Bounding the Lebesgue constant for Berrut s rational interpolant at general nodes

Transcription

1 Bodig the Lebesge costt for Berrt s rtiol iterpolt t geerl odes Le Bos Stefo De Mrchi Ki Horm Je Sido My, Abstrct It hs recetly bee show tht the Lebesge costt for Berrt s rtiol iterpolt t eqidistt odes grows logrithmiclly i the mber of iterpoltio odes. I this pper we show tht the sme holds for very geerl clss of well-spced odes essetilly y distribtio of odes tht stisfy certi reglrity coditio, icldig Chebyshev Gss Lobtto odes s well s exteded Chebyshev odes. Itrodctio For N, let X = {x, x,..., x } be set of distict iterpoltio odes i the rel itervl [, b], let B = {b, b,..., b } be correspodig set of crdil bsis fctios. The orm of the lier iterpoltio opertor which mps f C [, b] to g = b jf(x j is lso kow s the Lebesge costt [9, ] Λ(X, B = mx x b b j (x, oe is iterested i choices of odes X bsis fctios B sch tht Λ(X, B is smll, s this leds to smll bods o the pproximtio error f g, mesred i the mximm orm. I the cse of polyomil iterpoltio, whe B is the set of Lgrge bsis polyomils with respect to X, it is well kow tht the Lebesge costt grows logrithmiclly with for Chebyshev odes, bt expoetilly for eqidistt odes [4]. However, mch smller Lebesge costts re observed for rtiol iterpoltio t eqidistt odes [5, ]. I prticlr, Bos et l. [, 3] show tht the Lebesge costt for Berrt s rtiol iterpolt [] with bsis fctios b i (x = ( i x x i / ( j x x j, i =,..., grows logrithmiclly i the mber of eqidistt iterpoltio odes, tht the sme holds for the more geerl fmily of brycetric rtiol iterpolts which ws itrodced by Floter Horm [6]. Moreover, Horm et l. [7] show tht this behvior exteds to qsi-eqidistt odes with boded globl mesh rtio. I this work we show tht for essetilly y resoble choice of iterpoltio odes X, the Lebesge costt for Berrt s rtiol iterpolt, Λ(X = mx x b x x j ( j, ( x x j is boded from bove by c l( for some costt c >. More precisely, we derive this reslt for well-spced odes (Sectio, show tht sch odes c be geerted by distribtio fctio stisfyig reglrity coditio (Sectio 3. We coclde the pper by discssig severl exmples (Sectio 4. Uiversity of Vero (Itly, leordpeter.bos@ivr.it Uiversity of Pd (Itly, demrchi@mth.ipd.it Uiversity of Lgo (Switzerl, ki.horm@si.ch

2 . Prelimiries Withot loss of geerlity we ssme tht the odes i X re ordered, x < x < < x, tht x = x =, becse the Lebesge costt i ( is ivrit with respect to lier trsformtios of the iterpoltios odes. We frther deote the legth of the k-th sbitervl by h k = x k x k, k =,..., ( restrict or discssio to the iterpoltio itervl [, b] = [x, x ] = [, ]. Followig the pproch i [], let s itrodce the fctios N k (x = (x x k (x k x x x j ( j D k (x = (x x k (x k x x x j for x k < x < x k k =,...,, observe tht the Lebesge costt ( stisfies Λ(X = Or geerl gol is to estblish the bods mx k=,..., N k (x c h k l( D k (x c h k, N k (x mx x k <x<x k D k (x. where the costts c > c > re idepedet of x, k,. To this ed, it helps to recll from [] tht ( k N k (x = h k (x x k (x k x (3 x x j x j x j=k from [7] tht [ ( ( D k (x = (x x k (x k x x x k 3 x x k x x k x x k ( ( ] x x k x x k x x k3 x x k4 [( (x x k (x k x ( ] x x k x x k x k x x k x ( = h k (x x k (x k x, x x k x k x becse ll pired terms re positive for x k < x < x k. We remrk tht oe of the two terms i the lst fctor is missig i the cses k = k =, bt s the fctor will be smller the, we c sfely igore this detil i the sbseqet discssio. Well-spced odes Defiitio.. For ech N, let X be set of iterpoltio odes. We the sy tht X = (X N is fmily of well-spced odes, if there exist costts C, R, idepedet of, sch tht the three coditios x k x k C, j =,..., k, k =,...,, (5 x k x j k j (4

