ON RECURRENCE COEFFICIENTS FOR RAPIDLY DECREASING EXPONENTIAL WEIGHTS

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1 ON RECURRENCE COEFFICIENTS FOR RAPIDLY DECREASING EXPONENTIAL WEIGHTS E. LEVIN, D. S. LUBINSKY Abstract. Let, for example, W (x) = exp exp k x, x [ ] where > 0, k ad exp k = exp (exp ( exp ())) deotes the kth iterated expoetial. Let fa g deote the recurrece coe - ciets i the recurrece relatio xp (x) = A p + (x) + A p (x) for the orthoormal polyomials fp g associated with W. We prove that as! A = (log k ) = ( + o ()) where log k = log (log ( log ())) deotes the kth iterated logarithm. This illustrates the relatioship betwee the rate of covergece to of the recurrece coe ciets, ad the rate of decay of the expoetial weight at. More geeral o-eve expoetial weights o a o-symmetric iterval (a b) are also cosidered.. Itroductio ad Results Let a < 0 < b, Q (a b)! [0 ) be cotiuous, ad W = exp ( Q). The, provided all power momets exist, we may de e orthoormal polyomials p (x) = p W x = x + > 0 = 0 satisfyig the orthoormality coditios b a p p m W = m These orthoormal polyomials satisfy a recurrece relatio of the form xp (x) = A p + (x) + B p (x) + A p (x) Date April 6, 006. Research supported by NSF grat DMS0006 ad US-Israel BSF grat 00353

2 E. LEVIN, D. S. LUBINSKY where A = + > 0 ad B R, (We use uppercase for A rather tha the more commo lowercase, sice we wat to use the lower case for the Mhaskar-Rakhmaov- Sa umbers.) I the case whe (a b) = ( ), a classical result of Rakhmaov [8] implies that lim! A = ad lim! B = 0 ad hece W is a member of the Nevai-Blumethal class. The rate of covergece of A to ad B to 0 has bee studied for decades. May properties of the weight W (or more geerally a measure) ca be formulated i terms of series ivolvig A ad jb j. For example, it is kow [8] that Szegö s coditio is satis ed i log W (x) p dx > x if A A A > 0 I recet years, Barry Simo ad his collaborators have formulated results of this type that go way beyod Szegö s coditio [], [3], [8], [9]. For example, they cosider weights satisfyig the weaker coditio log W (x) p x dx > I this paper, we shall cosider weights that vaish so rapidly at the edpoits of the iterval that all of these coditios are violated. Whe (a b) is ubouded, the situatio is more complicated - see for example, []. I aalyzig expoetial weights W = e Q, a importat role is played by the Mhaskar-Rakhmaov-Sa umbers a < a < a < b, the roots of the equatios (.) (.) = 0 = a a a a xq 0 (x) p (x a ) (a x) dx Q 0 (x) p (x a ) (a x) dx For example, whe Q is covex ad Q (0) = 0, a are well de ed ad uique, ad a < 0 < a. Oe importat feature of the a is

3 the Mhaskar-Sa idetity RECURRENCE COEFFICIENTS 3 (.3) kp W k L[ ] = kp W k L [a a ] valid for all polyomials P of degree [5], [6], [7]. We de e the ceter of the Mhaskar-Rakhmaov-Sa iterval (.) = (a + a ) ad its half-legth (.5) = (a a ) I the special case that Q is eve, we have a = a = = 0 ad a is the root of the equatio = 0 a tq 0 (a t) p dt t I this paper, we show for a large class of expoetial weights that A = O C ad B = O C for some C > 0. Whe (a b) = ( ) ad approaches with a rate slower tha ay egative power of, this leads to A approachig with a rate slower tha ay egative power of. I this case, we ca give the exact rate of approach of A to. Oe special case of our results deals with expoetial weights o [ ] that decay rapidly at the edpoits, though with possibly differig rates. I the sequel, exp 0 (x) = x ad for k exp k = exp (exp ( exp ())) deotes the kth iterated expoetial. k log k = log (log ( log ())) deotes the kth iterated logarithm. Moreover, log 0 x = x ad for Theorem. Let k ` 0, with at least oe positive let > 0 ad 8 < exp exp k () exp k ( x ), x [0 ) (.6) W (x) = exp exp` () exp` ( x ), x ( 0]

