QUADRATURE FORMULAS ON THE UNIT CIRCLE BASED ON RATIONAL FUNCTIONS

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1 QUADRATURE FORMULAS ON THE UNIT CIRCLE BASED ON RATIONAL FUNCTIONS Adhemar Bultheel Dept. f Cmputer Scieces, K. U. Leuve, Belgium Erik Hedrikse Dept. f Mathematics, Uiversity f Amsterdam, The Netherlads Pabl Gzalez-Vera Dept. f Mathematical Aalysis, La Lagua Uiversity, Teerife, Spai Olav Njastad Dept. f Mathematics, Uiversity f Trdheim-NTH, Nrway Abstract Quadrature fulas the uit circle were itrduced by es et als m 989. O the ther had, Bultheel et a\' als csidered such quadratures by givig results ccerig ~ errr ad cvergece. I ther recet papers, a mre geeral situati was studied by the abve fur authrs ivlvig rthgal ratial fuctis the uit circle which geeralize the well kw Szeg6 plymials. I this paper, these quadratures are agai aalyzed ad results abut cvergece give. Furthere, a applicati t the Piss itegral is als made.. Prelimiaries I this paper we shall use the tati T = Iz : Iz I = }, D = Iz : Iz I< }, ad E = Iz : I z I > } fr the uit circle, the pe uit disc ad the exterir f the uit circle. Let Il be be a fiite Brel measure [-t,t]. I rder t estimate the itegral, IIIIf} = It f(ei8)dll(8) = I f(z)dll(z) = If(z)dll(Z) -t T (we have take the freedm t write the previus itegral i differet fs but havig i mid that itegrati will always be ver the uit circle i e f r ather), the s-called Szeg6 quadrature fulas were itrduced i []. (See, als [] fr a differet apprach). Such quadratures are f the f, I If} = \' A~)f(x~)), x~):;a!:x~), X~)ET ad A~» 0, j =,..., (.),L =l s that, II, t'" If} = IIf} fr all fe A -(-l).(-l).(fr every pair (p,q) f

2 itegers, p ~ q, A f the frm, p,q will dete the liear space f all Lauret plymials L c,z}, C, E [;, = ad A the space f all Lauret plymials ([3], [4]). It is kw that such quadrature frmulas (.) are f great iterest t slve the trigmetric mmet prblem r equivaletly the Schur cefficiet prblem (see, []). O the ther had, Waadeladl [5] recetly studied such quadratures fr the / Piss itegral, that is, whe the measure /-I. is give by, r mre geerally, I - r d/-l.(e)= t ' r E -rcse + r (0,) d/-l.(e)= ~ ~, ~ red, z = eie (.) Iz-rl Observe that by takig r=0, we have the rmalized Lebesgue measure da(e)= de/. Szeg quadratures fr such situati were als studied by Camach ad Gzalez-Vera i [6]. Fially, i [7] aspects ccerig... wjtir ( errr ad cvergece were aalyzed. I this paper, frmulas (.) will be agai csidered, but istead f Lauret plymilas, mre geeral ratial fuctis with prescribed ples t T will be used, givig lse t the Ratial Szeg frmulas which were 0\ earlier itrduced by urselves i [8] ad [9] ad wher,fue s-called Ratial Szeg fuctis play a fudametal rle. Fr cmpleteess, let {aj} 7 c D be a give sequece ad csider fr = 0,,.. the ested spaces ~ f ratial fuctis f degree at mst which are spaed by the basis f Partial Blaschke prducts {B } where B =, B = k 0 0 ~ B-l fr =,,... ad the Blaschke factrs are defmed as a a-z ~/z) = ~ la I ---= l-az By cveti, we set a / I a I = - fr a = O. Smetimes, we shall als write - - B (z) = W (z)/tt(z); = (-) a Ia.l, w = (z-a) ad Tt =(-a,z)., = Nte that if all the a. are equal t zer, the spaces ~ cllapse t the space

