Asymptotic Zero Distribution of Laurent-Type Rational Functions*

Transcription

1 joural of approxmato theory 89, 5888 (1997) artcle o. AT Asymptotc Zero Dstrbuto of Lauret-Type Ratoal Fuctos* N. Papamchael Departmet of Mathematcs ad Statstcs, Uversty of Cyprus, P.O. Box 537, Ncosa, Cyprus I. E. Prtsker - Departmet of Mathematcs, Uversty of South Florda, 4202 East Fowler Aveue, Tampa, Florda 33620, U.S.A. ad E. B. Saff Isttute for Costructve Mathematcs, Departmet of Mathematcs, Uversty of South Florda, 4202 East Fowler Aveue, Tampa, Florda 33620, U.S.A. Commucated by V. Totk Receved May 15, 1995; accepted February 27, 1996 dedcated to a. w. goodma o the occaso of hs 80th brthday We study covergece ad asymptotc zero dstrbuto of sequeces of ratoal fuctos wth fxed locato of poles that approxmate a aalytc fucto a multply coected doma. Although the study of zero dstrbutos of polyomals has a log hstory, aalogous results for trucatos of Lauret seres have bee obtaed oly recetly by Edre (Mchga Math. J. 29 (1982), 4357). We obta extesos of Edre's results for more geeral sequeces of Lauret-type ratoal fuctos. It turs out that the lmtg measure descrbg zero dstrbutos s a lear covex combato of the harmoc measures at the poles of ratoal fuctos, whch arses as the soluto to a mmum weghted eergy problem for a specal weght. Applcatos of these results clude the asymptotc zero dstrbuto of the best approxmats to aalytc fuctos multply coected domas, FaberLauret polyomals, LauretPade approxmats, trgoometrc polyomals, etc Academc Press * Research supported, part, by a Uversty of Cyprus research grat. - Research doe partal fulfllmet of a Ph.D. degree at the Uversty of South Florda. Research supported, part, by NSF Grat DMS Copyrght 1997 by Academc Press All rghts of reproducto ay form reserved. 58

2 ASYMPTOTIC ZERO DISTRIBUTION INTRODUCTION The lmtg behavor of zeros of sequeces of polyomals s a classcal subject that cotues to receve much atteto (see, e.g. [8, 11, 16]) because of ts applcatos fucto theory, umercal aalyss ad approxmato theory. Two of the fudametal results of the subject are the theorems of Jetzsch [9] ad Szego [17] o the zero dstrbuto of partal sums of a power seres. It s rather surprsg that although the study of zero dstrbutos of power seres sectos has a log hstory, aalogous results for trucatos of Lauret seres has bee vestgated oly relatvely recetly by Edre [5] who, partcular, proved the followg. Theorem A. Let A=[z: r< z <R] be the exact aulus of covergece for the Lauret seres f(z)= : k=& a k z k, where 0<r<1<R<. Let 4 1 =[m ] ad 4 =1 2=[ ] =1 be two sequeces of postve tegers tedg to, such that ad lm a &m 1m =r (1.1) Cosder all the zeros of the trucato lm a 1 = 1 R. (1.2) T, m (z)= : a k z k k=&m that le the agle % 1 arg z<% 2 (% 1 <% 2 % 1 +2?). The, as, there are (1+o(1)) % 2&% 1 m 2? of those zeros that have modulus <1 ad of them have modulus >1. (1+o(1)) % 2&% 1 2?

3 60 PAPAMICHAEL, PRITSKER, AND SAFF Moreover, for ay gve =>0, there are outsde the aulus Re &= z Re = at most o( ) zeros of T, m (z) of modulus >1 ad there are at most o(m ) zeros of modulus <1 outsde re &= z re =. Ths theorem s completely aalogous to Szego 's result for power seres [17]. It says that all but a eglgble proporto of the zeros of the T, m (z)'s accumulate o the crcles z =r ad z =R, ad that the argumets of these zeros that are close to oe of the crcles, are equdstrbuted the sese of Weyl. I ths paper we preset varous geeralzatos of Edre's result to the zero dstrbuto of certa sequeces of ratoal fuctos havg fxed locato of poles ad covergg locally uformly ftely-coected domas to a aalytc fucto f(0). As applcatos we descrbe the lmtg zero dstrbuto of Lauret-type approxmats. The ma tools of our vestgato are the theores of weghted potetals ad weghted polyomal zero dstrbutos developed [15] ad [11]. The paper s orgazed as follows. I Secto 2 we state ad dscuss our ma results. Secto 3 s devoted to applcatos such as zero dstrbutos of trgoometrc approxmats ad LauretPade approxmats. I Secto 4 we dscuss the theory assocated wth a weghted potetal problem ad, fally, Secto 5 we make use of ths theory order to prove the theorems of Sectos 2 ad MAIN RESULTS Let K be a bouded cotuum (ot a sgle pot) whose complemet cossts of a fte umber of domas. We deote by C the exteded complex plae, by [G l ] the set of bouded compoets of C "K ad by 0 the ubouded compoet. (It s clear that the G l ad 0 are smply coected domas ad that C "K=( G l)_0.) Fally, for each, 2,..., we assocate a arbtrary but fxed pot a l # G l. By the Rema mappg theorem there exsts a uque coformal mappg, l : G l D of G l oto the ope ut dsk D, ormalzed by the codtos, l (a l )=0 ad,$ l (a l )>0. The quatty R l :=1,$ l (a l ) s called the teror coformal radus of G l wth respect to a l. Smlarly, there exsts a coformal mappg 8: 0 D$ of the ubouded compoet 0 oto the exteror of the ut crcle D$=[z: z >1] ormalzed by 8()= ad lm z 8(z)z=1C, where C :=cap K s the logarthmc capacty (trasfte dameter) of K (cf. [19]). We shall keep the same otato, l (z) for the exteso of the coformal mappg, l : G l D oto the boudary G l the sese of Carathe odory's theory of prme eds [6]. Thus, for each, 2,...,, the mappg, l

