Asymptotics of recurrence coefficients for orthonormal polynomials on the line -Magnus s method revisited

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1 Asymptotics of recurrece coefficiets for orthoormal polyomials o the lie -Magus s method revisited S B Dameli 30 August, 2002 Abstract We use Freud equatios to obtai the mai term i the asymptotic expasio of the recurrece coefficiets associated with orthoormal polyomials p (w 2 ) for weights w = W exp( Q) o the real lie where Q is a eve polyomial of fixed degree with o egative coefficiets or where Q(x) = exp(x 2m ), m Here W(x) = x ρ for some real ρ > AMS(MOS) Classificatio: 42C05 Keywords ad Phrases: Asymptotics, Etire Fuctios of Fiite ad Ifiite Order, Erdős Weights, Freud Weights, Orthogoal Polyomials, ecurrece Coefficiets Itroductio The idea of this article arose out of two papers of Magus i 986, see [] ad [2] who used Freud equatios to study the asymptotic behaviour of recurrece coefficiets for orthoormal polyomials p (w 2 ) o the real lie where w = W(x)exp( x 2m ), m N ad W(x) = x ρ for some real fixed ρ > Usig these equatios, Magus established a importat special case of Freud s cojecture We refer the reader to [], [3], the refereces cited therei ad our discussio below for a comprehesive history of this cojecture Freud differece equatios are closely related to discrete Pailevé equatios ad have foud may applicatios i approximatio theory ad more recetly i mathematical physics, see [3] ad the may refereces cited therei Our mai objective i this paper wil be to show that that the methods of Magus i the above papers may be improved to obtai the mai term i the asymptotic expasio of the recurrece coefficiets associated with orthoormal polyomials p (w 2 ) for weights w = W exp( Q) o the real lie where Q is a eve polyomial of fixed degree with o egative coefficiets ad where Q(x) = exp(x 2m ), m Here W(x) = x ρ for some real ρ > The polyomial case (which we will deote

2 by Q P ) provides a alterative proof to a earlier case studied by Bauldry, Máté ad Nevai, see [2] I this case, sharper results tha ours have bee obtaied by Bleher ad Its, see [3] ad Deift, Kriecherbauer ad McLaughli i [4] whe ρ = 0 I these later papers, the aalyticity of w is exploited together with Lax-Levermore theory ad iema Hilbert techiques The expoetial case (which we will deote by Q E ) is ew eve i the case ρ = 0 Oe of the mai objectives i this paper is to show that our method of proof alows us to treat eve weights with varyig rates of smooth decay at ifiity ad with possible poits of o aalyticity The o eve aalogues of our results require further ideas ad will appear i a forthcomig paper We refer the reader to [3], [4], [0], [3], [7] ad the may refereces cited therei for a detailed accout of the history of this subject We fiish this sectio with a short descriptio of the structure of this paper ad some eeded otatio Because Q P ad Q E are of differet atures, we choose to structure the paper i a way that they are delt with separately This paper is thus orgaised as follows: I Sectio 2, we state our mai results I Sectio 3, we we illustrate that the recurrece coefficiets for Q P satisfy a differece equatio which admits uique solutios which may be well approximated ad we carefully approximate these solutios by sequeces ivolvig the scaled edpoits of the equilibrium measure for exp( 2Q P ) usig a improved techique which is of idepedet iterest I Sectio 4, we prove our mai result for Q P I Sectio 5, we show that the recurrece coefficiets for Q E are a solutio (for ρ > ) ad a uique solutio (for ρ = 0) of a Freud differece equatio ad i Sectio 6, we we idicate how to adapt the techiques used for Q P to show that these later solutios may be well approximated I this sectio, we also preset the proof of our mai result for Q E For ay two sequeces (b ) ad (c ) of oegative real umbers, we shall write b = O(c ), if there exists a costat C > 0 such that for all ad b Cc, b c if there exists two costats 0 < C C 2 < such that for all C b /c C 2 Similar otatio will be used for fuctios ad sequeces of fuctios 2 Mai result I this sectio, we state our mai result 2

