UNIVERSALITY LIMITS INVOLVING ORTHOGONAL POLYNOMIALS ON AN ARC OF THE UNIT CIRCLE

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1 UNIVERSALITY LIMITS INVOLVING ORTHOGONAL POLYNOMIALS ON AN ARC OF THE UNIT CIRCLE DORON S. LUBINSKY AND VY NGUYEN A. W stablish uivrsality limits for masurs o a subarc of th uit circl. Assum that µ is a rgular masur o such a arc, i th ss of Stahl, Totik, ad Ullma, ad is absolutly cotiuous i a op arc cotaiig som poit z 0 iθ 0. Assum, morovr, that µ is positiv ad cotiuous at z 0. Th uivrsality for µ holds at z 0, i th ss that th rproducig krl K z, t for µ satisfis K lim z 0 xp πis, z0 xp πi t K z 0, z 0 iπ S s t T θ 0, uiformly for s, t i compact substs of th pla, whr S z is th sic krl, ad T/π is th quilibrium dsity for th arc. si πz πz. I R I th thory of radom Hrmitia matrics, arisig from scattrig thory i physics, uivrsality limits play a importat rol. Thy ca b rducd to scalig limits for rproducig krls ivolvig orthogoal polyomials, which maks th aalysis fasibl. This has b compltd i a vry wid array of sttigs [], [3], [4], [8], [9], [0], [], [3], [4], [5], [0], [3]. I a rct papr, Eli Lvi ad th first author stablishd uivrsality limits for masurs o th uit circl [9]. I this papr, w cosidr istad subarcs of th uit circl. Our aalysis dpds havily o th work of Loid Goliskii, who providd a dtaild xpositio for Szgő-Brsti thory for such arcs, ad dducd asymptotics of orthogoal polyomials ad thir Christoffl fuctios [6], [7]. I tur, Goliskii s work dpdd havily o work of Akhizr []. Lt α 0, π ad lt our arc b { } α iθ : θ [α, π α]. Lt µ b a fiit positiv Borl masur o α or quivaltly o [α, π α] with ifiitly may poits i its support. Th w may dfi orthoormal Dat: Jauary 8, 03. Rsarch of first author supportd by NSF grat DMS008 ad US-Isral BSF grat ; Scod author supportd by Gorgia Tch 0 Rsarch Expric Program for Udrgraduats

2 DORON S. LUBINSKY AND VY NGUYEN polyomials φ z κ z +..., κ > 0, 0,,,... satisfyig th orthoormality coditios π π α α φ z φ m zdµ θ δ m, whr z iθ. W shall usually assum that µ is rgular i th ss of Stahl, Totik ad Ullma [], so that. lim κ/ cos α. Hr cos α is th logarithmic capacity of α. A simpl suffi cit coditio for rgularity is that µ > 0 a.. i [α, π α, but thr ar pur jump ad pur sigularly cotiuous masurs that ar rgular. Th th rproducig krl for µ is. K z, u φ j z φ j u. j0 To stat our rsults, w d som auxiliary fuctios: for θ [α, π α], w dfi λ θ [0, π] by th quatio.3 cos λ θ cos θ cos α. Obsrv that λ is a strictly icrasig cotiuous fuctio of θ, that maps [α, π α] oto [0, π]. W also lt si θ.4 T θ. cos α cos θ T θ / π is th dsity of th quilibrium masur for α i th ss of pottial thory. Fially, w d th sic krl: si πz.5 S z πz. Our mai rsult is: Thorm. Lt α 0, π, ad lt µ b a fiit positiv Borl masur o [α, π α] that is rgular. Lt J α, π α b compact, ad b such that µ is absolutly cotiuous i a op st cotaiig J. Assum morovr, that µ is positiv ad cotiuous at ach poit of J. Th uiformly for θ 0 J ad s, t i compact substs of th complx pla C, w hav K m iθ 0+ πs m, iθ 0 + π t m lim m K m iθ 0, iθ 0 iπ S s t T θ 0.