4 ASYMPTOTIC ZERO DISTRIBUTION 61 s defed o the closure G l,.e., l : G l D. Smlarly, for the exteror mappg we take 8: 0 D$. For our study of lmtg dstrbutos we shall utlze the measures ad + e (B) := (,B,0) (2.1) + l (B) := (a l,b,g l ),,...,, (2.2) for ay Borel set B/C, where (, B, 0) s the harmoc measure of the set B at the pot wth respect to 0, ad (a l, B, G l ) s the harmoc measure of B at the pot a l wth respect to the doma G l (cf. [13, 19]). We remark that + e s the same as the equlbrum measure for K the sese of logarthmc potetal theory. Aother coveet way to descrbe the above harmoc measures s to terpret them as premages of the ormalzed arclegth measure o the ut crcle [z: z =1] uder the correspodg coformal mappgs. That s, ad (, B, 0)=m(8(B&0)) (2.3) (a l, B, G l )=m(, l (B&G l )),,...,, (2.4) where dm=d%2? o [z: z =1]. We recall that + e ad + l,,...,, are compactly supported ut Borel measures,.e. &+ e &=&+ l &=1,,...,, ad supp + e =0, supp + l =G l [13]. The ma goal of ths paper s to study the lmtg zero dstrbuto of sequeces of ratoal fuctos that are detfed by a mult-dex N=(k, m 1, m 2,..., m ) ad have the form: k R N (z)= : j=0 t N z j j + : m l : j=1 s N l, j (z&a l) &j. (2.5) I other words, R N (z) salauret-type ratoal fucto whose poles are located at the fxed pots a l # G l,,...,, ad at # 0. Ift N k ad s N l, m l,,...,, are ozero, the the total umber of poles of R N (z) the exteded complex plae C s equal to N =k+ : m l, (2.6)

5 62 PAPAMICHAEL, PRITSKER, AND SAFF whch s the orm of the mult-dex N. We also ote that, ths case, R N (z)= tnp k N(z) > (z&a l), (2.7) m l where P N (z) s a moc polyomal of degree N whose zeros cocde wth those of R N (z). Next we troduce the ormalzed coutg measure the zeros of R N (z): & N := 1 N : P N (z j )=0 $ zj, (2.8) where $ z s the ut pot mass at z ad where all zeros are couted accordg to ther multplctes. The results of ths paper o zero dstrbutos are all stated terms of the weak* covergece of measures. We say that a sequece of Borel measures [+ ] =1 coverges to the measure +, as, the weak* topology f lm fd+ = fd+ for ay cotuous fucto f o C havg compact support. Throughout the paper we assume that k=k(), m 1 =m 1 (),..., m =m (), N=N(), for some creasg sequece 4 of tegers, ad that k(), m l (),,...,, as,#4. Furthermore, we assume that the followg lmts exst: m l lm N N = lm # 4 m l () N() =: : l,...,. (2.9) Ths ormalzato meas that we deal wth so-called ``ray sequeces'' of ratoal fuctos. Clearly, ad : l 0,,...,, lm # 4 k() N() =1& : : l, : : l 1.

6 ASYMPTOTIC ZERO DISTRIBUTION 63 Before statg our results ther full geeralty, t s coveet to cosder the specal case where K s the closure of a aular rego A bouded by two Jorda curves, wth oe curve teror to the other, so that C "K has oly oe bouded compoet whch we deote by G 1. Assume that a 1 =0 # G 1 ad cosder the weak* lmt of the ormalzed coutg measures & (k, m) the zeros of Lauret-type ratoal fuctos of the form k R (k, m) (z)= : j=0 m j z j + : j=1 t k, m s k, m j z &j, (2.10) where k=k(), m=m(). Also, accordace wth (2.9), we assume the exstece of the lmt lm k+m m k+m = lm # 4 m() =: : (2.11) k()+m() ad that k(), m(). The followg theorem s a specal case of Theorem 2.2 below. Theorem 2.1. Suppose that the sequece [R (k, m) (z)] # 4 defed by (2.10) coverges locally uformly a bouded aular rego A to a aalytc fucto that s ot detcally zero. Assume further that (2.11) holds ad hat lm # 4 lm # 4 t k, m k 1k = 1 C, (2.12) s k, m m 1m =R 1, (2.13) where C :=cap A ad R 1 s the er coformal radus of G 1 wth respect to the org. The & (k, m) * + w :=(1&:) + e +:+ 1, as, # 4, (2.14) where, from (2.1), (2.2), + e = (, },0) ad + 1 = (0, }, G 1 ). We remark that the lmt measure + w defed (2.14) s, fact, the equlbrum measure for a weghted potetal problem correspodg to a specal weght w (see Secto 4).

7 64 PAPAMICHAEL, PRITSKER, AND SAFF The ``ot detcally zero fucto'' assumpto s essetal Theorem 2.1. To show ths, we cosder the sequece R (, ) =z & 1 2 z, =1, 2,..., coverget to f#0 locally uformly A :=[z:12< z <1] as. Ths sequece satsfes codtos (2.12) ad (2.13) as well as (2.11) wth :=12. However, R (, ) has all zeros o [z: z =1-2]. I cotrast, f we cosder the modfed sequece [R (, ) +1] =1, the all the codtos of Theorem 2.1 hold ad ths sequece of ratoals has oe half of ts zeros accumulatg o z =1 ad the other half of ts zeros accumulatg o z =12 agreemet wth Theorem 2.1. Some examples of Lauret-type ratoal approxmats satsfyg the codtos of Theorem 2.1 are gve Secto 3. We also meto that t s possble to prove a result o the zero dstrbuto of the partal sums of Faber-Lauret seres to a fucto aalytc a doubly coected doma [18] va a applcato of Theorem 2.1. For the detals see [14]. As a mmedate corollary of Theorem 2.1 we obta Edre's result of Theorem A. Ideed, f A=[z:0<r< z <R<] s the exact aulus of covergece for the Lauret seres the the trucatos f(z)= : k=& a k z k, T, m (z)= : a k z k k=&m coverge locally uformly A to f(z)0, as. Also, we assume wthout loss of geeralty that the lmt : := lm m +m exsts (otherwse we ca apply our results to a subsequece, for whch ths lmt does exst). The codtos (2.12) ad (2.13) of Theorem 2.1 are satsfed vew of (1.1) ad (1.2). Thus, Theorem 2.1 gves & N * (1&:) + e +:+ 1 (2.15) the weak* sese as N = +m,oras. Sce ths case + e cocdes wth the ormalzed arclegth measure o [z: z =R] ad + 1 cocdes wth the ormalzed arclegth measure o [z: z =r], the result