3 Give a weight w as above, we may defie a uique system of orthoormal polyomials, see [8], satisfyig ad deote by p (w 2, x) := γ x +, γ := γ (w 2 ) > 0 p (w 2, x)p (w 2, x)w 2 (x)dx = δ m, m, 0 < x, (w 2 ) < x, (w 2 ) < x 2, (w 2 ) < x, (w 2 ) < the simple zeros of p (w 2 ) We write the three term recurrece of p (w 2 ) i the form xp (w 2, x) = α + p + (w 2, x) + α p (w 2, x), 0 Here, p (w 2 ) = 0, p 0 (w 2 ) = ( w(x)dx ) /2, α (w 2 ) = γ /γ > 0 ad α 0 = 0 are the associated recurrece coefficiets Let us put α := α(w) = (α ) = ad let 0 α α 0 α 2 J(α) = α 2 0 α 3 α 3 0 α 4 be the ifiite Jacobi matrix associated with the recurrece coefficiets α The asymptotic behaviour of the recurrece coefficiets is expressed i terms of the scaled edpoits of the equilibrium measure for exp( 2Q), = ( ) =, where a u = a u (exp( 2Q)) is the positive root of the equatio u = 2 π 0 a u tq (a u t) t 2 The umber a u is, as a real valued fuctio of u, uiquely defied ad strictly icreasig i (0, ), see [9] ad [7] Followig are our mai results: Theorem 2a Let w = W exp( Q P ) where Q P is a eve polyomial of fixed degree ad with o egative coefficiets The [ ( )] α = /2 + O, (2) dt 3

4 ad α + α = + O ( ), (22) Theorem 2b Let w = W exp( Q E ) where Q E (x) = exp(x 2m ), m The [ ( )] α log = /2 + O, (23) ad α + α ( log = + O ), (24) 3 The Freud equatio ad its approximatio for Q := Q P I this sectio we will show that the recurrece coefficiets geerated by w satisfy a Freud equatio whose solutios may be well approximated Let us write M Q(x) = x 2mj c j, m, x for some o egative sequece {c j } ad fixed M so that Q (x) = 2m M x 2mj jc j Formally applyig Q to the operator J(α) we lear that (Q (J(α))) i,j = Q (x)p i (x)p j (x)w(x)dx The right had side of the later equatio above is easily iterpreted as the spectral represetatio of the left had side Note that Q (J(α)) is a matrix Throughout, (J(α)) i,j will deote the elemet of the (i + )th row ad (j + )th colum of the matrix J(α) ad (Q (J(α))) i,j will deote the elemet of the (i + )th row ad (j + )th colum of the matrix Q (J(α)) Similar covetios will be used i the sequel Motivated by the discussio above, we defie a sequece of operators F ( N) actig o sequeces x = (x ) = of positive real umbers, by F (x) := x (Q (J(x))), =, 2, 3 (3) The Freud equatios for obtaiig the sequece α of recurrece coefficiets for the orthoormal polyomials with weight w = W exp( Q) from the operators defied above is the give by F (α) = + odd() =, 2, 3 (32) 4

5 where odd() is if is odd ad 0 if is eve It follows the quite easily usig the method of [2, pg 367] ad [, Lemma 23] that the followig holds: Lemma 3 The recurrece coefficiets α = (α, α 2, α 3, ) are a uique positive solutio of the Freud equatios (32) for the weight w We ow approximate F (α) by a sequece of operators related to F (a/2), where a = ( /2) = are the uique scaled edpoits of the equilibrium measure for exp( 2Q) The idea is similar to the otio of a cosistet approximat solutio first used i [] but the geeralizatio preseted here works for eve weights of polyomial ad of faster tha polyomial decay at ifiity ad allows for improved estimates Let us defie polyomials p := x + of degree, 0 by the recurrece xp (x) := + 2 p +(x) + 2 p (x), = 0,, where we may take p 0 = ad p = 0 By Favard s theorem ([8], Theorem 25 ), there exists a positive measure µ with all momets fiite such that these polyomials are orthogoal with respect to µ Without loss of geerality, we may assume that p are suitably scaled so that they are orthoormal with respect to µ ad satisfy the recurrece above We ote that p depeds o the scaled edpoits of the support of the equilibrium measure for exp( 2Q) Let 0 a /2 a /2 0 a 2 /2 J(a/2) = a 2 /2 0 a 3 /2 a 3 /2 0 a 4 /2 be the ifiite Jacobi matrix associated with the recurrece coefficiets ( /2) =0 The the Freud operator applied to the sequece (a /2, a 2 /2, a 3 /2, ) gives F (a/2) = 2 (Q (J(a/2))), (33) = /2 Q (x)p (x)p (x)dµ (x), We shall prove: Theorem 32 For, we have F (α) F (a/2) = O () (34) 5

