On the Approximation Properties of Bernstein Polynomials via Probabilistic Tools

Transcription

1 Boletí de la Asocacó Matemátca Veezolaa, Vol. X, No O the Approxmato Propertes of Berste Polyomals va Probablstc Tools Heryk Gzyl ad José Lus Palacos Abstract We study two loosely related problems cocerg approxmato propertes of Berste polyomals B fx of some fucto f o [0, 1]: the absece of Gbbs pheomeo at pots at whch f has jumps, ad the covergece of B f x towards f x. Both results are obtaed usg classcal probablstc tools. I partcular the proof of the secod statemet reles o the represetato of the dervatve B f x as the expectato of the fuctoal of a radom varable. 1 Itroducto. Let f be a real fucto defed o the terval [0, 1]. Let S,x be a Bomal radom varable wth parameters ad x, ad let E[X] deote the expected value of the radom varable X. I our prevous paper [3], we showed how to use the theory of large devatos to derve rates of covergece of the Berste polyomals, defed as: B fx = =0 x 1 x f = Ef S,x to the lmt fucto f, whe the f beg approxmated s Lpschtz cotuous. The rate O 1/3 obtaed wth Berste s classc probablstc proof, where all that s used s Chebyschev s equalty, was mproved to Ol / 1/. M. K. Kha [4] brought to our atteto the fact that the optmal rate amog Lpschtz fuctos s deed 1/. Moreover, P. Mathé, a ce hstorcal framework, showed [5] that f the fucto s Hölder cotuous wth expoet α for some 0 < α 1, the rate of covergece s α/. Some obvous questos come up whe we compare the approxmato of f gve by 1 wth approxmatos by, say, Fourer seres or other orthogoal expasos. For starters: how does B fx 0 behave whe f has a jump dscotuty at x 0? Ad more mportat, does the Gbbs-Wlbraham pheomeo see [Hewtt-Hewtt] take place as well? 1

2 6 H. Gzyl ad J. L. Palacos I ths paper we cotue usg probablstc tools to further study approxmato propertes of B fx whe f s ot cotuous. I fact, we show that the Gbbs pheomeo does ot occur for the approxmato of mootoe, pecewse smooth fuctos, havg both left ad rght dervatves at every pot, by Berste polyomals, cotrary to what happes wth the Fourer seres of such a fucto. Specfcally, we cosder a smple jump fucto see below ad prove that B fx 0 1 fx fx 0 0 as, ad also that the covergece s mootoe o both sdes of the dscotuty, as a cosequece of whch the characterstc overshoot of the Gbbs pheomeo does ot occur. We also provde estmates of the sze of the approxmato error ad the rate of covergece at the dscotuty pot. Later o we prove that for a bouded fucto f, B f x coverges to f x at all x at whch f x exsts. Let f be a sequece of approxmats to a pecewse smooth f that has rght ad left dervatves at every pot, say a partal sum of a trgoometrc seres, Bessel fuctos or Gegebauer polyomals. I geeral, the Gbbs pheomeo ca be descrbed by the followg behavor: If x 0 s a dscotuty pot of f, the lm f x = fx fx 0. O ay subterval [x 1, x ] for whch the fucto s cotuous, we have uform covergece: lm max f x fx = 0. x 1 x x O ay subterval cotag a sgle dscotuty x 0 of the fucto, we have Gbbs pheomeo: for small δ > 0 where lm max f x m f x x 0 x δ x 0 x δ C = π π 0 s x x = C fx fx 0 0. dx See the work by Bachma-Narc-Beckester, Dym-Mc Kea or Gray- Psky for precse statemets ad examples whch the Gbbs pheomeo occurs. Gbbs pheomeo arses whe approxmatg dscotutes usg smooth approxmats. I addto to the more recet paper by Gray ad Pky, Hewtt-Hewtt provde us wth a excellet survey wth may hstorcal detals. I [1], Gottleb ad Shu provde a short complemetary hstorcal revew, amog other terestg results whch we meto below.