8 ASYMPTOTIC ZERO DISTRIBUTION 65 (2.15) gves a more compact maer (usg weak* covergece) the coclusos of Theorem 1.1. Theorem 2.1 s a specal case of the followg more geeral result. Theorem 2.2. Let K be the closure of a multply coected Jorda doma,.e. 0 ad G l,,...,, are Jorda curves wthout commo pots. Suppose that the sequece [R N (z)] # 4 (cf. (2.5)) coverges locally uformly the teror K% of K to f(z)(0) ad (2.9) holds. If () lm # 4 t N k 1k = 1 C ad () lm # 4 s N l, m l 1m l =Rl,,...,, the the ormalzed zero coutg measures & N for R N satsfy () & N * + w the weak* sese as, # 4, where + w := \ 1& : : l+ + e+ : : l + l. (2.16) Coversely, suppose that : l >0,,...,, wth : l{1. If each a l has some eghborhood free of zeros of [R N (z)] # 4, the () mples () ad (). Our ext result s a ``oe-sded'' verso of Theorem 2.2. To state t we recall that the logarthmc potetal of a compactly supported Borel measure + s defed by U + (z) := log 1 z&t d+(t). Suppose that G s a ope bouded set ad supp +/G. The a measure +^ supported o G s called the balayage of + to the boudary of G f &+^ &=&+&, the potetal U +^ s bouded o G ad U +^ (z)=u + (z) q.e. o G (see Chapter IV of [10]). By q.e. (quas-everywhere) we mea that the above equalty holds for all z #G wth the possble excepto of a set of zero logarthmc capacty.

9 66 PAPAMICHAEL, PRITSKER, AND SAFF Theorem 2.3. If Theorem 2.2 we assume that codto () holds (but ot ecessarly ()), the for ay weak* lmt & of the measures & N, as, # 4, we have ad U & (z)=u + w (z), z G :=. & C"G =+ w C"G = \1& : where + w s gve by (2.16). Furthermore, G l (2.17) : l+ + e, (2.18) &^ N * + w, as, # 4, (2.19) where &^ N deotes the measure obtaed by balayage of the part of & N supported G := G l to G= G l.(here the part of & N supported outsde of G s kept fxed). Remark 2.4. Theorem 1 [5] follows from Theorem 2.2 appled to the partal sums of Lauret seres. Remark 2.5. I Theorem 2.3 we actually requre oly that k() as, # 4, so that m l (),,...,, may be bouded. Thus, f we cosder a sequece of polyomals [P N (z)] that coverges locally uformly K% to a ozero aalytc fucto such that () s satsfed, the we have : l =0,,...,, ad & N (B) 0, as, for ay compact B/C "0. The last statemet follows from Hurwtz's theorem ad the maxmum modulus prcple, accordg to whch [P N (z)] coverges locally uformly C"0. Thus, we obta from (2.19) of Theorem 2.3 that & N * + e as, # 4. It follows that ths case Theorem 2.3 reduces to certa results of Blatt, Saff ad Smka [2] o the zero dstrbuto of polyomals. It s of some terest to cosder the aalog of Theorem 2.2 for a set K wth empty teror. I ths case the codto of covergece for the sequece of ratoal fuctos ca be replaced by a weaker oe. Namely, we have the followg. Theorem 2.6. Let K be a arbtrary bouded cotuum (ot a sgle pot) wth empty teror ad suppose that C "K cossts of a fte umber

10 ASYMPTOTIC ZERO DISTRIBUTION 67 of compoets 0, G 1,..., G. Assume that for the ratoal fuctos R N of (2.5) we have (2.9) ad lm # 4 The codtos () ad () of Theorem 2.2 mply (). &R N & 1 N K =1. (2.20) By the orm (2.20) we mea the uform orm,.e., & f & K := sup f(z). z # K The last theorem ca be appled, for example, case whe K s a sgle Jorda curve. Ths provdes a exteso of Edre's result for Lauret seres to the stuato whe the aulus of covergece A degeerates to the ut crcle,.e. wth K=T=[z: z =1]. We preset ths result Theorem 3.1 We remark that t s possble to further relax the geometrc codtos mposed o the set K the theorems of ths secto. For example, oe could allow every compoet of C "K to be a ftely coected doma, where we assume, as before, that the umber of compoets s fte. The aalogues of our result ths case are straghtforward. Aother geeralzato s to allow ftely may fxed poles every compoet of C "K stead of just oe. 3. APPLICATIONS 3.1. Zeros of Fourer Sectos Cosder the Fourer expaso f(z)= : k=& a k e k% = : k=& a k z k (3.1) for a fucto f # L 2 (T), where T=[z: z =1]. Equalty (3.1) s wth respect to L 2 orm. Theorem 3.1. Let f # L 2 (T). Suppose that for the represetato (3.1), the fuctos g(z)= : k=0 a k z k (3.2)

11 68 PAPAMICHAEL, PRITSKER, AND SAFF (aalytc z <1) ad &1 h(z)= : k=& a k z k (3.3) (aalytc z >1) caot be aalytcally cotued to a ope set cotag the ut crcle. The there exst two sequeces of postve tegers [ ] =1 ad [m ] such that for the ormalzed coutg measure & =1 +m the zeros of the trucato we have T, m (z)= : a k z k, (3.4) k=&m & +m * as, where s the ormalzed arclegth measure o T,.e. d =d%2?. We gve a sketch of proof here. Sce both fuctos g(z) ad h(z) have sgulartes o T, there exst two subsequeces of postve tegers [ ] =1 ad [m ] =1 such that lm a 1 =1 ad lm a &m 1m =1. (3.5) From the Ho lder equalty we obta &T, m & 2-2? &T, m & 2? -+m &T, m & 2, (3.6) where &T, m & =max z # T T, m (z). It follows that lm &T, m & 1( +m ) = lm &T, m & 1( +m ) 2 =1, (3.7) because lm &T, m & 2 =& f & 2 {0. Theorem 3.1 s ow a cosequece of Theorem 2.6 because the measures + e ad + 1 of Theorem 2.6 are gve ths case by + e =+ 1 = ; thus for ay [ ] ad [m =1 ] =1, satsfyg (3.5) such that we obta lm m +m =:, + w =(1&:) + e +:+ 1 =(1&:) +: =.