6 For the proof of Theorem 32 we eed some preparatory material ad lemmas The Jacobi matrix for the Chebyshev polyomials of the secod kid U (x) = si( + )θ, x = cosθ, si θ which are orthoormal with respect to the weight 2 π x2 o [, ], is give by 0 /2 /2 0 /2 J 0 = J(/2) = /2 0 /2 (35) /2 /2 /2 We recall that 2 π x k U (x)u (x) x 2 dx = (J k 0 ),, k (36) Let /2 be the costat sequece ( /2, /2, /2, ), which should ot be cofused with the sequece a/2 = (a /2, a 2 /2, a 3 /2, ) The the Jacobi matrix for /2 is give by J( /2) = J 0, (37) ad F s ( /2) = 2 (Q ( J 0 )) s,s (38) Our mai idea for provig Theorem 32 is to approximate the sequece F s (a/2) by F s ( /2) whe s is close to This is achieved by the followig lemma Lemma 33 We have for M F (a/2) F ( /2) = O jc j a 2mj where for positive itegers k ad k, = a k x k p (x)p (x)dµ (x) 2/π 2mj, (39) x k U (x)u (x) x 2 dx (30) Proof: Observe that F (a/2) = a M 2 2mj x 2mj p (x)p (x)dµ (x) 6

7 ad F ( /2) = 2 The the lemma follows 2 Q ( x)u (x)u (x) x π 2 dx We ow proceed to estimate k, for positive itegers k ad ad to this ed we make use of a idea of [8] The author thaks Walter Va Assche i this regard Lemma 34 For positive itegers k ad C k, (2k ) (3) ( max +j k j k 2 + ) +j k ( ) k j 2 x,+k where x,+k is the largest zero of p +k Proof: First observe that k, = a k (J(a/2) k ), (J0 k ), Hece we will estimate the differece betwee the operators J(a/2) k /a k ad J0 k ecallig the idetity k A k B k = B j (A B)A k j, ad lettig e = (0, 0, 0,, 0,, 0, 0, ) be the th uit basis vector we write (J(a/2) k ), (J0 k ), [ ] = e ( ) k J(a/2)k J0 k e k [ ] ( ) k j = e J j 0 J(a/2) J 0 J(a/2) e a k Here e Jj 0 is the th row of Jj 0 which, by usig (36) has the form ( ) (J j 0 ),0 (J j 0 ), (J j 0 ),2 (J j 0 ),+j ( 2 = x j 2 U U 0 x2 dx x j U U x2 dx π π = 2 π x j U U 2 x2 dx ( π ) x j U U +j x2 dx 2 x j U (x)u j (x) x π 2 dx 2 x j U (x)u +j (x) x π 2 dx 0 7 ),

