SCALING LIMITS FOR MIXED KERNELS

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1 SCALING LIMITS FOR MIXED KERNELS DORON S. LUBINSKY A. Let µ ad ν be measures supported o, with correspodig orthoormal polyomials { p µ } ad {p ν } respectively. Defie the mixed kerel K µ,ν We establish scalig limits such as π ξ µ lim ξ ν ξ π a b π a b = S cos x, y = p µ j x pν j y. K µ,ν + B ξ, ξ + aπ ξ, ξ + bπ ξ where S t = si t is the sic kerel, ad B ξ depeds o µ, ν ad ξ. This t reduces to the classical uiversality limit i the bulk whe µ = ν. We deduce applicatios to the zero distributio of K µ,ν, ad asymptotics for its derivatives. Orthogoal polyomials, uiversality limits, scalig limits 4C05. I Let µ deote a positive measure o the real lie, with ifiitely may poits i its support, ad all fiite power momets. For 0, let p µ x = γ µ x +... deote the th orthoormal polyomial for µ, so that p µ p µ mdµ = δ m. The th reproducig kerel for µ is K µ x, y = p µ j x pµ j y. If µ has support [, ], the uder appropriate coditios o µ, there is the uiversality limit i the bulk, π ξ lim µ ξ K µ ξ + aπ ξ, ξ + bπ ξ = S π a b, where S t = si t t is the sic kerel, ξ,, ad a, b may be ay complex umbers. This limit arises i the theory of radom matrices, ad is kow to be true i various formulatios, i a wide array of settigs [], [4], [5], [9], [], [3], Date: August 6, 05. Research supported by NSF grat DMS3608

2 DORON S. LUBINSKY [8], [0]. It has applicatios to spacig of eigevalues of radom matrices, ad zero distributio of orthogoal polyomials, amogst other thigs. Let us recall how the coectio betwee reproducig kerels ad radom matrices begis. Oe starts with a probability distributio o the eigevalues x, x,..., x of Hermitia matrices, of the form P x, x,..., x = Z 0 x k x j dµ x j, j<k where Z 0 is a ormalizig costat. By usig colum operatios o a Vadermode determiat, we see that. x k x j = det [ x k ] j = γ µ [ j,k 0 γµ...γµ det p µ k x j ]. j,k j<k Next, we use the fact that square matrices ad their trasposes have the same determiat, to obtai x k x j = γ µ 0 γµ...γµ det [K µ x j, x k ] j,k. j<k Thus, P x, x,..., x ca be expressed i terms of reproducig kerels for the measure µ. That there is a similar expressio for the m poit correlatio fuctio is far deeper [4]. Now let ν be aother measure supported o the real lie, with th orthoormal polyomial p ν x = γ ν x +... ad th reproducig kerel K ν x, t. We defie the mixed kerel K µ,ν x, y = p µ j x pν j y. Suppose that we use. first for the measure µ ad the for the measure ν, ad oly the take trasposes, ad multiply. We lad up with the represetatio x k x j = γ µ 0 γµ...γµ γν 0γ ν...γ ν det [K µ,ν x j, x k ] j,k. j<k Thus there might be some iterest i aalysig scalig limits of the mixed kerel K µ,ν, ad that is the focus of this paper. Note that K µ,ν is the represetig kerel of the liear operator L give by L a j p µ j = a j p ν j. This liear operator is a isometry betwee the weighted L spaces L µ ad L ν. I the sequel, we use the otatio w µ, ν, ξ = π ξ µ ξ ν ξ. A simple, but useful, special case, is where we have poitwise asymptotics for both p µ ad p ν. This is outlied i:

3 3 Propositio. Assume that µ ad ν are positive measures with support,. Assume that I is a subiterval of,, i which µ ad ν are absolutely cotiuous, ad that as, uiformly for ξ = cos θ I, both. π ξ /4 µ ξ / p µ ξ = cos θ + ρ µ ξ + o ;.3 π ξ /4 ν ξ / p ν ξ = cos θ + ρ ν ξ + o ; where ρ µ ad ρ ν are cotiuous fuctios i I. Let J be a compact subiterval of the iterior of I. The uiformly for ξ = cos θ J ad a, b i compact subsets of R,.4 w µ, ν, ξ lim π b a = S K µ,ν ξ + aπ ξ, ξ + bπ ξ. π b a cos + ρ µ ξ ρ ν ξ If i additio, there exists C > 0 such that µ C ad ν C i I, this also holds uiformly for a, b i compact subsets of C. Here are two examples where the propositio is applicable: Example. Assume that µ ad ν have support [, ], ad satisfy Szegő s coditio, so that.5 log µ x dx > ad x log ν x dx >. x Assume i additio, that i a subiterval I of,, µ ad ν are absolutely cotiuous, that µ, ν are bouded above ad below by positive costats, ad for ξ I, I µ t µ ξ t ξ dt <, with the itegral covergig uiformly i ξ I, ad a similar coditio o ν. The it follows from results of Freud [6, p. 46, Table II, etrya], that the asymptotics. ad.3 hold uiformly i I we ote that it is stated oly poitwise i the table there. There are earlier results, uder more restrictive coditios o µ, i the books of Geroimus [8, p. 00] ad Szegő [9, p. 98]. I this example, as follows from [9, p. 99, eq...3], ρ µ ξ is give for ξ = cos θ by.6 ρ µ ξ = π 4π P V log µ cos t cot θ t π dt = π P V log µ t ξ t ξ t dt,