3 UNIVERSALITY LIMITS 3.6 Rmarks a I th cas α 0+, w s that T θ ad th right-had sid of.6 rducs to iπ S s t, which is th rsult of Lvi ad Lubisky [9]. b If J cosists of just a sigl poit θ 0, th th hypothsis is that µ is absolutly cotiuous i som ighborhood θ 0 ε, θ 0 + ε of θ 0, whil µ θ 0 > 0 ad µ is cotiuous at θ 0. c As i [9], [3], th mai ida i this papr is a localizatio pricipl, ad a compariso iquality. d As i [], this limit has implicatios for th spacig of th zros of th rproducig krl. It is kow that for a, th zros of K, a li o th uit circl [8, Thm..., p. 9]. Corollary. Assum th hypothss of Thorm., ad lt θ 0 J. For k, lt iθ k dot th kth closst zro of K iθ 0, to θ 0, with θ k > θ 0, whil iθ k dots th kth closst zro to θ 0, with θ k < θ 0. Th for larg ough, θ ±k xists, th zro iθ ±k is simpl, ad.7 lim θ ±k θ 0 ±πk T θ 0. This rsult should b compard to th clock thorms i [], [9], th stimats i [6], ad arlir work of Frud [5, p. 66]. W ca also dduc asymptotics for drivativs of th rproducig krl: w lt K j,k z, z φ j m z φ k m z. m0 Corollary.3 Assum th hypothss of Thorm., ad lt θ 0 J, z 0 iθ 0. For j, k 0,.8 lim z j k 0 K j,k z 0, z 0 j+k K z 0, z 0 [ + T θ0 j+k+ ] T θ0 j+k+. T θ 0 j + k + I th squl C, C, C,... dot costats idpdt of, z, u, θ, s, t. Th sam symbol dos ot cssarily dot th sam costat i diffrt occurrcs. W shall writ C C α or C C α to rspctivly dot dpdc o, or idpdc of, th paramtr α. [x] dots th gratst

4 4 DORON S. LUBINSKY AND VY NGUYEN itgr x. For squcs {c }, {d } of o-zro ral umbrs, w writ c d if thr xist positiv costats C, C idpdt of such that C c /d C. Giv masurs µ, µ #, w us K, K # to dot thir rspctiv rproducig krls. Similarly suprscripts, # ar usd to distiguish othr quatitis associatd with thm. W dot th th Christoffl fuctio for th masur µ by.9 Ω iθ /K iθ, iθ mi dgp π π α α P it dµ t / P Th papr is orgaisd as follows. I Sctio, w prov som of th rsults for a spcial wight cosidrd by Loid Goliskii i [6]. I Sctio 3, w prov Thorm. ad Corollaris. ad.3.. A S W A I this sctio, w cosidr th masur dµ θ W θ dθ, whr si α. W θ, θ [α, π α]. si θ cos α cos θ This is th spcial cas Ω of th wights cosidrd by Loid Goliskii [6, p. 37]. Goliskii [6, p. 44] providd a dtaild drivatio of xplicit formula for th corrspodig orthoormal polyomials {φ }: for,. φ iθ A θ i θ λθ + B θ i θ +λθ whr λ θ [0, π] is dtrmid by.3. Morovr, A θ { iλθ si α i θ + si α } g iθ.3 i si ; θ B θ { iλθ si α i θ + si α } g+ iθ.4 i si ; θ ad all w shall d to kow about g ± is that [6, p. 4, 35, 38] thy ar cotiuous ad.5 g ± iθ si θ/ si α/. Not that i [6], w chos Ω ad ρ Ω iθ siα/. W shall assum si θ/ that. holds v for 0 as this maks o diffrc to our asymptotics. W prov iθ.