8 where we used orthogoality Thus e Jj 0 spa{e j,, e +j } which is a subset of spa{e k,, e +k } wheever j k Similarly (J(a/2)/) k j e gives the ( )th colum of the matrix (J(a/2)/ ) k j, ad it has the form () k j xk j p (x)p k+j (x)dµ (x) () k j xk j p (x)p +k j 2 (x)dµ (x) 0 so that it belogs to spa {e k+j,, e +k j 2 } which for every j 0 is a subset of spa{e k,, e +k } Let P,k be the projectio matrix P,k = 0 k I 2k the it follows that (J(a/2) k ), (J0 k ), a k k = e J j 0 P,k ( ) J(a/2) J 0 P,k (J(a/2)/ ) k j e Observe that the matrix P,k (J(a/2)/ J 0 )P,k coverges to 0 for every fixed k as i ay orm Now, by the Cauchy-Schwarz iequality ad the relatio Ax 2 A 2 x 2 we see that k e J j 0 P,k ( ) J(a/2) J 0 P,k (J(a/2)/ ) k j e k P,k (J(a/2)/ J 0 )P,k 2 e Jj 0 2 (J(a/2)/ ) k j e 2 We begi with the estimate of e J j 0 2 = j x i 2, i= j 8

9 where x i = 2 π Here, for each j i j x i 2 π by Cauchy-Schwarz Hece x j U (x)u +i (x) x 2 dx U (x)u +i (x) x 2 dx, e Jj 0 2 2j + 2k Next we estimate (J(a/2)/ ) k j e 2 = where y i = k j i= k+j+ y i 2, ( ) k j x k j p (x)p +i (x)dµ (x) We recall the Gauss-Jacobi quadrature [8] P(x)dµ (x) = λ l, P(x l, ), l= for every polyomial P of degree at most 2, with λ l, = 2 p (x > 0, l l,)p (x l, ) The, sice x k j p (x)p +i (x) is a polyomial of degree k j+2+i 3 2(k + ) 4, y i = +k ( ) k j λ l,+k x k j l,+k p (x l,+k )p +i (x l,+k ) ( x,+k l= l= ) k j +k λ l,+k p 2 (x l,+k) l= +k λ l,+k p 2 +i (x l,+k) ( x,+k ) k j, 9

10 where we used Cauchy-Schwarz ad Gauss-Jacobi agai Thus (J(a/2)/ ) k j e 2 2k 2j ( ) k j x,+k Fially, we estimate 2k ( x,+k ) k j P,k (J(a/2)/ J 0 )P,k 2 P,k (J(a/2)/ J 0 )P,k ( max +j k j k 2 + ) +j 2 The, combiig our estimates, we have (3) ad the lemma is proved We ow use Lemma 34 to estimate (39) Lemma 35 For C we have F (a/2) F ( /2) = O() (32) Proof: First observe that for a give positive iteger k k ( ) k j k x,+k ( ) k j ( ) k j x,+k a+k = a a +k Next we observe that [6, Theorem 7] implies that x,+k +k /2 2 It is importat at this poit to metio that Freud s result holds for the special system of orthogoal polyomials geerated by the measure µ ad with recurrece coefficiets ( /2) = Thus we have usig the strict mootoicity of ( /2) = k ( ) k j k x,+k ( ) k j a+k k ( a+k ) k Isertig this estimate ito (3) ad usig (39) the gives for C F (a/2) F ( /2) (33) ( ) 2mj 2 = O j 2 c j a 2mj a+2mj max +i i 2mj We proceed to estimate (33) We make heavy use of the followig estimates which are well kow, see for example [9] 0

11 (a) Uiformly for u [v/2, 2v] ad v C a u a v u v (34) (b) For all v C ad u, (c) For u C, Now for C, a uv a v = O(u) (35) Q (j) (a u ) u a j, j = 0,, 2, 3 (36) u F (a/2) F ( /2) (37) M ( ) 2mj 2 = O j 2 c j a 2mj a+2j max +i a i 2mj = O a M ( j 3 c j a 2mj + 2mj ) 2mj 2 M = O a3 j 3 c j a 2mj 3 ( ) a 3 = O Q(3) ( ) = O() This establishes (32) ad the lemma We are ow i the positio to prove Theorem 32 Proof: Set for x M (x) := logx /2π π 0 Q(2x cos(θ))dθ The, see for example ([], Sectio 4), we always have Thus F ( /2) = ( /2)Q (J( /2)), = x x M (x) x=a/2 = 0 F ( /2) = 0 (38) Now recallig that the recurrece coefficiets are a uique solutio of (32) we may apply Lemma 35, (38) ad the triagle iequality to deduce the result