4 4 DORON S. LUBINSKY where P V deotes the Cauchy pricipal value itegral. That this is cotiuous follows from the assumed uiform covergece above. So.7 B ξ : = ρ µ ξ ρ ν ξ = 4π P V = π P V log µ t /ν t t ξ π π log µ cos t ν cos t cot θ t dt ξ t dt. The appearace of this cojugate fuctio type expressio suggests that for our scalig limits to hold, we eed µ /ν to satisfy a Szegő coditio, though quite possibly we do ot eed µ, ν to idividually satisfy a Szegő coditio. I such a case, the pricipal value itegral defiig B ξ will exist for a.e. ξ. Example.3 A geeralized Jacobi weight has the form m µ x = h x x t j τ j x α + x β, j= where α, β, τ,..., τ m >, ad < t <... < t m <, while h satisfies a Dii-Lipschitz coditio, that is Here 0 ω h; [, ] ; t dt <. t ω h; [, ] ; t = sup { h x h y : x, y [, ], x y t}, t > 0, is the modulus of cotiuity of h o [, ]. Badkov [, p. 38, Cor. ] proved that for such measures µ, defied by their absolutely cotiuous compoet, the asymptotic. holds, uiformly for x i a compact subiterval of, \ {t, t,..., t m }. Agai, ρ µ may be expressed i the form.6, ad the Dii-Lipschitz coditio ca be used to prove the cotiuity of ρ µ. So if both µ, ν are geeralized Jacobi weights, the result.4 holds i ay compact subiterval of, excludig all the zeros/sigularities of µ, ν, ad the formula.7 persists. Our mai result is that uder far more geeral coditios, the scalig limit holds i liear Lebesgue measure, meas: Theorem.4 Let µ ad ν be measures o, that satisfy the Szegő coditio.5 ad let B be as i.7. Let ε, R > 0. For, let H deote the set of ξ, for which.8 w µ, ν, ξ sup a, b R π a b S K µ,ν ξ + aπ ξ > ε. π a b cos + B ξ, ξ + bπ ξ

5 5 The.9 meas H 0 as. Note that a, b are allowed to be complex i.8. Whe µ = ν, our proof here gives a simpler proof of a special case of the mai result of [3]. There uiversality was show to hold i measure for arbitrary compactly supported measures. Corollary.5 Let µ ad ν be measures o, that satisfy the Szegő coditio.5, ad let B be as i.7. Let I be a closed subiterval of, ad assume that i some eighborhood of I, ad for some C >,.0 C µ C ad C ν C. The for ay R, p > 0, w µ, ν, ξ lim sup K µ,ν ξ + aπ ξ, ξ + bπ ξ I a, b R π a b π a b S cos + B ξ p dξ = 0.. We ca also deduce asymptotics for derivatives of K µ,ν. Give o-egative itegers l, m, defie the differetiated kerel ivolvig lth derivatives of p µ j ad mth derivatives of p ν j :. K µ,νl,m x, y = Defie for l, m 0,.3 τ l,m = ad.4 ρ l,m = { { p µ j l x p ν j m x. l m l+m+, l + m eve 0, l + m odd, l m+ l+m+, l + m odd 0, l + m eve. Corollary.6 Assume the hypotheses of Corollary.5. Let l, m 0. The for ay p > 0, l+m w µ, ν, ξ ξ lim K µ,νl,m ξ, ξ I.5 [ τ l,m cos B ξ ρ l,m si B ξ ] p dξ = 0. If i additio, we have the uiform limits as i Propositio., the this limit holds poitwise at ξ. Fially, uiform scalig limits imply results o zero asymptotics, as first oted