3 hold for ech set of odes X. x k x k C, j = k,...,, k =,...,, (6 x j x k j k R x k x k x k x k R, k =,..., (7 This defiitio icldes eqidistt odes (with smllest possible costts C = R =, bt it lso covers Chebyshev Gss Lobtto odes exteded Chebyshev odes, s show i Sectio 4. Theorem.. If X = (X N is fmily of well-spced odes, the there exists costt c > sch tht for y. Proof. We will ctlly show, more precisely, tht Λ(X c l( Λ(X (R ( C l(, (8 where R C re the costts defiig X s well-spced i Defiitio.. The clim the follows by lettig c = (R (C. Let x k < x < x k for some k. From (3 we get k x x k N k (x = h k (x k x (x x k x x j j=k k x k x k h k (x k x k (x k x k x k x j k x k x k = h k ( x k x j j=k x k x k x j x k It the follows from coditios (5 (6 i Defiitio. tht k C N k (x h k ( k j j=k ( C h k C j k with the costt c = (C >. To bod D k (x from below, we cosider the fctio ( g(x = (x x k (x k x j= x x k. x k x x j x j=k x k x k x j x k ( h k C l c h k l( j x k x = (x x k(x k x (x x k (x k x (x k x k = ( ( (x k x k x k x k x x k x k x k x k x h k h k h k = ( (. h k x x k h k x k x Let s ssme tht h k h k ote tht the other cse c be treted similrly, de to the ppret symmetry of g(x with respect to h k h k. It is the strightforwrd to verify tht we c bod g(x from bove by replcig h k with h k, h k h k h k g(x ( (, h k x x k h k x k x 3

4 tht this pper bod ttis its mximm t x = (x k x k /. Hece, h k h k h k g(x ( ( = ( h k x x k h k x k x h k h ( k h k h k / h k h k / It the follows from (4 the pper bod i coditio (7 tht D k (x h k h k h k h k = h k h k h k h k h k h k h k h k = = h k (h k h k (h k h k = h k h k h k. h k h k /h k h k R = c h k for the costt c = /(R >. Note tht the lower bod i coditio (7 is reqired to esre h k /h k R for k =,...,, which i tr is eeded i the other cse whe h k > h k. 3 Geertig well-spced odes Theorem. grtees tht the Lebesge costt for Berrt s rtiol iterpolt grows logrithmiclly i the mber of iterpoltio odes i cse they re well-spced, bt the qestio remis how geerl the ltter property relly is. It trs ot, thogh, tht beig well-spced is othig specil, s the three coditios i Defiitio. re stisfied by essetilly y resoble choice of iterpoltio odes. To this ed, let s cosider fmilies of odes which re derived by eqidisttly smplig some give fctio. Defiitio 3.. We sy tht fctio F C[, ] is distribtio fctio if it is strictly icresig bijectio o the itervl [, ]. Give distribtio fctio F some N, we defie the ssocited iterpoltio odes X = X (F by settig x k := F (k/, k =,...,. (9 By Defiitio 3., these odes re ordered stisfy or ssmptio x = x =. As show i Sectio 4., there exist distribtio fctios tht yield ode sets for which the Lebesge costt i ( grows fster th logrithmiclly with, bt this behvior c be rled ot by the followig reglrity coditio. Defiitio 3.. We sy tht distribtio fctio F is reglr, if F C [, ] F hs fiite mber of zeros T = {t, t,..., t l } [, ] with fiite mltiplicities. Tht is, F (t > for t [, ] T there exist positive rel mbers r, r,..., r l, sch tht G j (t = F (t/ t t j rj is cotios with lim t tj G j (t > for j =,..., l. We ow prove two properties of reglr distribtio fctios, which lter help to estblish the three coditios from Defiitio.. Note tht some of the techicl detils i the proofs re provided i the Appedix. Propositio 3.3. Let F be reglr distribtio fctio. The there exists costt C > sch tht for ll x, y, z [, ] with x > y z. F [x, y] F [x, z] C Proof. Sice F C [, ], the divided differece F [, v] is cotios bivrite fctio. By Defiitio 3., F [, v] for y (, v [, ] F [, v] = if oly if = v T. Hece, the trivrite fctio G(x, y, z = F [x, y] F [x, z] is cotios o [, ] 3 E, where E = {(x, y, x [, ] 3 : x T }. To costrct sfety mrgi rod the siglrities of G t E, let s fix δ >, sch tht the ope itervls I j = (t j δ, t j δ re mtlly disjoit for j =,..., l, cosider the set D := [, ] 3 l (I j [, ] I j. j= 4