5 UNIVERSALITY LIMITS 5 Thorm. Lt dµ θ W θ dθ b giv by., ad lt K m dot its mth rproducig krl. Th uiformly for s, t i compact substs of th complx pla, K m iθ 0+ πs m, iθ 0 + π t m lim m K m iθ 0, iθ 0 iπ S s t T θ 0. W bgi with ral s, t: Lmma. Lt θ 0 α, π α, s, t R, ad for m, dfi θ θ m, φ φ m by.6 θ θ 0 + πs m ; φ θ 0 + πt m. a Uiformly for s, t i compact substs of th ral li,.7 lim m m λ θ λ φ T θ 0 π s t. b Uiformly for s, t i compact substs of th ral li, lim m m K m iθ, iφ.8 c.9.0 A θ 0 iπ T θ 0 S + B θ 0 iπ +T θ 0 S A θ 0 si θ 0 si α B θ 0 si θ 0 si α s t s t T θ 0 ; + T θ 0. T θ 0 + T θ 0 d Uiformly for s i compact substs of th ral li,. lim m m K m iθ, iθ si θ 0 si α. ad. lim m m K m iθ, iθ W θ T θ 0. Uiformly for θ 0 i compact substs of α, π α, ad s, t i compact substs of C, K m iθ, iφ.3 lim m K m iθ 0, iθ 0 iπ S s t T θ 0..

6 6 DORON S. LUBINSKY AND VY NGUYEN Proof a Now by th dfiitio.3 of λ, cos α [cos λ θ cos λ φ] cos θ cos φ. Hc cos α λ θ λ φ λ θ + λ φ si si θ φ θ + φ si si. 4 4 Sic λ is cotiuous i α, π α, ad θ, φ θ 0 as m, w dduc that.4 cos α m [λ θ λ φ] si λ θ 0 θ φ m si θ o π s t si θ 0 + o. Fially, cos α si λ θ 0 cos α cos λ θ 0 cos α θ cos 0. Substitutig this ito.4 givs th rsult. b From.,.5 m K m iθ, iφ m [A θ i θ λθ + B θ i θ +λθ] m 0 [A φ i φ λφ + B φ i φ +λφ] Σ + Σ + Σ 3 + Σ 4,

7 UNIVERSALITY LIMITS 7 whr ths four sums ar spcifid blow: firstly by cotiuity of A, Σ m m 0 A θ A φ i λθ+λφ m A θ 0 i λθ+λφ + o m 0 A θ 0 m im λθ+λφ i A θ 0 i m λθ+λφ S λθ+λφ + o m π S A θ 0 iπ T θ 0 s t S T θ 0 π λ θ + λ φ + o λ θ + λ φ + o, by.7 ad th cotiuity of S at 0, whr S 0. Similarly, Nxt, Σ 4 m m 0 B θ B φ i +λθ λφ B θ 0 iπ +T θ 0 S Σ 3 m Hr as m, m 0 A θ 0 B θ 0 m s t A θ B φ i λθ λφ m 0 A θ 0 B θ 0 m θ φ + T θ 0 + o. i λθ λφ + o im λθ λφ i λθ λφ + o. λ θ λ φ λ θ 0 + o ad λ θ 0 π, 0, so th domiator i λθ λφ i Σ 3 is boudd away from 0. Thus Σ 3 o, ad similarly, Σ 4 o.

8 8 DORON S. LUBINSKY AND VY NGUYEN Combiig th abov asymptotics for Σ j, j,, 3, 4, givs th rsult. c Now iλθ 0 si α θ i 0 + si α si α + cos α cos θ0 λ θ si α cos α [ cos θ 0 cos λ θ 0 + si θ ] 0 si λ θ 0 cos θ 0 si θ 0 cos α θ cos 0, by dfiitio.3 of λ θ 0. W cotiu this as si θ 0 T θ 0. Th.9 follows from.3 ad.5..0 is similar. d This follows by sttig φ θ i.8 ad usig.9 ad.0. From.8 to., K m iθ, iφ K m iθ 0, iθ 0 T θ 0 iπ T θ 0 S lim m + This ca b cotiud as [ iπ T θ 0 π T θ 0 π + T θ 0 iπ +T θ 0 S iπ iπ T θ 0 π T θ 0 iπ cos iπ iπ + si π + cos π s t T θ 0 s t + T θ 0. T θ0 si π T θ0 ] T θ0 si π + T θ0 cos π T θ 0 { iπ T θ0 si π T θ 0 { iπ T θ0 + si π s t [ T θ 0 cos π s t T θ 0 π s t si π s t T θ 0 iπ S s t T θ 0. Proof of Thorm. } iπ T θ 0 } iπ T θ 0 π s t + i si π s t ]