12 4 The Proof of Theorem 2a I this sectio, we preset the proof of Theorem 2a We make use of ideas of [] Proof: Let us write Q (x) := 2m M jc j x 2mj, x Choose ε > 0, a positive iteger m ad a real positive sequece γ = (γ ) = with the property that there exist positive costats C ad C 2 such that uiformly for C γ C 2 (4) Moreover let σ := {σ, σ 2, } (42) be ay real sequece havig oly fiitely may o-zero terms Usig Favard s Theorem, we use the sequece γ to geerate orthoormal polyomials p with respect to some o egative eve mass distributio d The if J(γ) is the Jacobi matrix correspodig to the recurrece coefficiets γ, we kow from Sectio 3 that if j is atural umber fixed for the momet the we have for every x 2mj p (x)p (x)d (x) = ((J(γ)) 2mj ), (43) Thus we may defie the associated Freud equatios by G,j (γ) = γ ((J(γ)) 2mj ),, =, 2, (44) We observe for later use that this sequece depeds oly o the fiite umber of terms {γ mj+, γ +mj } Now with this sequece of operators we defie the Fréchet derivative by H j (γ) := G,j(γ) log γ 2 G,j(γ) log γ 2 We deote the quadratic form associated with {σ, σ 2, } G,j(γ) log γ 2 G,j(γ) log γ 2 (45) 2

13 by: (H j (γ), σ) := First usig (34) we kow that = k= +, G,j (γ) σ σ k log γk 2 (46) Thus it follows from ([], Theorem 53) that the matrix (45) is symmetric ad that, subsequetly, there exists C depedig oly o C ad C 2 i (4) such that a 2mj σ 2 C(H j (γ), σ) (47) = Here it is crucial that the sequece cotais fiitely may o-zero terms {σ, σ 2, } The Fréchet derivative is used i the appreciatio of the distace betwee two sequeces a ad a i terms of F (a ) ad F (a ) We joi them by a rectliear path log γ (t) = log a + t (log a log a ), 0 t,, =, 2, ad itegrate alog the matrix (45) More precisely, we proceed as follows: We specialize our choice of sequeces defied above i the followig way First set σ := log α 2 log(/2) 2, =, 2, 3 (48) ad put, σ := (0, 0,,0, {σ } N +mj =v+ mj, 0, 0 ) (49) where N ad v are positive itegers with N v fiite Thus σ is a real sequece with fiitely may o zero terms Next we set for 0 t log γ 2 (t) = log(/2) 2 + tσ =, 2, (40) Observe that γ (t) > 0 for every t (0, ) ad moreover ([0], (20)) esure that (4) holds uiformly alog the path (40) Thus uiformly alog this path, (47) will hold with the choice of γ ad the sequece σ Next observe that usig well kow results cocerig fuctios of several variables, (see for example [5], Chapter 3) ad recallig the defiitio of the path (40), we may write 0 k= σ k G,j (γ) log γ 2 k dt = G,j (α) G,j (a/2) (4) 3

14 To see this it is useful to recall that ad G,j (α) = α ((J(α)) 2mj ), G,j (a/2) = /2((J(a/2)) 2mj ), are the same polyomials depedig oly o 2mj 2 terms i differet variables α ad /2 Moreover if we wish to write either of the above formulae usig their spectral represetatio, they become G,j (α) = α x 2mj p (x)p (x)w(x)dx ad Thus ad G,j (a/2) = /2 x 2mj p (x)p (x)dµ (x) 2m 2m M jc j G,j (α) = F (α) M jc j G,j (a/2) = F (a/2) Now we multiply (4) by σ for v N ad so we obtai 0 N +mj =v k= mj+ G,j (γ) σ σ k log γk 2 dt = N σ (G,j (α) G,j (a/2)) (42) which holds uiformly alog the path give by (40) Thus we have that 0 = k= G,j (γ) σ σ k log γk 2 dt = =v N σ (G,j (α) G,j (a/2)) (43) =v Now we apply the operator 2m M jc j to (43) This gives recallig that N v is fiite M ( N ) G,j (γ) 2m jc j σ σ k 0 log γk 2 dt = O σ F (α) F (a/2) =v = k= Now recallig (47) which holds for each fixed j ad the fact that for each fixed j N N +mj σ 2 a 2mj σ 2 a 2mj =v =v+ mj 4