6 6 DORON S. LUBINSKY by Eli Levi [0]: Corollary.7 Let µ ad ν be measures supported o,, ad assume that at a give ξ, for which w µ, ν, ξ > 0, we have the limit.4, holdig uiformly for a, b i compact subets of the plae. Fix a C. Let j be a o-zero iteger. The for large eough, as a fuctio of v, K µ,ν ξ + aπ ξ, v has a zero v,j that satisfies.6 lim v,j ξ = a + j π ξ. Moreover, it has a zero ˆv,j that satisfies.7 lim ˆv,j ξ = a + j + [ ρµ ξ ρ π ν ξ ] π ξ. I additio, for large eough, the oly zeros of this fuctio for v i a ball ceter ξ, of radius O, have this form. Remark A iterestig special case occurs whe the hypotheses of Corollary.7 hold, ad µ k+ t + t.8 ν t =, t,, t with k a iteger. I this case Oe ca show that B ξ = k + π P V.9 B ξ = k + π, usig classical idetities for Szegő fuctios. +cos θ log +t t ξ t ξ t dt. Ideed if we cosider the Jacobi k+, "weight" o the circle, f θ = cos θ its Szegő fuctio [9, p. 77, eq. k+, +z 0..3] is z ad for z = e iθ, this equals i cot θ k+ k k+. = i cot θ The argumet of this is k + π, ad usig [9, p. 79, eq ],.9 follows. The.4 becomes πb a w µ, ν, ξ lim K µ,ν ξ + aπ ξ, ξ + bπ ξ = k+ si, πb a uiformly for a, b i compact subsets of the plae. The curious feature is that as a fuctio of b, the right-had side chages sig oly at b = a, where there is a simple zero. All other zeros are double. Those zeros attract zeros of the left-had side of total multiplicity. More itriguig is that this suggests the cojecture that perhaps x y K µ,ν x, y k+ 0 for all real x, y, at least for some special µ, ν related by.8. I the sequel, C, C, C,... deote positive costats idepedet of, a, b, u, v, z ad possibly other specified parameters. The same symbol does ot ecessarily

7 deote the same costat i differet occurreces. We prove Propositio. i Sectio, Theorem.4 i Sectio 3, ad Corollaries.5-.7 i Sectio 4. 7 We first give the. P P. Proof of Propositio. for real a, b Now if x = cos θ I ad y = cos φ I, our assumed asymptotic, ad elemetary trigoometry show that. w µ, ν, ξ K µ,ν x, y = cos jθ + ρ µ x cos jφ + ρ ν y + o = cos j θ + φ + ρ µ x + ρ ν y + cos j θ φ + ρ µ x ρ ν y + o cos ρ µ x + ρ ν y cos j θ + φ = si ρ µ x + ρ ν y si j θ + φ + cos ρ µ x ρ ν y + o. cos j θ φ si ρ µ x ρ ν y si j θ φ Next, if t is real, we ote the idetities cos jt = S t S t cos si jt = S t S t si t ; t. These follow easily by summig the fiite geometric series eijt ad takig real ad imagiary parts, ad recallig that S t = si t t. We ow apply these to the trigoometric sums i.. First, however, write ξ = cos θ 0 for some θ 0 0, π, ad The x = x = ξ + aπ ξ = cos θ ; y = y = ξ + bπ ξ = cos φ. cos θ cos θ 0 = x ξ = aπ ξ

8 8 DORON S. LUBINSKY Similarly si θ 0 θ θ 0 + o = aπ ξ θ θ 0 = aπ + o. φ θ 0 = bπ + o. Both these relatios hold uiformly i ξ J, ad a, b i compact subsets of the real lie. The cos j θ φ S θ φ = cos θ φ S θ φ b a b a = S π cos π + o. Similarly, si j θ φ b a b a = S π si π + o. Both hold uiformly for ξ J ad a, b i compact subsets of the real lie. Next, = cos j θ + φ S θ+φ cos S θ+φ θ + φ = S θ 0 + o S θ 0 + o O = O, uiformly for ξ J, as θ 0 0, π is bouded away from 0 ad π. A similar estimate holds for the si sum. We deduce that w µ, ν, ξ K µ,ν x, y = { cos ρµ ξ ρ ν ξ S b a π cos b a π si ρ µ ξ ρ ν ξ S b a π si b a π = b a b a S π cos π + ρ µ ξ ρ ν ξ + o, } + o uiformly for ξ J ad a, b i compact subsets of the real lie. Next, we exted this to complex a, b. For a give ξ, let w µ, ν, ξ f a, b = K µ,ν ξ + aπ ξ, ξ + bπ ξ.