4 E. LEVIN, D. S. LUBINSKY The (.7) A = h (log 8 k ) = + (log` ) =i ( + o ()) B = O (log k ) = + (log` ) = Note that the factors exp k () ad exp` () are iserted to esure cotiuity of the expoet at 0. Whe k = `, they ca be factored out ad dispesed with. Thus for the eve weight W (x) = exp exp k x x ( ) where k ad > 0, the theorem gives A = (log k ) = ( + o ()) Our geeral class of weights is give i De itio. Let a < 0 < b, ad W = e Q, where Q (a b)! [0 ) satis es the followig properties (a) Q 0 is cotiuous i (a b) f0g ad Q (0) = 0 (b) Q 00 exists ad is positive i (a b) f0g (c) (d) The fuctio lim t!a+ or b Q (t) = T (t) = tq0 (t) t 6= 0 Q (t) satis es, for some C > 0 ad > < T (s) C T (t) 0 < s=t < provided s t (a b) f0g. (e) There exists C > 0 such that Q 00 (x) jq 0 (x)j C jq 0 (x)j Q (x) The we write W F (C ). (f) Suppose i additio, that for each " > 0 a.e. x (a b) f0g (.8) T (x) = O (Q (x) " ) x! a+ or (.9) T (x) = O (Q (x) " ) x! b

5 RECURRENCE COEFFICIENTS 5 or both these hold. The we write W E (C ) Remarks (a) As examples of W E (C ), we metio the weights of Theorem.. Other examples are W = e Q, where ( exp` (jxj ) exp` (0) x [0 ) Q (x) = exp k jxj exp k (0) x ( 0] where k ` 0 with at least oe positive, ad >, ad (as above) exp k deotes the kth iterated expoetial. I the case k = ` = 0, W F (C ) E (C ). See [, pp. 8-9] for further orietatio. (b) O a ite iterval, the weight W = e Q, where > 0 ( x Q (x) = ) x [0 ) ( x ) x ( 0] belogs to F (C ) E (C ), sice both (.8) ad (.9) are violated. However, if k ad ( exp Q (x) = k ( x ) exp k () x [0 ) ( x ) x ( 0] the W = e Q E (C ), sice (.8) is ful lled. Basically o [ ], (.8) ad (.9) are satis ed oly for weights whose expoet Q grows faster tha ay positive power of ( x ) as jxj!. Our most geeral result is Theorem.3 Let W F (C ). The for some C > 0 (.0) A = O C ad B = O C Remarks (a) We ote that i [, Thm. 5., p. 0], we proved this for a more geeral class of weights, with o () istead of O C. (b) The same proof works for a larger class of weights, amely the class F lip i []. However, that class has a less explicit de itio, so is omitted. Whe the iterval is ite, ad we have iformatio o the rate of approach of to b a, the we ca tur this order relatio ito a asymptotic. This requires a little more otatio throughout this paper,

6 6 E. LEVIN, D. S. LUBINSKY we use the otatio i the followig sese give sequeces of real umbers fc g ad fd g, we write c d if for some positive costats C C idepedet of, we have C c =d C Corollary. Let W E (C ), ad (a b) be ite. The b a b a (.) A = ( + o ()) a + b b a (.) B = O Moreover, if we de e a t for all t by (.-.), the b a dt (.3) + tt (a t ) To make this asymptotic more explicit, we eed further hypotheses. Sice Q (x) sig (x) is strictly icreasig o (a b), ad maps that iterval oto ( ), it has a iverse, which we deote by Q [ ] ( )! (a b). Corollary.5 Let W E (C ), ad (a b) be ite. Assume also that for each (0 ), there exists A ad " (0 ) such that (.) u A ) b Q[ ] (u " ) b Q [ ] (u) with a similar assertio for egative u The b a b + a (.5) + A = b Q[ ] () + Q [ ] ( ) a ( + o ()) B = O b Q [ ] () + Q [ ] ( ) a

7 Remarks (a) I the special case (a b) = ( RECURRENCE COEFFICIENTS 7 ), the result becomes A = Q[ ] () Q [ ] ( ) ( + o ()) B = O Q [ ] () Q [ ] ( ) (b) The coditio (.) is ot always true for W E (C ). For example, if A > ad W = exp ( Q), where Q (x) = exp log x A x ( ) the W E (C ), but (.) fails. For this weight oe ca check that the coclusio of Corollary.5 is still true provided A >. It is ot, however true for < A <, sice Q [ ] () ad a decay at di eret rates i this case. We shall discuss this example i more detail i Sectio 5. So whe oe does ot assume somethig like (.), oe caot always reformulate Corollary. as Corollary.5. We give the idea of proof i the ext sectio, ad the techical details i Sectio 3. Throughout this paper, C C C deote positive costats idepedet of x. The same symbol does ot ecessarily deote the same costat i di eret occurreces.. The Idea of We use well kow represetatios of A B i the form A b x = p (x) p + (x) W (x) dx a B b x (.) = p (x) W (x) dx We split the iterval (a b) as (.) (a b) = I [ J [ K where a (.3) K = (a b) a + C a + C is the mai "tail" iterval (.) J = a + C a + C a + " a " cosists of small itervals ear a ad (.5) I = a + " a "