15 gives 2m M N jc j =v σ 2 a 2mj = O ( N ) σ =v ( N = O =v σ 2 ) /2 ( N ) /2 (44) =v We may ow set v = ad N = + so that (44) becomes 2m M jc j a 2mj + σ + 2 = O ( σ + ) Thus it follows from the above, (34) ad (36) that ( ) σ = O Q ( ) ( ) = O ecallig, as we may, see [0] that ( ) log α /2 α /2 fially gives (2) (22) follows from (2) ad (34) 5 The Freud Equatio ad its approximatio for Q := Q E I this sectio we will show that the recurrece coefficiets geerated by w also satisfy a Freud equatio whose solutios may be well approximated Let us write Q (x) = 2m x 2mj jc j, x where c j := j! Applyig this to the operator J(α) we obtai the formula Q (J(α)) i,j = Q (x)p i (x)p j (x)w(x)dx Set: F (x) := x (Q (J(x))), =, 2, 3 (5) 5

16 The Freud equatios for obtaiig the sequece α of recurrece coefficiets for the orthoormal polyomials with weight w = W exp( Q) from the operators defied above is the agai give by F (α) = + odd() =, 2, 3 (52) where odd() is if is odd ad 0 if is eve We prove: Theorem 5 The recurrece coefficiets α = (α, α 2, α 3, ) are a uique positive solutio of the Freud equatios (52) for the weight w with ρ = 0 ad are a positive solutio for all ρ > Our proof will be broke dow ito several steps We begi with Lemma 52 which follows as i [2, Lemma ] Lemma 52 For ρ >, the recurrece coefficiets α are a solutio of the Freud equatios (52) Proof We observe that by orthogoality we obtai (p p ) (x)w(x)dx = p (x)p (x)w(x)dx Now we write p (x) = γ /γ p (x) + π 2 (x), where π 2 is some polyomial of degree at most 2 The it is easy to see that (p p ) (x)w(x)dx = /α O the other had itegratio by parts gives (p p ) (x)w(x)dx = Q (x)p (x)p (x)w(x)dx ρ odd() α = Q (J(α)), ρ α odd() which shows that F (α) = + odd() for We ow show that α for exp( Q) is the oly positive solutio of (52) This requires a extesio of Theorem 22 i [] where the proof there relies heavily o the polyomial decay of the uderlyig weight The followig lemma is essetial A earlier special case is actually due to Nevai, [5] Lemma 53 Let µ be a mootoe icreasig fuctio o (, ) with all momets ν j (j = 0,, 2, ) fiite, ad assume P(x)Q (x)dµ(x) = P (x)dµ(x), (53) 6

17 for all polyomials P Further assume that µ is ormalized so that µ( ) = 0 ad µ(+ ) = w(x)dx The µ is absolutely cotiuous o (, ) ad µ (x) = exp( Q(x)) for all x Proof Defie q by q Q (q ) =, It is easy to see that for every fixed positive k > 0 ad uiformly for We first show that for eve ν = q k (log ) /2m x dµ(x) = O(q2+2 ) (54) Ideed, we write ν = x dµ(x) + x q 2+2 x dµ(x) = I + I 2 x >q 2+2 The I q 2+2 dµ(x) = O(q 2+2) as by assumptio all the momets of µ are fiite Next, observe that for x q 2+2 the mootoicity of xq (x) o (0, ) gives Now (53) implies so that xq (x) ( + ) q 2+2 Q (q 2+2 ) ( + ) = (2 + 2) ( + ) = + (55) x + Q (x)dµ(x) = ( + )ν, x [xq (x) ( + )] dµ(x) = 0, (56) for all 0 Thus by (55) ad (56) I 2 x [xq (x) ( + )] dµ(x) + x q 2+2 = x [xq (x) ( + )] dµ(x) + x <q 2+2 = x dµ(x) x + Q (x)dµ(x) x <q x <q 2+2 x dµ(x) = O(q2+2) x <q 2+2 7