9 9 Lemma. Assume the extra hypothesis i Propositio. that µ ad ν C i I. The for each ξ J, {f } are a ormal family, that is, {f a, b} are uiformly bouded for a, b i compact subsets of C. Proof We sketch the proof - full details i a very similar situatio are give i [, pp ]. Sice ξ is bouded below for ξ I, as are µ, ν, our assumed asymptotics for p µ, p ν give The also sup t I p µ t C ; sup p ν t C. t I sup K µ u, v C. u,v I Next, recall Berstei s iequality P z z + z P, L [,] valid for polyomials P of degree, ad z C. Let r > 0. Usig this o subitervals J of the iterior of I, separately i u ad v, yields for u, v J, complex a, b r, ad 0 r u + a, v + b C e C a + b. Kµ Here C ad C are idepedet of u, v,, r, a, b. A similar iequality holds for K, ν ad the Cauchy-Schwarz yields for u J, a, b r, ad 0 r, Kµ,ν u + a, u + b Kµ u + a, u + ā / Kν u + b, u + b / C e C a + b C e Cr. It the follows easily that {f a, b} is uiformly bouded for a, b i compact subsets of the plae. Proof of Propositio. for complex a, b We already have the result for real a, b. The uiform boudedess of the sequece {f } esures that they are a ormal family, while both sides of.4 are etire i a, b. Covergece cotiuatio theorems give the result i geeral. Remark Let us cast the three term recurrece relatios for p µ ad p ν i the followig form:. xp µ x = α µ + pµ + x + βµ p µ x + α µ p µ x ; xp ν x = α ν +p ν + x + β ν p ν x + α ν p x x,

10 0 DORON S. LUBINSKY with p µ = pν = 0. We also defie a "remaider term" [.3 j x, y = p µ j x pν j y αµ j+ +p µ j x pν j y α µ j+ α ν j+ α ν j α ν j+ y + α µ j+ αµ j α µ j+ β µ j α µ j+. βν j α ν j+ The the proof of the Christoffel-Darboux formula via the recurrece relatio also shows that.4 x y K µ,ν x, y = α µ p µ x p ν y p µ x pν y j x, y. Oe ca use this to prove scalig limits for K µ,ν ] x, y, at least whe j x, y coverges uiformly for x, y i a eighborhood of ξ. However, the method above is more geeral. 3. A B C E Our basic tool is cotaied i the followig theorem. For ay positive measure ω, ad δ 0, defie 3. L ω, δ = {x : ω x > δ}. Recall that meas deotes liear Lebesgue measure. Theorem 3. Let µ, ˆµ, ω be positive measures o the real lie with compact support. Let ad 3. η = + K ˆµ x, x dµ x Let ε, R > 0. The there exist sets E ad F, satisfyig γ ˆµ j γ µ j 3.3 meas E < η /4 + ε; meas F < ε, ad such that for x L µ, η /4 \E ad y L ω, 0 \F, ad all complex a, b, with a, b R, 3.4 K µ,ω K ˆµ,ω x + a, y + b. Here Ĉ is idepedet of, x, y, a, b, µ, ˆµ, η, but depeds o ω, ε ad R. The depedece of Ĉ o R is explicitly give i the proof of Theorem 3., but will ot be used i the sequel. We use i a essetial way, ideas from [3]. We begi with: Lemma 3. Let, ε > 0, ad P u, v be a cotiuous fuctio of two variables u, v C. Assume that V C is bouded, ad that for each v V, P u, v is a polyomial.

11 of degree i u. Let K R be bouded ad have positive Lebesgue measure, such that 3.5 P u, v for u K ad v V. The there is a set H,ε with meas H,ε < ε, ad 3.6 P x + z, v e C z /ε for all x K\H,ε, all z C, ad all v V. The costat C is idepedet of K, V,, x, z, v, P, ε. The set H,ε depeds o K,P,V but ot o x, z, v. Remark The essetial feature is that the same exceptioal set H,ε works for all v V. Proof Step : Some classical tools First recall the otio of the equilibrium measure ν ˆK of a compact set ˆK. This is a probability measure supported o ˆK that miimizes log z t dρ z dρ t amogst all probability measures ρ supported o ˆK. The Gree s fuctio g C\ ˆK z with pole at is a fuctio harmoic o C\ ˆK with boudary values 0 suitably iterpreted o ˆK, that behaves like log z + O as z. Readers lackig the potetial theory backgroud ca refer to [5], [7]. Next, the maximal fuctio of a measure such as ν ˆK is M [ dν ˆK] x = sup h>0 h x+h dν ˆK. x h Fially, H deotes the maximal Hilbert trasform, defied by H [ dν ˆK] x = sup t x dν ˆK t. ε>0 t x ε Step : Replace K by a set ˆK cosistig of fiitely may itervals I order to apply classical bouds o H, we wat K to be compact ad ν K to be absolutely cotiuous. This is true if K cosisted of fiitely may itervals i such a case ν K is eve aalytic i the iterior of K [7, p. 4]. So we replace K by a larger set ˆK cosistig of fiitely may itervals. First, recall that P is uiformly cotiuous i compact subsets of C, ad K,V are bouded. Choose a iterval I cotaiig K. The we ca fid fiitely may balls B, B,..., B m that cover V ad have ceters i V, ad such that 3.7 v, v B j sup P t, v P t, v. t I Suppose a j is the ceter of B j for each j. Now as P t, a j is a polyomial of degree i t, the set K j = {t R : P t, a j } cosists of at most itervals. Also, by our hypothesis 3.5, K K j. Let m ˆK = I K j. j=