5 As G is cotios, it hs pper bod C o the compct set D, we eed oly lyse wht hppes er the set E. To this ed, we first ote tht Defiitio 3. grtees m = mi j=,...,l ( if t I j F (t t t j rj M = mx j=,...,l ( sp t I j F (t t t j rj to be positive rel mbers. Sppose ow tht x, z I j [, ] for some j {,..., l} x > y z. The, so tht F [x, y] = x y F [x, z] = x z x y x z F (tdt F (tdt G(x, y, z = M x y m x z x y x z t t j rj dt = t t j rj dt = F [x, y] F [x, z] M m (r j =: C j by Lemm A.. The sttemet the follows by lettig C = mx(c, C,..., C l. M x tj t rj dt x y y t j m x tj t rj dt, x z z t j Propositio 3.4. Let F be reglr distribtio fctio. The there exist some ε > costt R > sch tht F [x, x s] R F [x s, x] R for ll s [, ε] x [s, s]. Proof. By rgmets similr to the oes sed i the proof of Propositio 3.3, the bivrite fctio G(x, s = F [x, x s] F [x s, x] is cotios, prt from the siglrities t {(x, : x T }. Note tht for x T, we hve F (x >, hece G(x, = F (x/f (x =. We cosider gi the mtlly disjoit itervls I j = (t j δ, t j δ for j =,..., l rod the zeros of F, fix ε = δ/, defie D := {(x, s : s [, ε], x [s, s] l j=i j }. As G is cotios positive o the compct set D, there exists some R > sch tht ( R G(x, s R, (x, s D, it remis to lyse the behvior of G er the siglrities. To this ed, let s [, ε] x I j [s, s] for some j {,..., l}. The, G(x, s = F (x s F (x F (x F (x s = with the costts m, M > from (, frther xs x xs F (tdt x x s F (tdt M t rj dt x m x x s t rj dt G(x, s M m 3rj =: R j by Lemm A.4. Similrly, we lso hve xs G(x, s m t rj dt x M x dt m M 3 (rj =, R x s t rj j so tht the sttemet follows by lettig R = mx(r, R,..., R l. 5