18 (54) the follows Next we show that (53) ad (54) imply that e itx Q (x)dµ(x) = it e itx dµ(x), (57) for every fixed t The idea follows ([], (28)) where we approximate e itx by its partial sums (itx)j /j! ad show that: ad lim, eve lim, eve eitx (itx) j /j! Q (x) dµ(x) = 0 (58) eitx (itx) j /j! dµ(x) = 0 (59) We prove (58) The proof of (59) is similar ad easier ecallig that Q (x) = 2m ad usig (54) ad the iequality e iu x 2jm jc j (50) (iu) j /j! u + ( + )!, u 8

19 we may write for ay 0 < δ < /2 e itx (itx) j /j! Q (x) dµ(x) (5) = O t + jc j x 2mj+ dµ(x) ( + )! = O t + jc j ν 2mj+ ] ( + )! = O t + + jc j q 2mj+ 4mj ( + )! +O t + jc j q 2mj+ 4mj+2+2 ( + )! j=+2 = O δ t + jc j + O t + ( + )! ( + )! ( δ t + ) ( ) t + = O + O ( + )! ( + )! = o(), j=+2 q 2mj+ 4mj+2+2 (j )! Thus (58) is established ad so (57) the follows easily Lemma 53 the follows quite easily usig the method of [, Theorem 22] ad the fact that Q is etire 6 Estimates for the Freud operators ad the Proof of Theorem 2b I this sectio, we prove Theorem 2b The idea followig Theorem 2a is to estimate F (α) by a sequece of operators actig o a/2 := { /2} Because Q grows faster tha a polyomial, we eed some modificatios i the methods of Theorem 2a We will ow idicate these modificatios below Step Firstly, as i Theorem 2a, defie as before polyomials p of degree, by 2xp (x) := + p +(x) + p (x) 9

20 The by Favard s theorem, there exists a positive measure µ with all momets fiite such that these polyomials are orthoormal with respect to µ Let 0 a /2 a /2 0 a 2 /2 J(a/2) = a 2 /2 0 a 3 /2 a 3 /2 0 a 4 /2 be the ifiite Jacobi matrix associated with the recurrece coefficiets ( /2) =0 The the Freud operator applied to the sequece (a /2, a 2 /2, a 3 /2, ) gives F (a/2) = 2 (Q (J(a/2))), (6) = /2 Q (x)p (x)p (x)dµ (x), We shall ow eed a related sequece of cousis defied by: F (a/2) =: (/2)2m jc j x 2mj p (x)p (x)dµ (x), (62) We shall prove: Theorem 6 For C, we have F (α) F (a/2) = O((log ) 3/2 ) (63) Step 2 To prove Theorem 6, we follow the method of Lemmas ad deduce that for C F(a/2) F ( /2) (64) ( ) 2mj 2 = O j 2 c j a 2mj a+2mj max +i i 2mj where /2 is the costat sequece ( /2, /2, ) We proceed to estimate (64) ad here we ow use that a u (log u) /2m ad the followig easily prove idetities, see for example, [9]: (a) Uiformly for u [v/2, 2v] ad v C a u a v u v log v (65) 20