12 ASYMPTOTIC ZERO DISTRIBUTION Zeros of LauretPade Approxmats wth the Fxed Deomator Degree For the fucto f(z)= : c k z k k=& aalytc A=[z: r< z <R] we cosder ts addtve splttg where s aalytc [z: z <R] ad f(z)= f + (z)+ f & (z), z # A, (3.8) f + (z)= c : k=1 f & (z)= c : &1 k=& c k z k c k z k s aalytc [z: z >r]. Followg [7], we troduce LauretPade approxmats of order (m, ). It s eough to cosder the case m for our purposes. Ths smplfes the defto of LauretPade approxmat of type (m, ) to the sum of classcal Pade approxmats of type (m, ) to f + (z) ad f & (z) about 0 ad respectvely. Let r (z) + m, (z)=p+ m, q + m, (z) be the classcal Pade approxmat of type (m, ) to the fucto f + (z) about 0,.e. We use the otatos f + (z) q + m, (z)&p+ m, (z)=o(zm++1 ). p + m, (z)= : m j=0 for the Pade umerator of type (m, ) ad q + m, (z)= : j=0 +, ( j) a m, z j +, ( j) b m, z j +, (0) for the (ormalzed) Pade deomator wth b m, =1.

13 70 PAPAMICHAEL, PRITSKER, AND SAFF We cosder ext the Pade approxmat of type (m, ) tof & (z) about,.e. where ad r & m, (z)= p& m, (1z) q & m, (1z), p & m, \ 1 m z+ = : j=0 q & m, \ 1 z+ = : j=0 &, ( j) a m, \ 1 j z+ &, ( j) b m, \ 1 j z+ &, (0) wth b m, =1. For p & m, (1z) ad q & m, (1z) we have f & (z) q & m, \ 1 z+ & p& m, \ 1 z+ =O \ m z. The LauretPade approxmat of type (m, ) to f(z) s the defed by where r m, (z) :=r + m, (z)+r& m, (z)=p m, (z) q m, (z), p m, (z) :=p + m, (z)q& m, (1z)+p& (1z) m, q+ m, (z) (3.9) s a Lauret polyomal of degree at most m ad q m, (z) :=q + m, (z)q& m, (1z) s a Lauret polyomal of degree at most. We ote that +, (0) or, sce b +, (m) p m, (z)=a &, (0) m, =b m, =1, &, (0) m, b m, z m +}}}+a +, (m) p m, (z)=a m, z m +}}}+a &, (m) +, (0) m, b m, &, (m) m, 1 z m 1 z m. (3.10)

14 ASYMPTOTIC ZERO DISTRIBUTION 71 Sce the zeros of r m, (z) cocde wth those of the umerator p m, (z), as gve by (3.10), t suffces to study the lmtg behavor of zeros of ths Lauret polyomal or Lauret-type ratoal fucto. Theorem 2.1 gves the ecessary tools for such a study. Suppose that f(z) has a meromorphc cotuato to the aulus A :=[z: r < z <R ] such that ths cotuato (whch we stll deote by f(z)) has exactly poles, couted accordg to multplctes, [z: r < z r] ad poles [z: R z <R ]. Thus, the total umber of poles of f(z) A s 2, where s a fxed postve teger. We assume that A s the largest aulus wth the above propertes ad that the codtos R <, r >0 hold. Theorem 3.2. Uder the above assumptos of f(z), there exst two subsequeces 4 1 ad 4 2 of dces such that the ormalzed coutg measure zeros of LauretPade approxmats of type (m, ) to f(z),.e., & m = 1 2m : $ zj, r m, (z j )=0 has weak* lmts satsfyg where & m z (R +r )2 * + 1 as m, m # 4 1, (3.11) d+ 1 = 1 2 d% 2? o z =R, ad & m z (R +r )2 * + 2 as m, m # 4 2, (3.12) where d+ 2 = 1 2 d% 2? o z =r. Ths meas that the weak* lmt of restrctos of & m, m # 4 1, o [z: z (R +r )2],asm, cocdes wth oe half of the ormalzed Lebesgue measure o z =R. O the other had, whe m, m # 4 2, the weak* lmt of the restrctos of & m o [z: z (R +r )2] s equal to oe half of the ormalzed Lebesgue measure o z =r. The results of Theorem 3.2 are aalogues of results [4] for the zero dstrbuto of classcal Pade approxmats wth fxed deomator degree.