6 F Figre : Lebesge costt for Berrt s rtiol iterpolt t eqidistt iterpoltio odes, geerted by the reglr distribtio fctio F (t = t for 5. Theorem 3.5. Let F be reglr distribtio fctio X the ssocited iterpoltio odes from (9 for y N. The the fmily of odes X = (X N is well-spced. Proof. We mst show the three coditios of well-spced odes i Defiitio.. For coditio (5, ote tht we my write x k x k = F ( k ( F k x k x j F ( ( k F j = F [ k, ] k k j F [ j, ] = F [ k, ] k k F [ j, ] k k j, the this coditio follows from Propositio 3.3 with x = (k /, y = k/, z = j/. Coditio (6 c be show logosly by oticig tht x k = x k, where X = { x, x,..., x } re the iterpoltio odes geerted by the reglr distribtio fctio F (t = F ( t. For coditio (7, we write, similr s bove, x k x k = F [ k, ] k x k x k F [ ], k coclde from Propositio 3.4 with x = k/ s = / tht this coditio holds for ll /ε, where ε is the costt itrodced i Propositio 3.4. Bt s the mber of remiig cses for < /ε is fiite, we c elrge the costt R from Propositio 3.4 so tht it lso covers these cses, the coditio (7 holds for ll N. 4 Exmples 4. Reglr distribtios The simplest exmple of reglr distribtio fctio is the fctio F (t = t, which geertes eqidistt odes (see Figre. For these odes, it hs lredy bee show by Bos et l. [] tht the Lebesge costt for Berrt s rtiol iterpolt grows logrithmiclly, their pper bod is eve tighter th or geerl pper bod i Theorem.. However, the reslts from the previos sectios llow s to hle more geerl ode distribtios. For exmple, Bos et l. [] report mericl tests which show tht the Lebesge costt grows logrithmiclly with for logrithmiclly distribted iterpoltio odes. As these odes re i fct geerted by the distribtio fctio F (t = l( t(e, which clerly is reglr becse F (t is strictly positive, Theorems. 3.5 ow provide proof of this behvior (see Figre. As secod exmple, cosider the fctio F 3 (t = ( cos(πt/, which is reglr distribtio fctio by Defiitio 3., s its first derivtive hs l = zeros t t = t = with mltiplicities r = r =. This fctio geertes the extrem of the Chebyshev polyomils (of the first kid, mpped to [, ],, k x k = ( cos(kπ//, k =,...,, which re lso kow s Chebyshev Gss Lobtto odes or Cleshw Crtis odes (see Figre 3. By Theorem 3.5, these odes re well-spced, so the Lebesge costt ( grows logrithmiclly with, de to 6

7 F Figre : Lebesge costt for Berrt s rtiol iterpolt t iterpoltio odes, geerted by the reglr distribtio fctio F (t = l( t(e for F Figre 3: Lebesge costt for Berrt s rtiol iterpolt t Chebyshev Gss Lobtto odes, geerted by the reglr distribtio fctio F 3 (t = ( cos(πt/ for 5. Theorem.. Followig the proofs of Propositios , sig δ = / ε = /4, the costts i Defiitio 3. tr ot to be C = π R = 9π/ for this ode distribtio, ledig to the pper bod Λ(X (9π/ ( 4π l(, ccordig to (8. Admittedly, this bod is ot very tight, it reqires frther ivestigtios to derive better pper bods. 4. No-reglr distribtios While the reglrity coditio for distribtio fctios i Defiitio 3. is clerly sfficiet for the ssocited odes to be well-spced the growth of the Lebesge costt to be logrithmic (see Theorems. 3.5, we believe tht this coditio is lso ecessry. To this ed, let s cosider the distribtio fctio {, t =, F 4 (t = ( exp( /t, < t. This fctio is C -flt t the origi hece ot reglr i or sese, becse its derivtive hs zero with ifiite mltiplicity t t =. For these odes, the mericl estimtes i Figre 4 idicte tht the Lebesge costt for Berrt s rtiol iterpolt grows expoetilly with. However, the lie betwee expoetil logrithmic growth seems to be very thi, s secod exmple demostrtes. The distribtio fctio F 5 (t = exp( /(t, t < /,, t = /, ( exp( /(t, / < t. is ot reglr, becse its derivtive hs zero with ifiite mltiplicity t t = /, we observe i Figre 5 tht the correspodig Lebesge costt grows expoetilly with. Bt this is tre oly for odd, while for eve the growth seems to be oly logrithmic. 7

8 F Figre 4: Lebesge costt for Berrt s rtiol iterpolt t iterpoltio odes, geerted by the o-reglr distribtio fctio F 4 i ( for 5. Note tht the ordite is plotted i logrithmic scle, so tht the lier growth observed i the plot correspods to expoetil growth of the vles themselves F Figre 5: Lebesge costt for Berrt s rtiol iterpolt t iterpoltio odes, geerted by the o-reglr distribtio fctio F 5 i ( for 35. As i Figre 4, the growth is expoetil, bt oly for odd, while the vles seem to grow logrithmiclly for eve. 4.3 Exteded Chebyshev odes Besides odes tht re geerted by reglr distribtio fctios, we c lso verify directly if give fmily of odes is well-spced. For exmple, the Chebyshev odes, mpped to [, ], which re lso kow s exteded Chebyshev odes [4] give by x k = [ ( k cos π / ( ] cos π, k =,...,, (3 re ot geerted by distribtio fctio, bt they re still well-spced, s the followig propositios cofirm. Therefore, it follows from Theorem. tht the Lebesge costt for Berrt s rtiol iterpolt t these odes grows logrithmiclly (see Figre 6, s lredy observed mericlly by Bos et l. []. Propositio 4.. For the exteded Chebyshev odes i (3, x k x k π /, j =,..., k, k =,...,, x k x j k j for y N. Tht is, coditio (5 of Defiitio. holds with C = π /. Proof. First ote tht x k x k x k x j = ( ( ( k 3 k cos π si π si π ( = ( (. k 3 k j k j π cos π si π si π ( k cos π ( j cos 8

