstein approximation

Transcription

1 O the covergece of dervatves of Berste approxmato Mchael S. Floater Abstract: By dfferetatg a remader formula of Stacu, we derve both a error boud ad a asymptotc formula for the dervatves of Berste approxmato. AMS subject classfcato: 41A10, 41A25. Key words: Berste approxmato, dvded dfferece, asymptotc formula, error boud. 1. Itroducto The Berste approxmato B (f to a fucto f : [0, 1] lr s the polyomal B f(x = where p, s the polyomal of degree, ( p, (x = x (1 x, = 0,...,. f p, (x, (1.1 Berste [ 1 ] used ths approxmato to gve the frst costructve proof of the Weerstrass theorem. Oe of the may remarkable propertes of Berste approxmato s that dervatves of B (f of ay order coverge to correspodg dervatves of f; see Loretz [ 7 ]. If f C k [0, 1] for ay k 0, the lm (B f (k = f (k uformly o [0, 1]. Other remarkable propertes are shape-preservato ad varato-dmuto [ 5 ]. These may propertes ca be vewed as compesato for the slow covergece of B (f to f. Wth the max orm o [0, 1], the error boud B (f, x f(x 1 2 x(1 x f, (1.2 gve Chapter 10 of [ 4 ], shows that the rate of covergece s at least 1/ for f C 2 [0, 1]. O the other had, the asymptotc formula lm ( B f(x f(x = 1 2 x(1 xf (x, (1.3 due to Voroovskaya [ 9 ], shows that for x (0, 1 wth f (x 0, the rate of covergece s precsely 1/. I ths ote we show that all dervatves of the operator B coverge at essetally the same rate by extedg both the error boud (1.2 ad the Voroovskaya formula (1.3. Frstly, the error boud geeralzes to: 1

2 Theorem 1. If f C k+2 [0, 1] for some k 0 the (B f (k (x f (k (x 1 ( k(k 1 f (k + k 1 2x f (k+1 + x(1 x f (k+2. 2 Secodly, Voroovskaya s formula ca be dfferetated : Theorem 2. If f C k+2 [0, 1] for some k 0, the uformly for x [0, 1]. lm ( (B f (k (x f (k (x = 1 d k 2 dx k {x(1 xf (x}, Thus the k-th dervatve of B (f coverges at the rate of 1/ whe the k-th dervatve of x(1 xf (x s o-zero. We remark that after completo of ths ote, t was foud that López-Moreo, Martíez-Moreo, ad Muñoz-Delgado [ 6 ] very recetly establshed Theorem 2 usg a completely dfferet approach. 2. Stacu s remader formula The tradtoal way to aalyze the error B (f f ad deed to derve both (1.2 ad (1.3 s to substtute the Taylor expaso f = f(x + x f (x +... to equato (1.1. To deal wth dervatves of B we wll stead borrow a dea from umercal dfferetato [ 2 ]. As s well kow, error formulas for umercal dfferetato ca be obtaed from dfferetatg Newto s remader formula for polyomal terpolato. Ths suggests fdg a aalogous remader formula for Berste approxmato ad subsequetly dfferetatg t. A atural remader formula for ths purpose s B f(x f(x = 1 1 x(1 x (, + 1 ], x f p 1, (x. (2.1 Here [x 0, x 1,..., x k ]f deotes the k-th order dvded dfferece of f at the pots x 0,..., x k, ad we ote that the rght had sde of (2.1 s vald at least for f C 2 [0, 1]. A more geeral form of ths formula for the remader tesor-product bvarate Berste approxmato was derved by Stacu [ 8 ], but does ot appear to be too well kow. It therefore seems worth offerg the followg proof, especally as t s shorter tha the orgal [ 8 ]. If oe recalls the detty 1 x(1 xp,(x = x p, (x, 2

3 gve Chapter 10 of [ 4 ], Stacu s formula follows smply from: ( B f(x f(x = f f(x p, (x =, x ] f 1 = x(1 x x p, (x = 1 x(1 x ( + 1 ] ], x f, x f p 1, (x. ( ], x f p,(x 3. Error aalyss I what follows t wll help to geeralze the operator B to s B,s,t f(x = (,..., + s ] t f p s, (x, (3.1 for ay s, t 0. We have B,0,0 = B ad the remader formula (2.1 ca be wrtte as B f(x f(x = 1 x(1 xb,1,1f(x. Dfferetatg ths k tmes ad usg the Lebz rule gves (B f (k (x f (k (x = 1 ( k(k 1(B,1,1 f (k 2 (x + k(1 2x(B,1,1 f (k 1 (x + x(1 x(b,1,1 f (k (x. Ths leads us to study the dervatves of B,1,1 f. Lemma 1. If f C r+2 [0, 1] for some r 0 the r+1 (B,1,1 f (r = r! j 1 j + 1 B,j,r j+2 f. Proof: Usg the formula (see Chapter 2 of [ 2 ] d r dx r, + 1 ], x f = r!, + 1 ], x, }. {{.., x f, } r+1 j=0 j=1 dfferetato of (3.1 wth s = t = 1 mples 1 r ( r (B,1,1 f (r (x = (r j! j = r! r j=0 ( 1... ( j j! (, + 1 j 1 ( j ] r j+1 f p (j 1, (x, + 1 ] r j+1 f p j 1, (x, (3.2 (3.3 3