9 UNIVERSALITY LIMITS 9 W alrady hav th rsult for ral s, t. For m, lt K m iθ 0+ πs m, iθ 0 + π t m.6 f m s, t K m iθ 0, iθ 0. This is a polyomial i iπs/m ad iπt/m. W shall show that {f m } is uiformly boudd for s, t i compact substs of C: that is, giv r > 0, thr xists C such that.7 sup sup m s, t r f m s, t C. Thus {f m } is a ormal family. I as much as th limit.3 holds for ral s, t, ad th right-had sid of.3 is a tir fuctio of s, t, it th follows from th pricipl of aalytic cotiuatio that th limit holds uiformly for s, t i compact substs of th pla. To prov.7, alog stadard lis, w ot first from.5 that for θ, φ α, π α, K m iθ, iφ m A θ + B θ A φ + B φ. Lt J b a compact subitrval of α, π α, ad J { iθ : θ J }. Th last iquality, ad cotiuity of A, B o J shows that sup z,u J m K m z, u C. Lt G dot th Gr s fuctio for C\J with pol at. From th Brsti-Walsh iquality [7, p. 56], it follows that for all z, u C, m K m z, u C mgz+gu. Morovr, G z 0 for z J, ad bcaus J is a "smooth" arc, G z iu C u, for z J ad u, whr J is ay compact subitrval of th itrior of J. It follows that for m m 0 r, K m iθ 0+ πs m, iθ 0 + π t m C C s + t C Cr. m S Lmmas 6. ad 6. i [9, pp ] for mor dtails. Fially, by., K m iθ 0, iθ 0 Cm. So w hav.7 ad th rsult.

10 0 DORON S. LUBINSKY AND VY NGUYEN 3. P T. W bgi with asymptotics for Christoffl fuctios: Lmma 3. Lt µ b a rgular masur o [α, π α]. Assum that µ is absolutly cotiuous i a op st cotaiig a compact st J α, π α, ad at ach poit of J, µ is positiv ad cotiuous. Lt A > 0. Th uiformly for a [ A, A], ad θ J, 3. lim Ω xp i θ + a µ θ /T θ. Morovr, uiformly for 0 A, θ J, ad a [ A, A], 3. Ω xp i θ + a. Rmarks a W mphasiz that w ar assumig that µ is cotiuous i J wh rgardd as a fuctio dfid o [α, π α]. b Asymptotics for Christoffl fuctios associatd with spcial masurs o th arc wr stablishd by Goliskii [7]. Totik [3], [4] stablishd asymptotics a.. o mor gral arcs ad curvs, that iclud 3. i th cas a 0. c It follows from Totik s rsults ad that abov, that π T θ is th dsity of th quilibrium masur i th ss of pottial thory for th arc. Proof W alrady kow this rsult for th spcial wight W θ dθ of th prvious sctio. Th xtsio to th gral cas is xactly th sam as for th whol uit circl i [9, pp , proof of Thorm 3.], so w omit th dtails. Nxt, w d a compariso iquality: Lmma 3. Lt r > 0 ad µ, µ b masurs o [α, π α], with µ rµ. Th for all ral θ, φ, K r K iθ, iφ /K iθ, iθ K iφ, iφ / [ K iθ, iθ K iθ, iθ ] / 3.3 rk iθ, iθ. Proof Lt µ # rµ, so that µ µ #. I [9, Thorm 4., pag 55-3], w showd