15 72 PAPAMICHAEL, PRITSKER, AND SAFF 4. POTENTIAL THEORETIC BACKGROUND I ths secto we dscuss the theory assocated wth some weghted potetal problems whch plays a crucal role the dervato of our results. I partcular, we show that the measure + w defed (2.16) arses a atural way as the extremal measure for a weghted eergy problem. Let K/C be a arbtrary compact set of postve capacty. For a Borel measure & wth compact supports S & /C"K such that &&&=:, 0:1, we cosder weght fucto w(z):= {eu& (z), 0, z # K, z K, (4.1) where U & s the logarthmc potetal of &. Note, that w(z) s cotuous as a fucto defed o K ad where w(z)=e &Q(z), z # K, (4.2) Q(z) :=&U & (z), z # K. (4.3) Let M(K) deote the class of all Borel measures + wth total mass &+&=1 supported o K, ad cosder the followg weghted eergy problem: ``For the weghted eergy tegral fd I w (+) := log 1 d+(z) d+(t), (4.4) z&t w(z) w(t) V w := f + # M(K) I w (+) (4.5) ad detfy the extremal measure + w # M(K) for whch the f (4.5) s attaed.'' We ote that w(z) s a cotuous admssble weght o K the sese of [12]. Therefore the followg s kow (cf. Theorem 3.1 of [12] ad Secto I.1 of [15]):

16 ASYMPTOTIC ZERO DISTRIBUTION 73 Theorem 4.1. For w as defed (4.1): (a) (b) V w s fte; _ a uque elemet + w # M(K) such that I w (+ w )=V w ad the logarthmc eergy of + w s fte,.e. &< log 1 z&t d+ w(z) d+ w (t)<+; (c) U + w (z)+q(z)f w, for q.e. z # K, where U + w (z) := log 1 z&t d+ w(t) ad F w :=V w & Q(t) d+ w(t); (d) U + w (z)+q(z)=f w, for q.e. z # supp + w. I fact, t s proved [15] that propertes (c) ad (d) characterze the extremal measure: If a compactly supported measure _ # M(K) has fte logarthmc eergy ad U _ (z)+q(z)=f q.e. o supp _, (4.6) U _ (z)+q(z)f q.e. o K, (4.7) the _=+ w ad F=F w. Usg ths result we ca explctly fd the extremal measure for the weght w(z) gve by (4.1). Theorem 4.2. Let K/C be a arbtrary compact set of postve capacty ad let & be a Borel measure wth compact support S & /C"K such that &&&=:, 0:1. The the soluto of the weghted eergy problem for w(z) as defed by (4.1), s gve by + w =(1&:) + K +&^, (4.8) where + K s the classcal equlbrum dstrbuto for K ad &^ s the balayage of & to K.

17 74 PAPAMICHAEL, PRITSKER, AND SAFF Furthermore, F w =(1&:) log 1 C + 0 g 0 (t, ) d&(t), (4.9) where C :=cap K ad g 0 (t, ) s the Gree fucto of the ubouded compoet 0 of C "K wth pole at. Remark. For the otos of the equlbrum dstrbuto, balayage ad Gree fucto see Chapter II ad IV of [10]. Proof. Recall, that for the weght w(z) gve by (4.1) Q(z)=&U & (z), z # K. Secto IV.1 of [10] yelds the exstece of the balayage &^ whch s the uque measure wth supp &^ /K ad &&^ & = &&& = :, such that ad U &^ ( z )=U & (z)+ g 0(t, ) d&(t) for q.e. z # K, (4.10) U &^ ( z )U & (z)+ g 0(t, ) d&(t) \z # C. (4.11) Sce U & (z) s uformly bouded o supp &^ /K, the last equalty mples by tegrato wth respect to &^ that &^ has fte logarthmc eergy. Ths meas that + :=(1&:) + K +&^ also has fte logarthmc eergy. Next, we recall from Frostma's theorem ([19, Secto III.3]), U + K (z)=log 1 C q.e. o K. The, t follows from (4.10) that q.e. o K U + (z)+q(z)=(1&:) U + K (z)+u &^ ( z )&U & (z) =(1&:) log 1 C + g 0(t, ) d&(t). It s clear that supp +/K/K ad &+&=(1&:) &+ K &+&&^ & =1&:+ :=1. Thus, by (4.6) ad (4.7), + must be the weghted equlbrum dstrbuto wth F w =(1&:) log 1 C + g 0(t, ) d&(t),.e. +=+ w. K

18 ASYMPTOTIC ZERO DISTRIBUTION 75 Let K be a bouded cotuum whose complemet cossts of a fte umber of domas ad, wth the otato troduced Secto 2, cosder the weght fucto z&a j w(z)={` &: j, z # K, j=1 0, z # C"K, (4.12) where, as before, : j 0, j=1,...,, j=1 : j1, ad for each j=1, 2,...,, a j s a fxed pot the bouded compoet G j of C"K. The w(z) s cotuous as a fucto defed o K ad where w(z)=e &Q(z), z # K, (4.13) Q(z):=: j=1 : j log z&a j. (4.14) Corollary 4.3. for The soluto of the mmal weghted eergy problem s gve by the measure Furthermore, w(z)= ` + w = \1& : z&a l &: l, z # K, F w = \1& : : l+ + e+ : where C s the logarthmc capacty of K. : l + l. (4.15) : l+ log 1 C, (4.16) Proof. For the weght w(z) defed by (4.12) we have that Q(z)=&U & (z), z # K, for the measure &= : : l $ al,

19 76 PAPAMICHAEL, PRITSKER, AND SAFF where $ t s the Drac $-measure at the pot t ad : :=&&&= : It s well kow (cf. [10, p. 222]) that the balayage of the ut pot mass $ al supported a l to K s just + l = (a l,},g l ),,...,. Therefore, ad so by Theorem 4.2, + w = \1& : &^ = : l =1 : l + l : l. : l+ + e+ : : l + l. Sce ths case g 0 (t,)d&(t)=0, formula (4.9) yelds F w = \1& : : l+ log 1 C, whch completes the proof. K 5. PROOFS 5.1. Lemmas Before we proceed wth the proofs of our ma results we eed several mportat lemmas. The frst of these s a aalogue of the well-kow BersteWalsh lemma for ratoal fuctos R N (z). Assume that E s a bouded cotuum whose complemet cossts of a fte umber of domas. Let us deote the bouded compoets of C "E by [G l] ad the ubouded compoet by 0. Cosder the coformal mappgs, l: G l D=[w: w <1] ad 8 : 0 D$=[w: w >1] wth ormalzatos respectvely, l(a l )=0,, $ l (a l )>0, where a l # G,,...,, ad 8 ()=, 8 $()>0. Lemma 5.1. that For the ratoal fucto R N (z) defed by (2.5) we have R N (z) &R N & 0 8 (z) k, z # 0 (5.1)