12 DORON S. LUBINSKY This set cosists of fiitely may itervals, some of which may reduce to a poit. As K ˆK, ad K has positive measure, some of the itervals i ˆK have positive legth. The ν ˆK is absolutely cotiuous. Moreover, for t ˆK ad v B j, we have that t I K j, so by 3.7, so P t, v P t, a j +, P t, v for t ˆK ad v V. Step 3: Apply Classical Potetial Theory Estimates By the Berstei-Walsh iequality [5, p. 56, Theorem 5.5.7], we have for ay x, for z C ad v V 3.8 P x + z, v e g C\ ˆK x+z. Next, by Lemma 4. i [3, p. 3, Lemma 4.], if x is a regular poit of the set ˆK, i the sese of potetial theory, we have the boud g C\ ˆK x + z 6 z M [ dν ˆK] x + Re z H [ dν ˆK] x, for all complex z. Iasmuch as ˆK cosists of fiitely may itervals, at most fiitely may poits are ot regular poits. Moreover, ˆK ad supp [ ν ˆK] are idetical except perhaps for fiitely may isolated poits i ˆK. Next, we use the fact that both the maximal fuctio ad the maximal Hilbert trasform are weak type,. That is, for λ > 0, [6, p. 37, Thm. 7.4] meas { x : M [ ] } 3 ν ˆK x > λ dν λ ˆK = 3 λ ; ad as ν K is absolutely cotiuous, [3, p. 30, Prop. 4.6; p. 34, Thm. 4.7], [7, p. 8 ff.] meas { x : H [ ] } C 0 dν ˆK x > λ dν λ ˆK = C 0 λ. Here C 0 is idepedet of ν ˆK ad λ, ad this is the oly place we eed ν ˆK to be absolutely cotiuous. Choosig λ = ε max {3, C 0}, we obtai a set H,ε of measure ε such that for x supp [ ν ˆK] \H,ε, all complex u, ad all v V, P x + z, v e 60 z max{3,c0}/ε. As K supp [ ν ˆK], except possibly for a set of measure 0, we are doe. Proof of Theorem 3. Now by orthogoality, = [ K µ,ω [K µ = x, t K ˆµ,ω x, t dω t ] dµ x x, x K µ,ˆµ γ ˆµ j γ µ j + x, x + K ˆµ x, x ] dµ x K ˆµ x, x dµ x = η.

13 Let δ = η /4 for which i the sequel. Also, let E, deote the set of x L µ, δ = L µ, η / K µ,ω x, t K ˆµ,ω x, t dω t η. By our lower boud of δ for µ i L µ, δ, we have Thus η δmeas E, [ K µ,ω x, t K ˆµ,ω x, t ] dω t dµ x η, E, 3.0 meas E, η δ = η /4. Next, for polyomials P of degree, ad all complex y, we have the Christofel fuctio iequality P y K ω y, ȳ P dω. Applyig this to P t = K µ,ω ad all complex y, 3. K µ,ω K ω y, ȳ x, t K ˆµ,ω x, t, we obtai for x L µ, δ \E,, x, y K ˆµ,ω x, y K µ,ω K ω y, ȳ η. x, t K ˆµ,ω x, t dω t Now we apply Corollary 4.3 i [3, p. 3]: there is a set F of liear Lebesgue measure less tha ε, depedig o ad ω, such that for ξ L ω, 0 \F ad all complex b with b R, Kω ξ + b, ξ + b C e C b /ε C e CR/ε. Here C ad C are idepedet of, b, ξ, R, µ, ˆµ ad C is idepedet of ε, but C does deped o ε. Thus for x L µ, δ \E,, ad for ξ L ω, 0 \F ad all b R, 3. Kµ,ω x, ξ + b K ˆµ,ω x, ξ + b C e CR/ε η /4. Next we apply Lemma 3. to the polyomial P t, v = K µ,ω t, v K ˆµ,ω t, v / C e CR/ε η /4 As our set K, we take L µ, δ \E,, ad as our set V, we take { ξ + b : ξ L ω, 0 \F ad b R }. Iasmuch as K ad V are bouded sets, ad P is cotiuous as a fuctio of two variables o C, the hypotheses of Lemma 3. are satisfied. The we obtai a set H,ε with meas H,ε < ε, ad K µ,ω K ˆµ,ω x + z, ξ + b. C e CR+C0 z /ε η /4