11 that UNIVERSALITY LIMITS K K # iθ, iφ /K iθ, iθ K iφ, iφ / [ K iθ, iθ K# iθ, iθ ] / K iθ, iθ. It is asily s from th dfiitio of th orthoormal polyomials ad rproducig krl that Th th rsult follows. K # z, w r K z, w. Proof of Thorm. Lt ε 0, ad θ 0 J. By cotiuity of µ ad W at θ 0, w ca choos δ > 0 such that for θ θ 0 δ Lt ε µ θ µ θ 0 ε ; ε W θ W θ 0 ε. c ε µ θ 0 W θ 0 ad dfi two w masurs µ ad µ # o [α, π α] by dµ # θ W θ dθ; dµ θ W θ dθ i θ θ 0 < δ; dµ θ W θ dθ + c dµ θ i [α, π α] \ θ 0 δ, θ 0 + δ. Th µ µ # ad cµ µ i [α, π α]. Morovr, by our asymptotics for Christoffl fuctios i Lmma 3., uiformly for s i a boudd ral itrval, K iθ 0 +πs/, iθ 0+πs/ lim K iθ 0 +πs/, iθ 0+πs/ µ θ 0 W θ 0 c ε ; K lim K # iθ 0 +πs/, iθ 0+πs/ iθ 0 +πs/, iθ 0+πs/. Morovr, uiformly for s i a boudd itrval, K iθ0+πs/, iθ 0+πs/ K # iθ0+πs/, iθ 0+πs/ iθ0+πs/, iθ 0+πs/. K

12 DORON S. LUBINSKY AND VY NGUYEN Th Lmma 3. applid to µ # ad µ givs, with r, ad θ θ 0 +πs/, φ θ 0 + πt/, K # K iθ, iφ /K # iθ, iθ K # iφ, iφ / [ K # K iθ, iθ ] / iθ, iθ K # iθ, iθ 0 as ; ad Lmma 3. applid to µ ad µ with r c givs, K c K iθ, iφ /K iθ, iθ K iφ, iφ / [ K iθ, iθ K iθ, iθ ck iθ, iθ [ C ε ] / C [3ε] /. ] / Hr C is idpdt of s, t, a, b,, ε. Combiig ths last two iqualitis givs, for larg ough, ck K # iθ, iφ / Cε /, ad rcallig th dfiitio of c, ad th fact that K O, also µ θ 0 W θ 0 K K # iθ, iφ / Cε /. Hr th lft-had sid is idpdt of ε, so w dduc lim sup µ θ 0 W θ 0 K K # iθ0+πs/, iθ 0+πt/ / 0. Usig Lmma 3. o K iθ 0, iθ 0 ad K # iθ 0, iθ 0 oc mor, ad Thorm., w obtai K iθ 0 +πs/, iθ 0+πt/ lim K iθ 0, iθ 0 K # lim iθ 0 +πs/, iθ 0+πt/ K # iθ 0, iθ 0 iπ S s t T θ 0. Th limit holds uiformly for s, t i a ral itrval. W still hav to stablish it for complx s, t. To do this, w ca procd as i th proof of Thorm.: lt f m b dfid by.6. W agai d to show th uiform bouddss.7. But i som itrval J cotaiig θ 0, w hav µ C, ad cosqutly, i a slightly smallr itrval J, K m iθ, iφ Cm, θ, φ J, m.