21 (b) For all v C ad u a uv a v = O(u C log v ) (66) (c) For u C, Q (j) (a u ) u(log u)j /2 a j u, j = 0,, 2, 3 (67) Now applyig the estimates (a)-(c) above, we see that for C, F (a/2) F ( /2) (68) ( ) 2mj 2 = O j 2 c j a 2mj a+2mj max +i a i 2mj [ ] ( j 3 c j a 2mj + 2mj ) Cj log = O log +O j=[ ] j 2 c j (log ) C+ +O j 2 c j (log j) Cj+ = O j= a3 [ ] log ( ) a 3 = O log Q(3) ( ) = O((log ) 3/2 ) j 3 c j a 2j 3 + O() + O() (68) together with the method of Theorem 2a gives Theorem 6 Step 3 Put σ := log α 2 log(/2) 2, =, 2, 3 (69) The usig the defiitio of F, the fact that Q is etire, Theorem 6 ad the method of Theorem 2a, we obtai that 2m ( jc j a 2mj + σ + 2 = O σ + (log( + )) 3/2) 2

22 Thus it follows from the above ad (67) that Usig that ( ) (log ) 3/2 σ = O Q ( ) ( ) log = O ( ) log α /2 α /2 fially gives (23) (24) follows from (23) ad (65) Ackowledgemets The author thaks the editors ad the referee for helpig to improve this mauscript i may ways This research is supported, i part, by a Georgia Souther esearch grat efereces [] W Bauldry, A Máté ad P Nevai, Asymptotic expasios of recurrece coefficiets of asymmetric Freud polyomials, i Approximatio Theory V, C Chui, L Schumaker ad J Ward, eds, Academic Press, New York, 987, pp [2] W Bauldry, A Máté ad P Nevai, Asymptotics for the solutios of systems of smooth recurrece equatios, Pacific J Math 33 (988), pp [3] P Bleher ad A Its, Semiclassical asymptotics of orthogoal polyomials, iema-hilbert problem ad uiversality i the matrix model, Aals of Mathematics, 50(999), pp [4] P Deift, T Kriecherbauer, K McLaughli, S Veakides ad X Zhou, Strog asymptotics of orthogoal polyomials with respect to expoetial weights, Commu Pure Appl Math 52(999), pp [5] W Flemig, Fuctios of Several Variables, Texts i Mathematics, Spriger-Verlag, 987 [6] G Freud, O the greatest zero of orthogoal polyomials, J Approx Theory 46 (986), pp 5 23 [7] G Freud, O the coefficiets i the recursio formulae of orthogoal polyomials, Proc oy Irish Acad, Sect A () 76 (976), pp 6 [8] G Freud, Orthogoal Polyomials, Akadémiai Kiadó, Budapest, 97 [9] A L Levi ad D S Lubisky, Orthogoal Polyomials for Expoetial Weights, Spriger-Verlag,

23 [0] D S Lubisky, H N Mhaskar, ad E B Saff, A proof of Freud s cojecture for expoetial weights, Costr Approx 4 (988), pp [] A P Magus, O Freud s equatios for expoetial weights, J Approx Theory 46 (986), pp [2] A P Magus, A proof of Freud s about orthogoal polyomials related to x ρ exp( x 2m ) I: Orthogoal Polyomials ad their Applicatios (C Breziski et al, eds) Lecture Notes i Mathematics, Vol 7 Berli: Spriger-Verlag, 985 [3] A P Magus, Freud s equatios for orthogoal polyomials as discrete Pailevé equatios, preprit [4] H N Mhaskar, Itroductio to the Theory of Weighted Approximatio, World Scietific, 996 [5] P Nevai, Persoal commuicatio [6] E A akhmaov, Strog asymptotics for orthogoal polyomials, I: Methods of Approximatio Theory i Complex Aalysis ad Mathematical Physics, (A A Gochar, E B Saff, eds) Lecture Notes i Mathematics 550, Spriger-Verlag, Berli, 993, pp 7 97 [7] V Totik, Weighted approximatio with varyig weights, Lecture Notes i Mathematics 300, Spriger-Verlag, Berli, 994 [8] W Va Assche, Weak covergece of orthogoal polyomials, Idag Math NS 6 (995), pp 7 23 Departmet of Mathematics ad Computer Sciece, Georgia Souther Uiversity, Post Office Box 8093, Statesboro, GA 30460, U S A dameli@gsucsgasouedu homepage: dameli 23