3 f plymials f degree. We als itrduce the fllwig trasfrmati f*(z) = f(i/i), which allws t defie fr f E $ the superstar cjugate as 0 * f (z) = B (z)f 0 0* Let w the sequece (4) : = O,l,... } be btaied by rthrmalizati f the sequece (B : = O,l,... ) with respect t the ier prduct iduced by the measure /., amely (z) 'S --:ib <f,g>/. = t -tf(e )g(e )d/.(s) These fuctis are uiquely determied by the requiremet that the leadig cefficiet k i 4>(z) = ~ k.b.(z) j=l * is psitive. We the have k = 4> (a). 0 0 Fially, i rder t summarize the mai result give i [9] fr the Ratial Szeg frmulas, let us itrduce the fucti spaces f the frm fl =..t: + $ = ( P/r t ; PE }, p ad q beig egative itegers p.q p* q p q p+q Observe that..t: 0* = spa (, B *,...,B 0* } = spa (, lib,..., lib}. 0 Therefre, fl = spa (lib, lib,...,b,, B,..., B }. p.q P p-l q = -t). Whe all the a. are equal t zer, the B (z) = l ad e has (fl k fl = spa ( l: k = -p, -p+l,...,q) = A. Furthermre, fr WET, set p,q * X (z,w) = 4>(z) + w4>(z), s that the fllwig hlds (see, [9]), 0 0 Therem i) X (z,w) has simple zers which lie the uit circle. ii) Let x,..., x -p,q be the zers f f (z,w). The, there exist psitive 0 0 umbers A,..., A such that the frmula I (f} = \' A.f(x.) is exact, that IS, 0 0,L = I (f} = III(f} /""" fr all f E fl I this case, fl is said t be a maximal dmai f validity Mrever, it was als prved i [9] that the ly quadrature frmulas with such a maximal dmai f validity are just thse es give i Therem, where the weights A.'s are give by 3

4 A = L,(z)dl(z) (.3) L,(z) E $ = f7l defmed by the iterplati cditis L,(x,) = I 0,- (Actually, a mre geeral iterplatig fucti space f7l, p ad q beig p,q egative itegers such that p+q = -I, ca be csidered, s that the resultig quadrature frmula des t deped p ad q. See [9]).. A alterative apprach I this secti we shall give a alterative apprach t get the abve quadrature frmulas usig Hermite iterplati i the space f7l (cmpare -I,-I with the apprach give by Markv fr the classical Gauss frmulas [0] ad with the e give i [7] fr the Szeg frmulas the uit circle i the \ plymial case). Writig A = {a,} 00 ad A= {I / a,: a,e A}, it is easily I \ see that f7l represets a Chebyshev system ay set X c a::- (AuA), s that.m \ give the distict des {x,: j=.,...,} c a:: - (AuA), there exists a uique fucti Q E f7l hldig, ~ t.w\r -I,-I Q(x,) = f(x,), i =,,...,; Q'(x,) = f'(x,), i =,,...,- These are - cstraits, which crrespds t the dimesi f f7l -I,-I I rder t determie such Hermite ratial iterplat, we ca write -I where cditis Detig H j,o ad Q(z) = L H. (z)f(x,) + L H. I(Z)f' (x,), I, 'I, = = H belg t f7l ad satisfy j,l -I,-I H. (x,) = 0.., ~ i,j ~ ; H: (x,) = 0, ~i~, ~j~- (.),0,0 H. (x,) = 0, ~~-, ~j~; H: (x,) = 0.., I ~ i,j ~ - (.),, N (z) = (z-x,), we set fr i =,,..., I L~(z) = - E $ IN (Z)It (x.) Z-Xi t -(Z)[N'(X,)] - which satisfies L~(x,) = 0, ~ i '* j ~. Defie L (z) = L\z) ad,o Z - x L. (z) = L~(z) + A, i L~(z) E $,fr i =,,...,-,0 Z - X -I,-I the iterplati with A, E a:: chse such that V (x,) =. Furthermre we set,0 z-x L. (Z) = (x, -x ) _i L~(z) E $, i =,,...,-, z-x - 4