20 ASYMPTOTIC ZERO DISTRIBUTION 77 ad R N (z) &R N& G l, z # G l,,...,, (5.2), l(z) m l where the orms are Chebyshev orms. Proof. The proof s stadard: we cosder the fucto h(z)= R N (z)[8 (z)] k ad observe that t s aalytc everywhere 0. Thus, by the maxmum modulus prcple, we have that R N (z) 8 (z) " R k N =&R N & 0, z # 0. 8 k"0 Ths gves (5.1). Smlarly, by cosderg the fucto h(z)=r N (z)[, l(z)] m l, whch s aalytc G l, we obta the same way that R N (z) }, l(z) m l &RN, m l l & G l =&R N& G l, z # G l,,...,, ad (5.2) follows. K As a cosequece, we have the followg: Lemma 5.2. Let the cotuum K satsfy the assumptos of Theorem 2.2, ad assume that the sequece [R N (z)] # 4 coverges locally uformly K% to f(z)(0). The ad lm # 4 lm # 4 &R N & 1k 0 =1 (5.3) &R N & 1m l G l =1,,...,. (5.4) Proof. Sce K s the closure of a ftely coected Jorda doma K%, a argumet smlar to that of [20, p. 96], shows that for ay R>1 we ca fd a cotuum E/K% such that 0/E R, where E R deotes the cotuum bouded by the level curve 8(z) =R ad E. (It s eough to take E to be a Jorda curve suffcetly close to 0.) The, by the locally uform covergece of R N (z) tof(z)0k%, we have that &R N & E & f&r N & E +& f & E 2 &f& E

21 78 PAPAMICHAEL, PRITSKER, AND SAFF for suffcetly large. Thus, from (5.1), &R N & 8 (z) =R 2 &f& E R k, ad, by the maxmum modulus prcple, ths yelds lm sup Hece, o lettg R 1 we get that &R N & 1k 0 lm sup &R N & 1k E R R. lm sup &R N & 1k 0 1. (5.5) Smlarly, by takg E to be a Jorda curve close to G l, ad usg (5.2) we fd that Hece, &R N &, l (z) =1R 2 &f& E R m l,,...,. lm sup &R N & 1m l G l 1,,...,. (5.6) If we assume that lm f &R N & 1k 0 =q<1, the we ca coclude that there exsts a subsequece [R N ] # 4$/4 uformly coverget to zero o 0. It the follows from (5.1), wth E=0, that ths covergece takes place the strp betwee 0 ad the level curve defed by 8(z) =R for some R>1,.e. lm # 4$ R N (z) 1k lm # 4$ &R N & 1k 0 } R qr, z # [z # t:1 8(t) R]. Thus, we oly eed to take R such that qr<1. But ths mples the exstece of aalytc cotuato of f(z) through 0, whch vashes detcally the strp. Thus, f(z) must vash everywhere, cotradctg our assumpto that f(z)0. Ths proves (5.3). The same argumet ca be appled the case of G l,,...,, to prove that lm f &R N & 1m l G l <1 s mpossble for ay,...,, thus yeldg (5.4). K

22 ASYMPTOTIC ZERO DISTRIBUTION 79 Lemma 5.3. have that For the leadg coeffcets of R N (z) defed by (2.5) we t N k 1 C k &R N& 0, C=cap K, (5.7) ad s N l, m l R m l l &R N & Gl, (5.8),...,. Proof. Followg the proof of Lemma 5.1, ad the maxmum modulus prcple we have R N (z) 8(z) k&r N& 0, z # 0. Ths gves (5.7), f we pass to the lmt as z. Smlarly, by passg to the lmt wth z a l R N (z) }, l (z) m l &RN & Gl, z # G l, we obta (5.8). K Corollary 5.4. Let K ad [R N ] # 4 be the same as Lemma 5.2. The lm sup t N k 1k 1 C (5.9) ad lm sup s N l, m l 1m l Rl,,...,. (5.10) Proof. Iequaltes (5.9) ad (5.10) follow mmedately from (5.3), (5.4), (5.7) ad (5.8). K Lemma 5.5. Wth the assumptos of Theorem 2.2, the moc polyomal P N (z), (2.7), s asymptotcally extremal o K wth respect to the weght w(z) defed by (4.12),.e. lm # 4 where F w s gve by (4.16). &w N P N & 1 N K =e&f w, (5.11)

23 80 PAPAMICHAEL, PRITSKER, AND SAFF Proof. Sce K s a cotuum ad a l K,,...,, there exst two costats d 1 ad d 2 such that Cosder lm sup d 1 z&a l d 2, \z#k,,...,. &w N P N & 1 N =lm sup =lm sup lm sup "\ ` "\ ` 1 _ =C 1& : l =e &F w. N z&a l l+ &: N z&a l l+ &: P N (z) " 1 N K > (z&a l) m l t N k 1 N " t N ` (z&a l ) (m l N )&: k l"k R N (z) " 1 N K }&R N & 1 N K (For the above we made use of Theorem 2.2(), (2.9), (5.3) ad (5.4) order to deduce that lm " ` # 4 (z&a l ) (m l N )&: l"k=1, lm # 4 &R N & 1 N K =1, & ad 1 1& lm t N # 4 k 1 N =C : l. + By Corollary 4.5 of [12] ad Corollary 4.3 we have lm f &w N P N & 1 N K e&f w, (5.12) from whch the lemma follows mmedately. K Lemma 5.6. Let & N be the measure defed by (2.8). The, wth the assumptos of Theorem 2.2 we have that for ay closed subset B/( G l)_0. & N (B) 0, as, # 4, (5.13)