14 4 DORON S. LUBINSKY for all x L µ, δ \ E, H ε, all z C, ad all v = ξ + b V. I particular, for a R, K µ,ω K ˆµ,ω x + a, ξ + b C e C3R/ε η /4. Replacig ξ by y, settig Ĉ = C e C3R/ε ad E = E, H,ε gives the result. 4. P T.4 We shall approximate the measures µ, ν i Theorem.4, by measures ˆµ, ˆν to which Propositio.. ca be applied. We begi with Lemma 4. Let R > 0, ε 0, ad µ be as i Theorem.4, with the additioal restrictio that µ is absolutely cotiuous. Let ω be a measure satisfyig ω > 0 a.e. i,. There is a measure ˆµ with support,, that satisfies the Szegő coditio as i.5, that has ˆµ aalytic i + ε, ε, ad i additio satisfies the followig: there exists 0 ad for 0, a set Ẽ such that for x, \Ẽ, ad all complex a, b with a, b R, 4. K µ,ω K ˆµ,ω while 4. meas Ẽ 3ε. x + a, x + b ε, Proof We shall use Theorem 3., ad break the proof ito several steps. We shall fix a auxiliary small parameter ε 0, ε. Later o, we shall choose this small eough depedig o the give R ad ε. Step : Limits that idicate how to choose ˆµ We first ote that if our approximatig measure ˆµ is absolutely cotiuous, ad satisfies for some A >, 4.3 Aπ x ˆµ x A π, x,, x the by the mootoicity of Christoffel fuctios/ reproducig kerels, ad bouds for the Christoffel fuctios of the Chebyshev weight, we have [6, pp. 03-5, Lemma 3., Theorem 3.4] C A K ˆµ x, x C A, x [, ], where C ad C are idepedet of ˆµ,, x, A. Moreover, we have by results of Maté, Nevai, ad Totik [4, p. 449, Theorem 8 ], lim K ˆµ x, x = π x ˆµ x, a.e. x,. Hece, by Lebesgue s Domiated Covergece Theorem, lim K ˆµ x, x dµ x = π µ x x ˆµ x dx.

15 5 It is oly here that we eed µ to be absolutely cotiuous. Next, Szegő asymptotics for leadig coeffi ciets of orthogoal polyomials [9, p. 309, eq..7.], [6, p. 04, p. 45] yield γ ˆµ lim γ µ = exp π The the average also has the same limit: γ ˆµ j lim γ µ = exp j π so 4.4 lim η = η := + log ˆµ x dx µ x x. log ˆµ x dx µ, x x π µ x x ˆµ dx exp log ˆµ x dx x π µ. x x Step : Choosig ˆµ We wat η to be small. Let δ 0, ε be so small that all of 4.5 µ x + dx < ε x 8 ; S x δ x δ log µ x x δ dx x < ε 8 ; log x dx < ε x 8. Next let M > 0 be a umber so large that if S = {x [ + δ, δ ] : log µ x M}, 4.8 log µ dx x < ε x 8 ; µ dx x + < ε x 8. For x δ, defie M, if log µ x < M; g x = log µ x, if log µ x M; M, if log µ x > M. S The g M, ad we ca fid a polyomial P such that P M i [ + δ, δ ], ad 4.9 e 3M δ We set ad +δ g x P x ˆµ x = dx x < ε 8. x, δ x < ˆµ x = e P x, x < δ. Note that ˆµ satisfies 4.3 for some A > 0, so 4.4 holds. Also, ˆµ is aalytic i + ε, ε, recall that δ < ε < ε.

16 6 DORON S. LUBINSKY Step 3: Estimates ivolvig log µ x log ˆµ x Now, δ log µ x log ˆµ x dx +δ x [ +δ, δ ]\S + S < ε g x P x log µ x + M log µ x S dx x dx x dx x < ε. Here we have used 4.9, 4.8 ad that log µ M i S. Also, by 4.6, 4.7, log µ x log ˆµ x dx x δ x ε 8 + log x dx < ε x 4. Thus 4.0 x δ log µ x log ˆµ x dx < 3 x 4 ε. Usig the iequality e t t for t, we obtai 4. exp π log ˆµ x dx µ < 3 x x π 4 ε < ε 4. Step 4: Estimates ivolvig µ ˆµ Next, for x [ + δ, δ ] \S, log µ x log ˆµ x 3M, so the iequality e t t e t, t R, gives = [ +δ, δ ]\S [ +δ, δ ]\S e 3M = e 3M [ +δ, δ ]\S [ +δ, δ ]\S x x µ x ˆµ x dx e log µ x log ˆµ x dx log µ x log ˆµ x dx x g x P x dx x < ε 8, by 4.9. Also, if S + = {x [ + δ, δ ] : log µ x M} µ x S + x ˆµ x dx = µ x S + x e M dx µ x dx < ε x 4, S +