3 Approxmato Propertes of Berste Polyomals 7 I the ext secto we shall show that the Berste polyomal approxmat satsfes ad above, ad that t does ot overshoot at a pot of dscotuty, other words, that holds wth C = 1 wheever f s a fte sum of jump fuctos, that s fuctos that chage oly at jump pots. We exted these results to a mootoe f wth a fte umber of jumps wth a mor modfcato o. I the last secto we show how to wrte B f x as a expectato of a fuctoal of a bomal varable, ad study the covergece of ths expectato to f x. No Gbbs pheomeo for Berste polyomals. Let us cosder a smple jump fucto { a x < x0 fx =. b> a x x 0 The Berste polyomal for ths fucto s B fx = ap S,x < x S,x 0 + bp x S,x 0 = a + b ap x 0, for whch the followg bouds obvously hold: a = B f0 B fx B f1 = b. 3 It ca also be proved that B f s a creasg fucto o accout of the fact that f x y the for 0 k oe has y 1 y = P S,y k. x 1 x k P S,x k = k 4 Both sdes of 4 are equal to 1 whe k = 0. For other values of k, oe subtracts form each sde the approprate terms that preserve the equalty. A more elegat way to prove 4 s to cosder depedet copes of the bvarate 0 1-valued varables X, Y, 1, wth jot dstrbuto dctated by the probabltes P X = Y = 1 = x, P X = 1, Y = 0 = 0 ad P X = Y = 0 = 1 y. If we defe Z = =1 X ad W = =1 Y, the the dstrbutos of Z ad W are those of, respectvely, S,x ad S,y, ad by costructo {Z k} {W k}, so that P S,x k = P Z k P W k = P S,y k. We wll deal ow wth uform covergece o a terval [x 1, x ] [0, 1] {x 0 }. Cosder frst x < x 0. By Chebyschev s equalty S,x P x S,x 0 P x x 0 x

4 8 H. Gzyl ad J. L. Palacos ad therefore x1 x x 0 x 1 4x 0 x, 5 B fx a = fx uformly the terval [x 1, x ]. A smlar argumet holds for x [x 1, x ] such that x 0 < x 1. O the other had, f x = x 0 we have S,x0 B fx 0 = a + b ap x 0. Now, f we cosder X, 1, to be depedet Beroull varables wth parameter x 0 we ca wrte S,x0 =1 P x 0 = P X x 0 0 = P X x 0 x0 1 x 0 0 1, =1 as by the cetral lmt theorem CLT, ad B fx 0 a + b = a b S,x0 P x 0 1 0, as, that s, B fx 0 coverges to the average of the rght ad left lmts of f at x 0. These calculatos take care of propertes ad for a jump fucto as descrbed. To verfy that codto holds wth C=1,. e., that the approxmato by Berste polyomals fts well, ote that o accout of B f beg creasg we have max B fx m B fx = B fx 0 + δ B fx 0 δ, x 0 x δ x 0 x δ ad lettg ted to fty ad usg property o x 0 + δ ad x 0 δ we obta lm max B fx m B fx = fx 0 + δ fx 0 δ x 0 x δ x 0 x δ = fx fx 0 0. Commets These results ca be exteded obvously to a fte sum of smple jump fuctos. The exteso to ay mootoe fucto wth ftely may jumps [0, 1] -whch ca be wrtte as a sum of a cotuous mootoe fucto plus a fte sum of smple jump fuctos- s ow straghtforward. Notce that

5 Approxmato Propertes of Berste Polyomals 9 whe dealg wth approxmatos by Berste polyomals, tems ad metoed the troducto, rema vald, But tem has to be replaced by If x 0 s a dscotuty of f the lm lm δ 0 max B fx m B fx x 0 x δ x 0 x δ = fx fx 0 0. whch ca be rephrased as sayg that there s o Gbbs pheomeo for approxmato by Berste polyomals. Ths coects wth the followg geeral problem, for whch there seems to be o geeral aswer: Suppose we wat to approxmate a jump by a sequece of approxmats. Whch propertes of the approxmatg sequece cause the overshoot pheomeo? Typcally, as approxmato by trgoometrc polyomals, the approxmatg sequece cossts of partal sums of a orthogoal seres. Is t orthogoalty whch causes the overshootg? If so, the oe should ot woder that Berste polyomals do ot show t. Is t smoothess? There seems to be o geeral theorem ths drecto, but seemgly Gbbs pheomeo arses from approxmatg a jump by a system whch s desged to allow approxmatos of arbtrary order. The latter statemet s supported by results of V. L. Velk [7], who showed that sple approxmato yelds the same pheomeo, f the smoothess of the sples teds to. Berste polyomals, although smooth, allow approxmato oly up to order 1, o matter how smooth the fucto s, see for example the results secto 1.6 of the book by G. G. Loretz. I secto 4.16 of [10] t s show how approxmatg by Cesaro sums, klls the overshootg. Ad ths ssue s explored further secto 3 of [1] as part of the problem of devsg rapdly coverget methods for recostructg local behavor value of a fucto at a pot from global data Fourer coeffcets. I what we dd above, we dd ot pay atteto to ssues related to speed of covergece. As we dd [], we ca make some precse statemets vokg the followg large devatos result foud for stace [9] Lemma. For a bomal radom varable S,x ad a > 0 arbtrary P S,x x > a e a. 6 The, for example, the boud O 1 of the uform covergece obtaed wth Chebyschev equalty 5 may be mproved to the expoetal boud e x0 x. Lkewse, whe verfyg that codto above holds wth C=1, we ca argue that the speed of covergece to the lmt s expoetal. The lemma wll also be used the proof of proposto below.