8 8 E. LEVIN, D. S. LUBINSKY is the "mai part" of (a b). Here C > 0 ad " 0 0 are costats idepedet of. Usig restricted rage iequalities, we show that for some C C > 0 x (.6) p (x) p + (x) W (x) dx = O exp C K with a similar tail estimate for the itegral for B. Next, we ca use global bouds o p W ad p + W, amely jp W j (x) j(x a ) (a x)j = C x (a b) that were established i [] to show that x (.7) p (x) p + (x) W (x) dx = O C 3 J for some C 3 > 0, with similar estimates for itegrals arisig from B. The it remais to deal with the itegrals x I = p (x) p + (x) W (x) dx ad I J = I We make the substitutio u = L (x) = x x p (x) W (x) dx, x = L [ ] (u) = + u that maps [a a ] oto [ ], ad J oto [ + " " ] so that (.8) (.9) I = " J = " u p p + W L [ ] + " u p W L [ ] + " (u) du (u) du If " is small eough, asymptotics i [] show that for x = cos [ + " " ] ad m = + (.0) = = (p m W ) L [ ] (cos ) (si ) = r cos ((m ) + ) + O C

9 RECURRENCE COEFFICIENTS 9 Here = ( ) is a explicitly give fuctio. The I = (.) = = = + C 5 C 5 C 5 C 5 C 5 C 5 C 5 C 5 cos cos cos ( + ) d + O C 6 cos [cos ( + ) + cos ] d + O C 6 cos cos ( + ) + + cos cos cos ( + ) d + O C 6 d + O C 6 We show that the itegral i the last right-had side is O C for some C > 0, by usig a Riema-Lebesgue type lemma. To do this, oe uses the fact that = f () for some smooth icreasig f, makes a substitutio t = f [ ] (), ad the applies results o the degree of trigoometric polyomial approximatio. This is the most techical part of the proof. Similar cosideratios apply to J. Combiig the above estimates (.6), (.7) ad (.) gives the result. 3. of Theorem.3 I this sectio, we esh out the techical details for the ideas give i the previous sectio. We begi with the tail itegrals i (.6), employig a restricted rage iequality. Throughout we assume that W F (C ) ad use the otatio a T itroduced i Sectio, as well as that from Sectio - i particular, I J K of (.) to (.5). Ideed our mai task is to rigorously estimate the itegrals over I J ad K. We also let = T (a ) ja =3 j We ote that i the case of a ite iterval [a b], the factor ja j, so ca be dropped. Lemma 3. Let 0 < p ad W F (C ). The for some C C C 3 > 0 for all, ad all polyomials P of degree kp W k Lp(K ) C exp C 3 kp W k Lp(a a )

10 0 E. LEVIN, D. S. LUBINSKY This is a easy cosequece of Theorem.(a) ad (b) i [, p. 96]. Takig i (b) there = C, with C < so small that 3 < T (a ) which is possible by Lemma 3.7 there [, p. 76, eq. (3.39)], we obtai kp W k Lp((ab)[a (+ )a (+ + )]) C exp C 3C= kp W k Lp(ab) Sice [, eq. (3.39), p. 76], if C > 0 is small eough, 8 s 9 =3 < = C T (a ja j = ) = O C for some C > 0, the result follows, usig also Theorem.(a) i [, p. 96]. Lemma 3. For some C C C 3 > 0 (3.) x p (x) p + (x) W (x) dx C exp C 3 (3.) K K x p (x) W (x) dx C exp C 3 We ote rst that x p (x) p + (x) I W (x) dx + " Ideed, x + " for x I ad we ca apply the Cauchy- Schwarz iequality. Similarly, x p (x) W (x) dx + " I We ow apply Lemma 3. with the weight W = e Q, ad p = we also use the fact that a + for W is the same as a + for W, while [, (3.5), p. 8] a + =a = + O = + O T (a ) with a similar relatio for a =a. This gives the result. Next we go a little iside the Mhaskar-Rakhmaov-Sa iterval

11 RECURRENCE COEFFICIENTS Lemma 3.3 There exist C C such that (3.3) x p (x) p + (x) W (x) dx C C (3.) J J We use the boud x p (x) W (x) dx C C jp m (x)j W (x) j(x a ) (a + x)j = C x (a b) It is valid for m = ad +. See [, (5.), p. 3], ad replace by + there. The we see that x p (x) p + (x) J W (x) dx dx C p jx a j ja + xj J sice [, eq. (3.5), p. 8] C max "= C = a (+) = + O The most di cult part is the ext dealig with the itegral over I. We use Lemma 3. Let (3.5) L (x) = x, L [ ] (u) = + u There exists " > 0 such that uiformly for m =3, ad uiformly for jxj " x = cos (3.6) = = (p m W ) L [ ] (x) x = r cos m + + f () + O "