17 7 by 4.8, while if S = {x [ + δ, δ ] : log µ x M} µ x S x ˆµ x dx = µ x S x e M dx dx < ε x 4, x δ S by 4.8 agai. Fially, µ x x δ x ˆµ x dx = µ x + dx ε x 4, x δ by 4.5. Combiig the above iequalities gives π µ x x ˆµ x dx < ε 8 + ε 4 + ε 4. 4 < ε. Together with 4., ad recallig 4.4, this gives η < ε + ε = 3 ε. Step 5: Completio of the Proof Choose 0 such that for 0, η < ε. µ x x dx x The usig Theorem 3., for the give ε, R > 0, there exist for > 0, sets E ad F, such that for x L µ, η /4 \E, \F, ad all complex a, b, with a, b R, K µ,ω K ˆµ,ω x + a, x + b Ĉη/4 Ĉε /4. Here, Ĉ depeds o R, ε, but is idepedet of, x, a, b, µ, ˆµ. Crucially, also, ε is idepedet of R ad ε. Set Ẽ = E F, \L µ, ε /4. By 3.3, meas Ẽ ε /4 + ε + meas, \L µ, ε /4. We ow choose ε so small that this last right-had side is less tha 3ε, while also Ĉε /4 < ε.. Proof of Theorem.4 for absolutely cotiuous µ ad ν We assume that µ ad ν are absolutely cotiuous. Let ε, R > 0. Choose a measure } ˆµ approximatig µ as i Lemma 4. with correspodig exceptioal sets {Ẽ as i 4.. { Similarly } choose a measure ˆν approximatig ν ad correspodig exceptioal sets F. We shall also use the estimate 4.0 ad its aalogue for ν, as well as the umber δ < ε from the proof of Lemma 4.. Sice the pair ˆµ, ˆν satisfies

18 8 DORON S. LUBINSKY the requiremets of Propositio., ad Example., for ξ i compact subsets of + ε, ε, ad uiformly for a, b i compact subsets of the plae, we have w ˆµ, ˆν, ξ lim K ˆµ,ˆν ξ + aπ ξ, ξ + bπ ξ b a π b a 4.3 = S cos + ρˆµ ξ ρˆν ξ, where 4.4 ˆB ξ := ρˆµ ξ ρˆν ξ = 4π P V π π log ˆµ cos t ˆν cos t cot θ t dt. Next, uiformly for ξ i [ + ε, ε], ad complex a, b with a, b R, w ˆµ, ˆν, ξ K µ,ν ξ + aπ ξ, ξ + bπ ξ a b π a b S cos w ˆµ, ˆν, ξ [ K µ,ν K ˆµ,ν ] ξ + aπ ξ, ξ + bπ ξ w ˆµ, ˆν, ξ [ + K ˆµ,ν K ˆµ,ˆν ] ξ + aπ ξ, ξ + bπ ξ w ˆµ, ˆν, ξ + K ˆµ,ˆν ξ + aπ ξ, ξ + bπ ξ a b S [ + a b π a b S cos + ˆB π a b ξ cos + B ξ] = : w ˆµ, ˆν, ξ T + T + o + T 3, 4.5 where we have used 4.3. Here, by Lemma 4. applied to µ, ˆµ, ν, we have for a, b R, ad ξ, \Ẽ, 4.6 T = K µ,ν K ˆµ,ν ξ + aπ ξ, ξ + bπ ξ ε, ad by that lemma applied to µ, ˆµ, ˆν, we have for a, b R, ad ξ, \ ˆF, 4.7 T = K ˆµ,ν K ˆµ,ˆν ξ + aπ ξ, ξ + bπ ξ ε. Next, [ T 3 = a b π a b S cos + ˆB π a b ξ cos + B ξ] ˆB ξ B ξ si ˆB ξ B ξ B ξ π a b cos + ξ ˆB