24 ASYMPTOTIC ZERO DISTRIBUTION 81 Proof. If B/0, the Lemma 5.6 s mpled by Lemma 5.5 ad Theorem 2.3(a) of [11]. I case of B/G l, 1l, the proof of ths lemma s smlar to the proof of Lemma 4.1 of [11] or ca be drectly reduced to t by the trasformato u=1(z&a l ). K 5.2. Proofs of Theorems 2.2, 2.3, ad 2.6 Proof of Theorem 2.2. Let _ be ay weak* lmt of & N,.e. & N * _ for some subsequece 4$/4. By the locally uform covergece of [R N (z)] # 4 to f(z)0 we obta from the Hurwtz theorem that _(B)=0 (5.14) for ay closed set B/K%. The, Lemma 5.6 ad (5.14) mply that supp _/K. Clearly, _ # M(K). We kow from Corollary 4.3 that supp + w =K. By the property (d) of Theorem 4.1 ad (5.11) we obta log z&t d& N(t)& log z&t d+ w(t)= N (5.15) uformly o K, where = N 0 as, #4. Usg the prcple of domato (Theorem II.3.2 [15] ad Secod Maxmum Prcple [10], p. 111) we coclude that (5.15) holds for all z # C. If we let $>0 ad cosder &~ N, the ormalzed coutg measure zeros of P N (z) that are closer tha $ to K, the by (5.15) log z&t d&~ N(t)& log z&t d+ w(t)= N +o(1) for every z such that dst(z, K)>$. But &~ N * _,as,#4$, therefore we have for every z K & log 1 z&t d_(t)+ log 1 z&t d+ w(t)0. (5.16) The fucto u(z) :=U + w (z)&u _ (z) s harmoc C "K. If we show that u(z) has a zero some compoet of C "K, the u(z) vashes detcally there by (5.16) ad the maxmum prcple [19]. Frst, we cosder 0, whch s the ubouded compoet of C "K. Its ot dffcult to see that lm u(z)= lm log 1 z z z&t d(+ w&_)(t)=0=u().

25 82 PAPAMICHAEL, PRITSKER, AND SAFF Cosequetly, u(z)#0 0. (5.17) Usg the assumpto that the sequece [R N (z)] # 4 coverges locally uformly to f(z)0, we ca choose a pot z 0 # K% such that lm # 4 R N (z 0 )= f(z 0 ){0. Ths mples the exstece of some eghborhood of z 0 that does't cota zeros of R N (z) by the Hurwtz theorem, for # 4 large eough. Thus, we have by the weak* covergece U _ (z 0 )= lm # 4$ = lm # 4$ = lm # 4$ U & N (z 0 ) = lm \ k # 4$ = \1& : 1 N log 1 P N (z 0 ) log tn k 1 N > z 0&a l &m l N R N (z 0 ) 1 N N log tn 1k k + : : l+ log 1 C + : m l N log 1 z 0 &a l + : l log 1 z 0 &a l. (5.18) For the potetal of the weghted equlbrum dstrbuto we obta U + w (z 0 )= \1& : = \1& : : l+ log 1 : l+ log 1 C + : z 0 &t d+ e(t)+ : : l log Comparg (5.18) ad (5.19) we coclude that : l log 1 z 0 &t d+ l(t) 1 z 0 &a l. (5.19) U + w (z)=u _ (z), \z # K%. (5.20) Sce both potetals are cotuous the fe topology (see Secto V.3 of [10]) ad the boudary of 0 the fe topology s the same as the Eucldea boudary [15], the we obta by (5.17) that U + w (z)=u _ (z), \z #0. Thus, we have U + w (z)=u _ (z), \z # C"G, (5.21)

26 ASYMPTOTIC ZERO DISTRIBUTION 83 where G= G l. By the ucty theorem [15] we get _ 0 =+ w 0. Now, we apply the fe topology argumet to the doma C "G to coclude that U + w (z)=u _ (z) o (C "G )=. G l. Sce U + w (z) ad U _ (z) cocde o K, whch cotas the supports of both measures, we obta U + w (z)=u _ (z), \z # C, by the maxmum prcple for harmoc fuctos. Aother applcato of the ucty theorem yelds _#+ w. Let us tur to the proof of the coverse part Theorem 2.2. We have that & N * _ as, # 4, where _=+ w. Ths mples U _ (z)=u + w (z), \z # C. (5.22) By the same argumet as (5.18) we get for z 0 # K% such that f(z 0 ){0, U + w (z 0 )= \1& : : l+ log lm # 4 t N 1k k + : : l log 1 z 0 &a l. The last equalty ad (5.19) mmedately gves codto () of Theorem 2.2. I order to prove () we use the assumpto that every a l # G l has some zero free eghborhood, so that by the weak* covergece U + w (a l ) = lm # 4 1 N log 1 P N (a l ) = lm # 4 1 N log } t N k s N l, m l > j{l a l &a j j} m = lm \ k N log tn 1k k + m l N log 1 +log ` a s N l, m l 1m l &a j l j N )+ &(m j{l # 4 = \1& : j{l j=1 : j+ log 1 C +: 1 l log lm ml s N l, m l 1m l 1 + : : j log a l &a j. (5.23)

27 84 PAPAMICHAEL, PRITSKER, AND SAFF We observe that the fucto g(t)=(, l (t)&, l (a l ))(t&a l ) s aalytc G l ad cotuous o G l. Also, g(t){0, \t # G l. Ths mples that log g(t) s harmoc G l ad cotuous o G l wth log g(a l ) =log,$(a l ) =log 1 R l. Now we calculate U + w (a l )= \1& : j=1 = \1& : j=1 : j+ log 1 C + : j=1 j{l : j log 1 z&a l d+ j(z) : j+ log 1 C + : 1 : j log a j &a l +: l log }, l(z)&, l (a l ) z&a l = \1& : j=1 j{l } d+ l(z) : j+ log 1 C + : 1 : j log a j &a l +: l log,$ l (a l ). (5.24) By (5.22) codto () of Theorem 2.2 ow follows from (5.23) ad (5.24),,...,. K Proof of Theorem 2.3. The proofs of (2.17) ad (2.18) are the same as the proof of Theorem 2.2 so we eed oly to preset the proof of (2.19). By (2.17) ad (2.18) we have U & G (z)=u + w G (z), \z G. (5.25) Let _ be a weak* lmt of the balayages of & N G evelope theorem [15] we have to G. The, by the lower lm f # 4$ N G (z)=u _ (z) (5.26) ad lm f U & N G (z)=u & G (z), (5.27) # 4$ where both equaltes hold q.e. C"G for a subsequece 4$/4. Sce N G (z)=u & N G (z), \z # C"G,