9 Figre 6: Lebesge costt for Berrt s rtiol iterpolt t exteded Chebyshev odes for 5. Moreover, for the cosidered rges of j k, we lwys hve k j, hece k j π π. Now, if i dditio kj π π, the sig the ieqlity φ/π si(φ φ for φ [, π/], we hve ( ( k x k x k π π ( ( = π k x k x j k j k j π π π π 4 k j k j π k j. O the other h, if kj π > π k kj, the π > π > π, so ( k si π (, k j si π becse si(φ is decresig o [π/, π]. Therefore, i this cse, ( si x k x k π ( π x k x j k j si π π k j π = π k j. Note gi tht coditio (6 i Defiitio. is etirely symmetric, hece we leve ot the detils. Propositio 4.. For the exteded Chebyshev odes i (3, x k x k x k x k, k =,...,, for y. Tht is, coditio (7 of Defiitio (. holds with R =. Proof. We first observe tht ( ( ( ( ( k k 3 k k cos x k x k π cos π si π si π si π = ( ( = ( ( = (, x k x k k k k k cos π cos π si π si π si π frther, with θ = kπ [ π π, π ], x k x k = si ( θ π x k x k si(θ ( ( π π = cos cot(θ si. 9

10 As cot(φ is strictly decresig o (, π, we coclde cot(θ cot ( π, hece ( ( ( ( x k x k π π π π cos cot si = cos. x k x k Similrly, ( ( x k x k π cos cot π π ( π si = ( π x k x k sec where we sed the trigoometric idetity cos(φ cot(π φ si(φ = sec(φ/ with φ = π/(. By (8, we therefore get the pper bod Λ(X 3 3π l( o the Lebesge costt for Berrt s rtiol iterpolt t exteded Chebyshev odes. Ackowledgmets This work hs bee doe with spport of the 6% fds, yer of the Uiversity of Pd, s well s those of the Uiversity of Vero. Refereces [] J.-P. Berrt. Rtiol fctios for grteed experimetlly well-coditioed globl iterpoltio. Compt. Mth. Appl., 5(: 6, 988. [] L. Bos, S. De Mrchi, K. Horm. O the Lebesge costt of Berrt s rtiol iterpolt t eqidistt odes. J. Compt. Appl. Mth., 36(4:54 5, Sept.. [3] L. Bos, S. De Mrchi, K. Horm, G. Klei. O the Lebesge costt of brycetric rtiol iterpoltio t eqidistt odes. Nmer. Mth.,. To pper. [4] L. Brtm. Lebesge fctios for polyomil iterpoltio srvey. A. Nmer. Mth., 4: 7, 997. [5] J. M. Cricer. Weighted iterpoltio for eqidistt odes. Nmer. Algorithms, 55( 3:3 3, Nov.. [6] M. S. Floter K. Horm. Brycetric rtiol iterpoltio with o poles high rtes of pproximtio. Nmer. Mth., 7(:35 33, Ag. 7. [7] K. Horm, G. Klei, S. De Mrchi. Brycetric rtiol iterpoltio t qsi-eqidistt odes. Dolomites Res. Notes Approx., 5: 6,. [8] T. J. Rivli. The Lebesge costts for polyomil iterpoltio. I H. G. Grir, K. R. Ui, J. H. Willimso, editors, Fctiol Alysis its Applictios, volme 399 of Lectre Notes i Mthemtics, pges Spriger, Berli, 974. [9] T. J. Rivli. A Itrodctio to the Approximtio of Fctios. Dover, New York, 98. [] J. Szbdos P. Vértesi. Iterpoltio of Fctios. World Scietific, Sigpore, 99. [] Q. Wg, P. Moi, G. Iccrio. A rtiol iterpoltio scheme with sperpolyomil rte of covergece. SIAM J. Nmer. Al., 47(6: ,.