19 9 For ξ = cos θ, ˆB ξ B ξ = 4π P V = 4π P V π π π π [ log ˆµ cos t ˆν cos t log µ cos t ν cos t [ log ˆµ cos t µ cos t log ˆν cos t ν cos t ] cot θ t dt ] cot θ t dt. Next, we use the fact that the cojugate fuctio operator is weak type, [3, p. 60, Thm. 6.8], so that for λ > 0, { meas θ 0, π : π 4π P V log ˆµ cos t π µ cos t cot θ t } dt > λ C π 0 λ log ˆµ cos t µ cos t dt C 0 λ ε C 0 λ ε, ad similarly π { meas θ 0, π : 4π P V π π log ˆν cos t ν cos t cot θ t } dt > λ C 0 λ ε. Here C 0 is a absolute costat, ad we are usig 4.0 for µ, ν. We choose λ = ε, ad let { G = ξ, : ˆB ξ B ξ > ε /}, so that by the above iequalities, measg C 0 ε /. Now let H = Ẽ F G [ + ε, ] [ ε, ]. We see that for large eough, we have meas H ε + C 0 ε / + 4ε 6 + C 0 ε /. Moreover, for 0, ξ, \H, ad a, b R, give w ˆµ, ˆν, ξ K µ,ν ξ + aπ ξ, ξ + bπ ξ a b π a b S cos w ˆµ, ˆν, ξ ε + ε + ε /. Fially, w ˆµ, ˆν, ξ is close to w µ, ν, ξ except o a set of small measure. Ideed, 4. shows that { meas x, : µ } x ˆµ x > ε/ < πε /. A similar iequality holds for ν ad ˆν. Sice ε < ε, we are fiished. Proof of Theorem.4 for the case where µ, ν may have sigular parts Let µ ac ad ν ac deote the absolutely cotiuous parts of µ ad ν respectively. The coclusio of Theorem.4 holds for K µ ac,νac. We write K µ,ν = [K µ,ν K µ ac,νac K µ,νac ] + [K µ,νac K µ ac,νac ] + B ξ

20 0 DORON S. LUBINSKY ad use Theorem 3. to show that each of the terms i [] is small outside a set of small measure. Let us illustrate o the first term. Now sice ν ac ν, we have K νac x, x K ν x, x for x R, by the variatioal property of reproducig kerels alog the diagoal. The K ν x, x dν ac x K νac x, x dν ac x =. We apply Theorem 3. to the measures ν, ν ac, µ. The η of 3. becomes η = + K ν x, x dν ac x γ ν j γ νac j γ ν j γ νac j Now Szego s theorem for the leadig coeffi ciets gives, as above, Hece By Theorem 3., give R, ε > 0, lim j γ ν j γ νac j. =. lim η = 0. sup K µ,ν K µ,νac a, b R = sup K ν,µ K νac,µ a, b R sup K µ,νac K µ ac,νac a, b R x + a, x + b x + a, x + b < ε, outside a set of small measure. A similar estimate holds for x + a, x + b. 5. P C.5-.7 Proof of Corollary.5 Our hypothesis.0 o µ ad ν esures that uiformly for x i I, we have Cauchy-Schwarz the shows that K µ x, x C ad K ν x, x C. sup x,y I, Kµ,ν x, y C. Also w µ, ν, ξ is bouded above for ξ I. Combiig the covergece i measure i Theorem.4 ad this last boud easily yields the result. Proof of Corollary.6

21 The mai idea is to use Cauchy s estimates for Taylor series of aalytic fuctios. First we expad the right-had side of.4 as a double Taylor series i a, b. Now π a b π a b Ψ a, b = S cos + B ξ si πa b = S π a b cos B ξ si B ξ. πa b Here usig Euler s formula si u = i e iu e iu, the biomial expasio, ad some elemetary maipulatio, we see that S π a b = si πa b πa b = l,m=0 l,m=0 a l b m l! m! πl+m τ l,m ; a l b m l! m! πl+m ρ l,m, where {τ l,m }, { } ρ l,m are defied respectively by.3 ad.4. Now let, w µ, ν, ξ a, b = K µ,ν ξ + aπ ξ, ξ + bπ ξ Ψ a, b. Usig a double Taylor series expasio o K µ,ν, ad those above, we see that = a, b l+m a l b m l! m! πl+m wµ,ν,ξ ξ K µ,νl,m ξ, ξ l,m=0 τ l.m cos B ξ ρ l,m si B ξ. Fially, Cauchy s iequalities show that for a give l, m, l+m π l+m w µ, ν, ξ ξ K µ,νl,m ξ, ξ τ l.m cos B ξ ρ l,m si B ξ sup a, b /r l+m. a, b r Now we itegrate for ξ over I, ad use Corollary.5. For poitwise covergece, Cauchy s estimates directly give the result. Proof of Corollary.7 This follows from Hurwitz s theorem, as S πb a = 0 whe, ad oly whe b = a+j, while cos πb a + B ξ = 0, whe ad oly whe b = a+j + π B ξ. Ackowledgemet The author thaks a aoymous referee for observig that i the origial versio of the article, the proof of Theorem.4 failed whe µ, ν had sigular parts.

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