6 10 H. Gzyl ad J. L. Palacos 3 Covergece of B fx towards f x I the prevous secto we showed that B f s creasg whe f s a creasg jump fucto by drect computatos wth the bomal dstrbuto. We ca show geeral that whe fx s creasg resp. decreasg, the the dervatve B f x of B fx s postve resp. egatve, ad therefore B f s also creasg resp. decreasg. I fact, we ca provde the followg Proposto 1. The dervatve of the Berste polyomal B f ca be expressed as B f x = E S,x x D, fx, 7 x1 x where f D, fx = S,x f x S,x x. Proof. Dervg 1 wth respect to x we obta = B f x = = 1 x ES,xf 1 x1 x E =0 =0 S,x f f 1 = x1 x E 1 [ S,x x x 1 1 x x 1 x 1 1 x E S,x f [ S,x S,x xf f S,x [ S,x x f ] S,x fx + fx S,x ] fx 1 = x1 x E = E S,x x f S,x fx x1 x S,x x = E S,x x D, fx, x1 x ]

7 Approxmato Propertes of Berste Polyomals 11 as clamed From ths s clear that B f x s postve resp. egatve case D, fx s postve egatve, ad ths occurs wheever f s creasg resp. decreasg. Furthermore, f x s a pot where the dervatve exsts, D, fx f x as, ad the squared term the tegral 7, by the CLT, coverges to the square of a stadard ormal varable, so we should have B f x f x whe s large. Ths ca be made rgorous va a probablstc proof whch fact does ot use the CLT, as follows: Proposto. Assume sup x [0,1] fx = M <. The for ay x such that f x exsts we have lm B f x = f x. Proof. Defe A, fx = D, fx f x. The we ca wrte B f x = E S,x x D, fx x1 x = f x + E S,x x A, fx, 8 x1 x S,x x sce E = 1. x1 x Now, sce f x exsts, gve ɛ > 0, there exsts δ > 0 such that f S,x x < δ the A, fx < ɛ. We prove that the absolute value of the secod summad 8 goes to zero by splttg ad boudg t by E S,x x A, fx 1 x1 x { S,x x <δ} +E S,x x A, fx 1 x1 x { S,x x δ}, 9 where 1 A stads for the dcator fucto of the set A. The frst summad 9 ca be boud by ɛe S,x x 1 x1 x { S,x ɛ, 10 x <δ}

8 1 H. Gzyl ad J. L. Palacos whereas the secod summad 9 ca be boud by M δ [ S,x + f x 4 E x 1 { M δ + f x 4 P S,x S,x x δ} ] x δ, 11 where the last equalty uses the fact that the square of the dstace of the pots S,x usg 6 by ad x the terval [0, 1] s less tha 1. Now 11 ca be boud, M δ + f x e δ, whch goes to zero as, fshg the proof We should meto that a smlar result s stated as a problem of chapter VII of Feller s classc [13]. What s ot clear s whether a represetato lke 7, from whch our proof follows, was kow to hm. Akowledgemets It s a pleasure to thak the referees for ther commets. They helped us mprove our presetato. Refereces [1] E. Parze, Moder Probablty Theory ad ts Applcatos, New York, Wley, [] H. Gzyl ad J. L. Palacos, The Weerstrass approxmato theorem ad large devatos, Amerca Mathematcal Mothly, [3] M. K. Kah, O the Weerstrass approxmato theorem ad large devatos, upublshed mauscrpt [4] P. Mathé, Approxmatos of Hölder cotuous fuctos by Berste polyomals, Amerca Mathematcal Mothly, [5] A. Gray ad M. A. Psky, Gbbs pheomeo for Fourer-Bessel seres, Exposto. Math., [6] E. Hewtt ad R.E. Hewtt,The Gbbs-Wlbraham pheomeo: a epsode Fourer aalyss, Arch. Hst. Exact Sc., 1: 1979/ [7] V.L. Velk, A lmt relato for dfferet methods of approxmatg perodc fuctos by sples ad trgoometrc polyomals, A. Math., 13:

9 Approxmato Propertes of Berste Polyomals 13 [8] G.G. Loretz, Berste Polyomals, Vol. 8 of Math. Expos., Uv. of Toroto Press, Toroto [9] N. Alo ad J. Specer, The Probablstc Method, Wley, New York, 199. [10] G. Bachma, L. Narc ad E. Beckeste, Fourer ad Wavelet Aalyss, Sprger-Verlag, Berl, 000. [11] H. Dym ad H. Mc Kea, Fourer Seres ad Itegrals, Acad. Press, New York, 197 [1] D. Gottleb ad C. Shu, O the Gbbs pheomeo ad ts resoluto, SIAM Rev, 39:4, 1997, [13] W. Feller, A Itroducto to Probablty Theory ad ts Applcatos, Vol. II, Wley, New York, Heryk Gzyl ad José Lus Palacos Depto. de Cómputo y Estadístca Uversdad Smó Bolívar Caracas, Veezuela hgzyl@reaccu.ve; jopala@cesma.usb.ve