28 ASYMPTOTIC ZERO DISTRIBUTION 85 by the defto of balayage, we obta from (5.26) ad (5.27) U _ (z)=u & G (z) q.e. C"G. The last equalty together wth (5.25) yelds U _ (z)=u + w G (z) q.e. C"G. But supp _/G ad supp + w G /G, therefore both potetals are cotuous C"G ad U _ (z)=u + w G (z), \z # C"G. Cosequetly, the measures must be detcal by the Carleso ucty theorem (see [3] ad [15]): Thus, by (2.18) ad (5.28) we have _=+ w G. (5.28) &^ N * & C"G + _ = + w C"G + + w G = + w, as, # 4. K Proof of Theorem 2.6. therefore s omtted. Ths proof follows that of Theorem 2.2 ad 5.3. Proof of Theorem 3.2 Frst, we show that p m, (z) coverges to some aalytc fucto g locally uformly A, wth g0. To ths ed we recall that by the addtve splttg (3.8) the fucto f + (z) must have a meromorphc cotuato wth precsely poles [z: R z <R ] because f & (z) s aalytc [z: z >r]. Smlarly, f & (z) has a meromorphc cotuato wth precsely poles [z: r < z <r]. Suppose that [z + j ] k+ are the poles of f + j=1 (z) wth the correspodg multplctes [l + j ] k+ j=1 such that k + : l + j j=1 =. From the classcal de Motessus de Ballore's theorem [1] we have lm r + (z)= f + m, (z), m

29 86 PAPAMICHAEL, PRITSKER, AND SAFF where the covergece s locally uform [z: z <R ]"[z + j ] k+ j=1, ad where So, lm q + m, (z)=q+ (z), (5.29) m Q + locally uformly [z: z <R ]. Smlarly, we have (z) :=`k+ j=1\ 1& z + l j. z + + j p + (z) f + m, (z) Q + (z), as m, (5.30) p & m, \ 1 z+ f & (z) Q & \ 1, as m, (5.31) z+ locally uformly [z: z >r ], ad q & m, \ 1 z+ Q& \ 1 z+ `k & & l := j j=1\ 1&z& j, z + as m, (5.32) locally uformly C, where [z & j ] k& j=1 are the poles of f & (z) [z: r < z r] wth multplctes [l & j ] j=1 such that k & : j=1 l & j =. Takg to accout (5.29), (5.30), (5.31) ad (5.32) we obta from (3.9) p m, (z) ( f + Q + Q& + f & Q & Q+ )(z), as m, locally uformly A. But g(z):=(f + Q + Q& +f& Q & Q+ )(z) =Q + (z) Q& (z)( f + (z)+ f & (z)) = f(z) Q + (z) Q& (z) for ay z # A. Thus, p m, (z) g(z), as m, (5.33)

30 ASYMPTOTIC ZERO DISTRIBUTION 87 locally uformly A, where g(z) s aalytc A does't vash detcally. To fsh the proof we eed to fd the mth root behavor for the leadg coeffcets of p m, (z). Applyg the result of [4, p. 263] to f + (z) ad p + (z) we obta that \4 m, 1 /N such that lm a +, (mk) m k, 1m k = 1. (5.34) m R m # 4 1 The (3.11) follows from (5.33), (5.34) ad Theorem 2.3 (cf. (2.18)). Usg the trasformato w=1z we proceed the same maer for f & (z) ad p & m, (z) to deduce that _4 2 /N such that lm a &, (ml) m m l, 1m l =r. m # 4 2 Aother applcato of Theorem 2.3 (2.18) yelds (3.12). K REFERENCES 1. G. A. Baker, Jr., ``Essetals of Pade Approxmats,'' Academc Press, New York, H. P. Blatt, E. B. Saff, ad M. Smka, JetzschSzego type theorems for the zeros of best approxmats, J. Lodo Math. Soc. 38 (1988), L. Carleso, Mergelya's theorem o uform polyomal approxmato, Math. Scad. 15 (1964), A. Edre, Agular dstrbuto of the zeros of Pade polyomals, J. Approx. Theory 24 (1978), A. Edre, Zeros of partal sums of Lauret seres, Mchga Math. J. 29 (1982), G. M. Goluz, ``Geometrc Theory of Fuctos of a Complex Varable,'' Traslatos of Mathematcal Moographs, Vol. 26, Amerca Math. Socety, Provdece, RI, W. B. Gragg ad G. D. Johso, ``The LauretPade Table,'' Iformato Processg, Vol. 74, ``Proc. 74th IEIP Cogress, Amsterdam,'' pp , North-Hollad, Amsterdam, R. Grothma, Ostrowsk gaps, overcovergece, ad zeros of polyomals, ``Approxmato Theory VI (New York)'' (L. L. Schumaker, C. K. Chu, ad J. D. Ward, Eds.), pp. 14, Academc Press, New York, R. Jetzsch, ``Utersuchuge zur Theore Aalytscher Fuktoe,'' Ph.D. thess, Berl, N. S. Ladkof, ``Foudatos of Moder Potetal Theory,'' Sprger-Verlag, Berl, H. N. Mhaskar ad E. B. Saff, The dstrbuto of zeros of asymptotcally extremal polyomals, J. Approx. Theory 65 (1991), H. N. Mhaskar ad E. B. Saff, Weghted aalogues of capacty, trasfte dameter ad Chebyshev costat, Costr. Approx. 8 (1992), R. Nevala, ``Aalytc Fuctos,'' Sprger-Verlag, New York, I. E. Prtsker, ``Covergece ad Zero Dstrbuto of Lauret-Type Ratoal Fuctos,'' Ph.D. thess, Uversty of South Florda, 1995.

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