4 where s the forward dfferece operator w.r.t.. Now otce that, + 1 ] + 1, x,..., x f = = 2 ad cotug to apply mples j, + 1, x,..., x ] f =, + 2 ], x,..., x f, + 1, + 2 ], x,..., x f, (j + 1 j,..., + j + 1, + 1, x,..., x ] f ], x,..., x f. Substtutg ths detty to equato (3.3 ad replacg j by j 1 gves the result. Due to Lemma 1, we have for f C r+2 [0, 1] ad r 0, r+1 (B,1,1 f (r r! j f (r+2 (r + 2! j=1 = 1 2 f (r+2. Theorem 1 ow follows from applyg ths boud to equato (3.2. To prove Theorem 2 we study the covergece of the operators B,s,t. Lemma 2. If f C s+t [0, 1] for some s, t 0 the lm B,s,tf = f (s+t uformly o [0, 1]. (s + t! Proof: We exted Davs s proof of Berste s theorem, amely the proof of Theorem of [ 3 ]. Let q := s + t. The for each, 0 s, there s some ξ the smallest terval cotag x, /,..., ( + s/ such that ad t s suffcet to show that,..., + s ] t f = f (q (ξ q! s S := (f (q (ξ f (q (xp s, (x 0. Let ɛ > 0. Sce f C q [0, 1], δ > 0 such that y x < δ mples f (q (y f (q (x < ɛ. Let I be the set of all, 0 s, for whch x δ < / < ( + s/ < x + δ, ad splt S to the two terms, C = I (f (q (ξ f (q (xp s, (x, D = I (f (q (ξ f (q (xp s, (x. 4

5 Now for I we clearly have ξ x < δ, ad so C I ɛp s, (x ɛ. Regardg D, otce that x s x + s s x s +, ad smlarly + s x s x + s + s s x s +, ad therefore { max x 2 + s 2} 2, x s x + O(1/, uformly for x [0, 1]. It follows that D 2 δ 2 f (q { max x 2 + s 2}, x p s, (x I 2 s δ 2 f (q = Thus lm S ɛ for ay ɛ > 0. s x 2p s, (x + O(1/. 2 ( sδ 2 f (q x(1 x + O(1/. Due to Lemmas 1 ad 2, we have for f C r+2 [0, 1] ad r 0, lm (B r+1,1,1f (r = r! j f (r+2 (r + 2! = f (r+2 2 j=1 uformly o [0, 1], ad Theorem 2 follows from multplyg equato (3.2 by ad lettg. Refereces 1. S. N. Berste, Démostrato du théorème de Weerstrass fodée sur le calcul de probabltés, Comm. Kharkov math. Soc. 13 (1912, S. D. Cote ad C. de Boor, Elemetary umercal aalyss, McGraw-Hll, P. J. Davs, Iterpolato ad approxmato, Dover, R. A. DeVore ad G. G. Loretz, Costructve approxmato, Sprger Verlag, Berl,

6 5. Goodma T. N. T., Shape preservg represetatos, Mathematcal methods Computer Aded Geometrc Desg, T. Lyche ad L.L. Schumaker (eds., Academc Press, New York, 1989, A. J. López-Moreo, J. Martíez-Moreo, F. J. Muñoz-Delgado, Asymptotc expresso of dervatves of Berste type operators, Red. Crc. Mat. Palermo., Ser. II, 68 (2002, G. G. Loretz, Zur theore der polyome vo S. Berste, Matematceskj Sbork 2 (1937, D. D. Stacu, The remader of certa lear approxmato formulas two varables, SIAM J. Num. Aal. 1 (1964, E. Voroovskaya, Détermato de la forme asymptotque d approxmato des foctos par les polyômes de M. Berste, Doklady Akadem Nauk SSSR (1932, Mchael S. Floater Cetre of Mathematcs for Applcatos Departmet of Iformatcs Uversty of Oslo Postbox 1053, Blder 0316 Oslo, NORWAY mchaelf@f.uo.o 6