11 A Appedix Lemm A.. Let, v, r R with < v r. The, v r r v r (r t r dt v r, if < v, (4 t r dt vr, if < < v, v. (5 r Proof. Usig the ssmptio < v i (4, the two bods c be estblished by oticig tht t r dt = t r dt = t r dt v r dt = v r t r dt = v r r v r v r = vr r r r. Similrly, the pper bod i (5 follows from the give coditios o v, becse v t r dt = ( ( t r dt t r dt = ( r v r ( v r v r = vr r r r. The lower bod i (5 is eqivlet to ( r v r which is tre, sice v ( >. vr ( r v r ( v r, Lemm A.. Let, b, c, r R with b < c r. The, ( c /( t r c dt t r dt (r. c b c Proof. For or coveiece, let B := c b b c By Lemm A., we the hve c r, b < c, c r /(r, b < < c, b c, B ( b r /(r, b < < c, b c, ( b r, b < c b t r dt A := c c t r dt. c r /(r, < c, c r /((r, < < c, c, A ( r /((r, < < c, c, ( r /(r, < c, where the first two cses i both ieqlities follow directly from (4 (5, respectively. The other two cses c be derived similrly by oticig the ppret symmetry of A B with respect to chgig the sigs of, b, c. Combiig ll cses, we fid B A c r c r /(r = r, b < c, c r c r /((r = (r, < b < c, c, c r ( r /((r c r c r /((r = (r, < b < c, c, c r /(r c r /((r =, b < < c, b c, c, c r /(r ( r /((r ( b r /(r ( r /((r ( b r ( r /(r cr /(r c r /((r =, b < < c, b c, c, ( br /(r ( b r /((r =, b < < c, b c, ( b r ( b r /(r = r, b < c,

12 where we sed the fct tht the coditios of the lst two cses imply b, lso c i the secod-lst cse. Note tht this covers ll possible cses of, b, c with b < c, s ssmed i the sttemet of the lemm. Lemm A.3. Let, v, s R with, v, s. The, ( v s s s (( v s s. Proof. Let s fix v cosider the fctio f( = ( v s s. It follows from the ssmptio s tht f ( = s(( v s s for. Hece, f( is icresig, so ( v s s = f( f( = ( v s ( s = s (( v s s. Lemm A.4. Let, b, r R with b >, r. The, 3 (r b Proof. Sbstittig t with s i both itegrl gives b t r dt = (b ( / t r dt t r dt 3 r. (6 b s r ( ds = ( ( b s r ds t r dt = ( s r ( ds = ( b b ( b ( s r ds, s the bods i (6 re symmetric with respect to mltiplictive iversio, we c ssme withot loss of geerlity tht. We first cosider the cse b, so tht b < < b b / t r dt t r dt = b b / t r dt t r dt = ( br r b r ( b r. The pper bod i (6 the follows from Lemm A.3 with = b, v = b, s = r, becse ( b r ( b r r ( r ( b r ( b r r r r ( b r ( b r ( b r r ( r ( r ( b r = ( b r r r ( b r r 3 r. For the lower bod, otice tht the covexity of f( = r for r implies ( br ( ( br ( b r ( b = r = ( b r r r ( b r ( b r r r ( b r 3 (r. I the secod cse with < b we hve b / t r dt t r dt = b b /( t r dt ( t r dt b t r dt = ( br r r (b r.

13 The pper bod i (6 the follows by sbstittig b with c, where c >, cosiderig { ( c r (3c r, c (3 r = ( c r (3 r (3c r r, c To estblish the lower bod, otice tht ( c r r 3 r ( r c r ( c r r r c r 3 r. ( b r = ( (b r ( r (b r = r r (b r r (b r, hece ( b r r r (b r = ( b r r r (b r 3 (r. 3