5 It is simple t check that ad L. (x,) = 0.., Ig,j~; L: ~(x,)= 0, I:::;j~-I, I:::;i~ I. I,] 'b L, (x,) = 0, I:::;i~-I, I:::;j~; L: (x,) = 0.., I:::;i,j~-I (.4),, I Frm (.3) ad (.4), we ca set fr i = I,,...,-I -(_a.,z) CO (x) H (z) = TI -l [L (z) + j.ll (z)] E a i,o TI-(-a.x. - ) c -l (Z) i,o i i, -l,-l I where j.l, is uiquely determied by the cditi H: (x.) = 0 ad ),0 TI-( I-a,z -) CO () x H (z) = -l L (z) E a. I.0 TI-(I -.X. a. ) CO -l (z),o -l,-l NV I satisfies the requiremets (.). Similarly, the fuctis TI-(_a.,z) c (x) H. (z) = -l L (z) E a. TI-(-a,x, - ) c -l (z) i,},\ ''\ -l,-l I " satisfy the cditis (.)..- (.3) (.5) Oce the iterplatig fucti Q(z) E a-l,-l has bee characterized, e gets /\ -l I {f} = Q(z)dj.l(z) = L A~)f(x,)+ L ~~)f'(x,) j = j= where A~)= H.,O(z)dj.l ad. ~~) = H.,(z)dj.l. Therefre /\ I If) ca (.6) csidered as a quadrature frmula which makes use f values f the fucti f /\ ad its derivative. Clearly I {f} has a dmai f validity a. Hwever, -l,-l a adequate chice f the des {x,} ca greatly simplify frmulas (.6). * Ideed, whe {x,} are the zers f (j> + w(j> (I w I = ), e has Therem /\ The quadrature frmula I {f} give by (.6) reduces t a -pit Ratial Szeg (r a R-Szeg, fr shrt) fm:ula whe the des are the zerc~f (j> + w<j>( Iwi = ) """- ""'- Prf. We write X = (j>+ wf = N (z)i (z), N E ad N (x.) = 0, j = I,...,. Nte that N is t ecessarily mic. By the characterizati therem fr R-Szeg frmulas (Therem ), it suffices t shw that ~~)= 0, I fr i =,..., -l. But ~~) I = H,,(z)dj.l, where H,, is give by (.5). be 5

6 Hece, we have t prve IT-( -a,z - ) z-x, 00. -I (z) z-x L(z)dl j = 0, Frm the defmiti f L~, this itegral ca be writte as (up t a cstat factr) N (z) N (z) N (z t (z) (z- X - -I.)c (z) dl - t (z) -I (-& z) N (z) -; d (,)00 (7:) l = -I = X (z)h *(z)dl = <X,h>l (-x z)... (I-x z)(z-a) where h(z) = - E % ad h( a ) = O. Thus, h belgs (l-x,z)t (z) -I - t % (\ -t(a) (-t(a)= I f E ~ f(a) = 0 D. Nw, by the rthgality -I, L'"' ~ prperties fr X, it fllws that <X,h>l = 0.0 Remark Nte that the same result ca be btaied if the fllwig iterplati prblem is csidered. Fid Q. E q(, (i =,,...,) such that -I,-I Q.(X.) = f(x.), j =,,...,; Q:(x.) = f'(x.), ~j~ (i:r!:j) (.7) We ca cclude that a -pit R-Szeg frmula IIf} is give by IIf} = = Q,(z)dl, where Q. is the uique sluti t the iterplati prblem (.7). * ad Ix,} are the zers f X (z) = 4> + w4> (I wi = ). Certaily, the give apprach culd be useful i rder t give a expressi fr the errr Elf} = = Il If} - IIf} = Il If - Qj}' 3. A applicati t the Piss itegral We shall w characterize the -pit R-Szeg frmulas fr the measure l iduced by the Piss itegral kerel give by (.). I this sese, the first step is btaiig the rthrmal system 4>}, = 0,,... We kw 4>= 0 ad fr =,,..., 4> has t verify the cditis (i) 4> E % - %; -l <4>,4>>l = ad (ii) <4>,B k>l = 0, k = 0,,...,-. Frm (i) e fmds 4> (z)(-lrl ) <4>,B >l = 4>(z) BSZ) dl = da k k B (z)( z-r)( - fz) k 6