13 UNIVERSALITY LIMITS 3 W ca ow mimic th proof giv i Thorm. to show th uiform bouddss.7, ad th apply ormality ad aalytic cotiuatio. Proof of Corollary. This is a asy cosquc of Hurwitz s thorm: th fuctio iπs S st θ 0 has simpl zros wh ad oly wh st θ 0 is a itgr. It follows from th uiform covrgc i Thorm., ad Hurwitz Thorm, that for larg ough, K iθ 0 +πs/, iθ 0 has a simpl zro s±k, with lim s ±kt θ 0 ±k. Morovr, ths ar th oly zros of K iθ 0 +πs/, iθ 0 i a boudd ighborhood of 0. Now obsrv that θ ±k θ 0 + πs ±k /, so θ ±k θ 0 πs ±k ±πk T θ 0 + o. Proof of Corollary.3 W bgi with th idtity This asily yilds S x si πx πx 0 iπxy + iπxy dy. iπ S s t T θ 0 [ ] iπs+yt θ0 iπt+yt θ0 + iπs yt θ0 iπt yt θ 0 dy. 0 W ow us th Maclauri sris for th xpotial fuctio o ach trm i th last li, ad th itgrat with rspct to y. O multiplyig ad dividig by a suitabl powr of, w obtai iπ S s t T θ 0 iπs j iπt k j0 k0 j! k! T θ 0 j + k + [ + T θ0 j+k+ ] + T θ0 j+k Nxt, th asymptotic i Thorm. ca also b rcast i th form K z 0 + πis, z0 + πi t 3.5 lim iπ S s t T θ 0, K z 0, z 0

14 4 DORON S. LUBINSKY AND VY NGUYEN uiformly for s, t i compact sts. To stablish this, o uss that πis/ + πis + O, togthr with bouds such as K,0 z, z C, uiformly for z z 0 C /, for ay giv C > 0. This lattr stimat may asily b dducd from Cauchy s stimats for drivativs, ad th fact that K z, z C 3 for z z 0 C / - as i th proof of Thorm., this follows from th Brsti-Walsh growth lmma for polyomials. Fially, w ot that Taylor sris xpasio givs, z0 K z 0 + πis K z 0, z 0 j,k0 K j,k + πi t z 0, z 0 πis j πit k K z 0, z 0 j+k. j! k! z j k 0 This, th Taylor sris 3.4, ad th uiform covrgc 3.5 giv th rsult. R [] N. I. Akhizr, O Polyomials Orthogoal o a Circular Arc, Dokl. Akad. Nauk SSSR, 30960, [i Russia]; Sovit Math. Dokl., 960, [] J. Baik, T. Krichrbaur, K. T-R. McLaughli, P.D. Millr, Uiform Asymptotics for Polyomials Orthogoal with rspct to a Gral Class of Discrt Wights ad Uivrsality Rsults for Associatd Esmbls, Pricto Aals of Mathmatics Studis, 006. [3] P. Dift, Orthogoal Polyomials ad Radom Matrics: A Rima-Hilbrt Approach, Courat Istitut Lctur Nots, Vol. 3, Nw York Uivrsity Prs, Nw York, 999. [4] P. Dift, T. Krichrbaur, K. T-R. McLaughli, S. Vakids ad X. Zhou, Uiform Asymptotics for Polyomials Orthogoal with rspct to Varyig Expotial Wights ad Applicatios to Uivrsality Qustios i Radom Matrix Thory, Commuicatios i Pur ad Applid Maths., 5999, [5] G. Frud, Orthogoal Polyomials, Akadmiai Kiado, Budapst, 97. [6] L. Goliskii, Akhizr s Orthogoal Polyomials ad Brsti-Szgő mthod for a circular arc, J. Approx. Thory, 95998, [7] L. Goliskii, Th Christoff l Fuctio for Orthogoal Polyomials o a Circular Arc, J. Approx. Thory, 0999, [8] A.B. Kuijlaars ad M. Valss, Uivrsality for Eigvalu Corrlatios from th Modifid Jacobi Uitary Esmbl, Itratioal Maths. Rsarch Notics, 3000, [9] Eli Lvi ad D.S. Lubisky, Uivrsality Limits Ivolvig Orthogoal Polyomials o th Uit Circl, Computatioal Mthods ad Fuctio Thory, 7007, [0] Eli Lvi ad D.S. Lubisky, Uivrsality Limits i th bulk for Varyig Masurs, Advacs i Mathmatics, 9008,

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