7 = ~ <I>_ (_z_) dz tl T B k(z)( z-r)(-rz) The demiatr B k(z)(z-r)(i-rz) vaishes at z = ex..,i =,,...,k ad z = r, which are all iside the uit disc D ad the ther zer!r is i E. If this itegral has t vaish, <I>shuld be zer at z = ex..,i =,,...,k ad z = r. This gives i cmbiati with cditi (i) that <I>shuld have the fllwig frm (z-r)b <I>(z) = k z- ~ E % ad k == Fr :5:; k :5:; -l, it is easily see that k (- r ) B (z) (z) <<I>,B > = /k ~ = (k == 0) k I..l T z - ex. I -rz - (3.) where B = B / B, ad thus B (ex.) = 0. The cstat k is determied by /k k /k = <<I>,<> = Ik.!.::.l!..L s that k = I..l I-lex. l-lrl The leadig cefficiet is fud as fllws * l-rz S that <I>(z) = k --ad thus I-a z cefficiet has t be psitive, [ - ex. ] / <I>(z)=k-rz (3.) * -ex. - z B(Z) I-a r <I> * (ex.) = k. Sice the leadig I-lex. We ca check sme particular k = exp(iy), y = -arg( - a r) [I-lex. - r ]/ cases. () r = 0, which delivers the Lebesgue measure. The zb (z) y =-arg(l-ar)=-arg(i)=oad<l>(z)=vl-lex. z-ex. This crrespds with the result i []. Whe all the ex.,are equal t zer, we recver the well kw result that <I>(z) = z. () r = ex.. Oe the has Y = -arg( - I ex.i ) = ad <I>(z) = B (z) (3) r == 0, ex. = 0, k =,,... The k 7

8 (z-r)b (z) 4>(z) = 0 _ = (z_r)z-l = ZO + rz- Zr, " This was btaied by Waadeladt'i a recet paper [5].,/ \ * The equati X (z) = 4>(z) 0 + w4>(z) 0 which prvides the des takes the f X (z) = k [(z-r)b (z)/(z-ex)] + ~K ( - rz)/( - a z) = r equivaletly \ [(z-r)b (z)/(z-ex )] + w (- rz)/( - a 0z) = 0, with w = w K 0/k0 E T. Usig B (z) = 0 0(z)/w 0(z), we get X (z) = [(z-r)tr (z) + w (z)(-rz)]l (z) = N (z)/ (z), N E (3.3) The des x. satisfy N (x) = O. Whe r = 0, e btais 0 zb (z) = I ex Iw I a (3.4) Nte that whe ex. = 0, this reduces t ZO = -wad the des x. are uifrmly distributed T, see [6]. Let us suppse that =, ex = ex = /, the (3.4) gives z - (+w)z + w = 0 ad the iterplati des x, x will be: Fr w = -: x = x =- ' Fr w = : x = (+v3i)/, x = (-v3i)/ Fr w = -i: x = [(I+v7) - (-v'7)i]/4, x = [(-v7) - (+v7)i]/4 Fr w = i: x = [(-v7) + (+v7)i]/4, x = [(+v7) + (-v7)i]/4 Fr the weights, e has i the geeral case X(z) - ex 0z _ I r I A. = IL.(z)dll(z) = m I ~ dz tl T Z-Xj I-a x. X'(x.)(z-r)(l-rz) 0 = _X_( _r) _-_a_r X'(X} r-x. - v.. iv x. where X (z) = N (z)/ (z) is give by (3.3). Sice X (r) = w(-lri)/(-a r), we fid _ w(-l j ). _ A , -,,..., X'(x.) (r-x.)(i-a x.) O O 8

9 O the ther had, sice X (x.) = 0, we get X'(x.) = N'(x.)It (x.) where N (z) = (z-r)ll (0 (z) + wt (z)(-rz) (3.5) - - Use N (x.) = 0, t fmd frm (3.5) that t (x.) = - (I-a x.) (x,-r)ll (0 which implies (x.)iw(l-rx.) - (3.6) Nw use k a, I t~(x.) = - tk(x.) L _ j = - a,x, I t btai frm (3.5) ad (3.6) ad (O'(x.) = (Ok(x.) L k x,_a, k j=l A = (-lri) / (I-Ix.) + (x,-r) L x -a - [ -[ k= j k s that fr j=i,,..., k A = (-lrl ) / (3.7) [ - Irl + IX,-rl - L l-ia I ] k=l I\-akl where the psitivity f the weights is clearly expsed. Agai, whe all the a, I are equal t zer e gets A- -~ '-, - ~, - l-irl + (-l)lx,-rl,,..., If we set x, = exp(i8.) ad r E (0,), the Therefre Ix. - rl = lexp(i8.) - rl = + r -rcs8, A_I - L!:. - (3 8), - ' -,,.,. l-irl + (-)( + r -rcs8,) The same expressi was btaied by Waadelad([5] fr the plymial case / If r = 0 (Lebesgue measure), it fllws frm (3.8) that A = - fr j =,...,. That meas that the crrespdig Szeg frmula fr the plymial case has all its des equally spaced the uit circle ad all its weights are equal t - (Cmpare with the results give i [6]). I the special case r = a, (3.8) yields 9

10 A = (- a ) / I [ x. _ a \' l-ia k I ] k~ IX j _ak (3.9) Sme ccludig example: assume that a k = a fr k =,,..., ad take als r = a. The * * <I>(z) = B (z), <I>(z) = ad X = <I>+ w<l>= B (z) + w Because w B = t;, with t;,(z) = (a/ I a I )( a - z)/( - az), the des x. are slutis f --_- = - wlal / a = wet [a --azz ] - \ Settig r. = (~)I/, j =,,...,, we get x. = (a - r)!( - (ir). As fr the weights, e gets frm (3.9) Aj = +, j =,,...,. / 4. Cvergece Let I If}, =,,..., be a sequece f R-Szeg frmulas (take it accut that fr each I If) represets a e-parameter family f quadrature frmulas), that is // I {f} = \' A~)f(x~\ x~);;/:x~)(i;;/:j),x~)et ad A~)> 0, j=i,,...,... I = where the weights A~)are give by (.3). I this secti, we shall study the cvergece f such quadratures fr ay fucti f i the class R~(T) f the itegrable fuctis T with respect t the measure ~. Fr this purpse a first result we shall eed is Lemma Let us defie fl = fl = $ + -t: ad fl = fl, the fl is dese i the ~ ~ ~ class C(T) f ctiuus fuctis T, iff L (- a I) = ~. Prf. This is a direct csequece f the "clsure criteri" discussed i Addedum A. f [ p. 44].0 Wew are w ready t prve a first result assertig the cvergece i the class C(T). Ideed, Therem 3 e has Let f be a ctiuus fucti T, the lim I If} = III If} = f(z)d~(z) -7~ t'" 0

11 Prf. Let e be a give real psitive umber. Take e' = e / l where l 0 = dl(z) By Lemma, there exists RN E ~ such that Ifez) - RN(z) I < E', V Z E T Assume > N ad write I (f) = L A~)f(x~»), the Hece, =. I,. I"'" If} - I (f) = III I"'" {f-r N } + I {RN-f}. I I {f} - I {f} I ~ I f(z)-r (z) I dl(z) + \ A~)I f(x~»)- R (x~»)i ~ l e' = e.l N.f.. N 0 = (Recall that A~» 0, j =,,... ad j = L A~)= l)' Assume w f a cmplex fucti defied the uit circle T which is itegrable with respect t the measure l. We ca write fez) = f (8) + if (8), z = exp(i8) (4.) where f.(8), j =, are real valued fuctis defmed the iterval [-,].. Let us first suppse f is a ctiuus fucti, r equivaletly f. (j =,) are ctiuus fuctis. Frm Therem 3, we ca write -7OO lim I {f.} = III I""' {f.}, j =, (4.) Nw parallellig rather clsely the argumets give i [3 pp.7-9] it ca be see that (4.) is als valid fr itegrable fuctis because f the fact that ay sequece f itegra~~4 rules with psitive weights which cverges fr all ctiuus fuctis/ cvergves fr all itegrable fuctis with respect t a fmite Brel measure l [-,]. Let w \r E Rl(T~ Frm (4.), e ca write I'I(f} I"'" = III{f ""' } + illi{f ""' } ad I (f} = I l{f } + ii {f } (4.3) Thus, by (4.) ad (4.3) the ext crllay immediately Crllary Uder the same hypthesis as Therem, e has fllws Remark -7OO lim IIf} = I" I"'" (f) fr ay fer l(t) Whe all the a.are equal t zer,the the Blaschke cditi \(- f.. a )=00 hlds trivally, ad the cvergece f the Szeg quadrature frmulas itrduced i [] is guarateed i the class Rl(T) (see als [7] fr a direct

12 prf). O the ther had, the special case whe the sequece {ex, } csists f a fiite umber p f pits cyclically repeated (see, e.g. [4]) the Blaschke cditi als hlds ad therefre the cvergece f the crrespdig quadrature prcess is assured. Refereces [] W.B. es, O. Njastad ad W.. Thr. Mmet thery, rthgal plymials ad ctiued fractis assciated with the uit circle. Bull Ld Math. Sc., :3-5 (989) [] G. Freud. Orthgal plymials, Pergam Press (97) [3] W.B. es, O. Njastad ad W.. Thr. Orthgal Lauret plymials ad the strg Hamburguer mmet prblem.. Math. Aal. Appl. 98, (984) [4] O. Njastad ad W.. Thr. The thery f sequeces f rthgal L-plymials. Det Kgelige Nrske Videskabers Selskab, : 54-9 (983) [5] H. Waadelad. A Seg quadrature frmula fr the Piss itegral. I Prceedigs IMACS'9 (Dubli) -3 (99) [6] P. Gzalez-Vera ad M. Camach. A te para-rthgality ad birthgality. Det Kgelige Nrske Videskabers Selskab. T appear [7] A. Bultheel, P. Gzalez-Vera, E. Hedrikse ad O. Njastad. Orthgality ad quadrature the uit circle. IMACS aals cmputig applied mathematics. Vl. 9, 05-0 (99) [8] A. Bultheel, P. Guzalez-Vera, E. Hedrikse ad O. Njastad. A Seg thery fr ratial fuctis. Techical Reprt TW-3, K.U. Leuve, Dept. f Cmputer Sciece (990) [9] A. Bultheel, P. Gzalez-Vera, E. Hedrikse ad O. Njastad. Orthgal ratial fuctis ad quadrature the uit circle. Numerical Algrithms. T appear. [0] V.I. Krylv. Apprximate calculatis f itegrals. Traslated by A. H. Strud. Macmilla, New Yrk (96) [] M.M. Djrbashia. A survey the thery f rthgal systems ad sme pe prblems. I P. Nevai, editr, Orthgal plymials: Thery ad practise, vl. 94 f series C: Mathematical ad Physical Scieces, 35-46, NATO-AS!, Kluwer Academic Publishers (990) [] N.. Achieser. Thery f apprximati. Frederick Ugar Publ. C.,New Yrk (956) [3] P.. Davis ad P. Rabiwitz. Methds f umerical itegrati. Secd ad

13 Editi. Academic Press (984) [4] A. Bultheel, P. Gzalez-Vera, E. Hedrikse ad O. Njastad. Orthgal ratial fuctis similar t Szeg plymials. IMACS aals cmputig ad applied mathematics. Vl. 9, (99) 3