12 E. LEVIN, D. S. LUBINSKY Here (3.7) f () = cos (t) dt (3.8) (t) = L [ ] (t) t [ ] ad (x) = p (x a )(a x) a Q 0 (s) a s This is Theorem 5.3 i [, p. 03]. We shall also eed some estimates o Lemma 3.5 (a) (3.9) (t) dt = (b) Uiformly for ad t ( ) p t (3.0) (t) ( t + t ) + t + t where (3.) t = ja tj t T (a t ) Q 0 (x) ds p x (s a ) (a s) (c) For some C > 0 ad for all u v [ ] (3.) (u) p u (v) p u v v C ( u + t ) + u + t (a) This is (.75) i [, p. 7]. (b) This is Theorem.0(IV) i [, p. 7]. Note that our class F (C ) is cotaied i the class F Lip there. (c) This is Theorem 6.3(a) i [, p. 8] with (x) = jxj = =

13 RECURRENCE COEFFICIENTS 3 Lemma 3.6 Let " > 0 ad The I = J = x I I x p (x) p + (x) W (x) dx p (x) W (x) dx I = + K + O C J = (L M ) + O C where (3.3) (3.) (3.5) K = L = M = " " " " " " The substitutio x = L [ ] (cos ) gives I = cos ( + " ) = cos ( " ) cos ( + " ) cos cos ( " ) cos si (( + ) f + ()) d cos si (( + ) f + ()) d cos si cos (( + ) f + ()) d cos p p + W L [ ] (cos ) si d cos cos + ( + ) f + () + ( + ) f + () d + O C by Lemma 3., applied with replaced by + ad m = +. Absorbig part of the itegral ito the order term, we cotiue this as I = = + " " " h cos cos ( + ) f + () i + cos d + O C " cos si (( + ) f + ()) d + O C

14 E. LEVIN, D. S. LUBINSKY Similarly, J = cos ( + " ) cos ( " ) cos p W L [ ] (cos ) si d = cos ( + " ) cos cos cos ( " ) + ( + ) f + () = " cos si cos (( + ) f + ()) + cos si (( + ) f + ()) " d + O C d + O C Now we study properties of the fuctio f de ed by (3.7), with a view to showig K L M! 0 as!. Lemma 3.7 (a) f is a strictly icreasig cotiuous fuctio that maps [0 ] oto [0 ] (b) For ad [0 ] (3.6) C si f 0 () C (c) For ad [0 ] (3.7) jf 0 () f 0 ()j C (d) (3.8) (3.9) C f () C 3, j j max si si h 0 i C ( ) f () C ( ) 3, (e) Let g = f [ ] (3.0)! = h i deote the iverse fuctio of f. The h C t g (t) C t =3 t 0 i C ( t) g (t) C ( t) =3 t h i (f) For h C g 0 (t) C t t 0 i (3.) h i (3.) C g 0 (t) C ( t) t

15 (g) For ad 0 < s < t < RECURRENCE COEFFICIENTS 5 (3.3) jg 0 (s) g 0 (t)j C js tj = max s 5 ( t) 5 (a) The ormalizatio (3.9) shows that f maps [0 ] oto [0 ]. (b) We see from (3.7) ad Lemma 3.5(b) that for [0 ] ad f 0 () = (cos ) si C si f 0 () C (c) This follows from Lemma 3.5(c). (d) Now for 0 f () = C 0 f 0 (t) dt 0 C 3 Similarly, our upper boud o f 0 gives f () C si t dt The boud ear is proved similarly. (e) Firstly settig = g (t) i the bouds i (d) gives h C g (t) t C g 3 (t), g (t) 0 i C ( g (t)) t C ( g (t)) 3, g (t) h i Here the costats are idepedet of, t. Moreover, for each xed " > 0, g i [" "], uiformly i. The the result follows. (f) For t 0 we kow that g (t) C, ad hece the bouds of (a), (e) give g 0 (t) = f 0 (g (t)) C si g (t) Ct Similarly, the bouds of (a) give for t [0 ] g 0 (t) = f 0 (g (t)) C

16 6 E. LEVIN, D. S. LUBINSKY For t we kow that g (t) C ad hece the bouds of (b), (e) give g 0 (t) = f 0 (g (t)) C si g (t) C ( g (t)) C ( t) (g) For 0 < s < t jg 0 (t) g 0 (s)j = f 0 (g (t)) f 0 (g (s)) jf 0 (g (s)) f 0 (g (t))j jf 0 (g (t)) f 0 (g (s))j jg (s) g (t)j max si g (s) si g (t)! = g 0 (t) g 0 (s) js tj s = C t s C js tj = s 5 s by the Mea Value Theorem ad the bouds of (f). Similarly for s < t < jg 0 (t) g 0 (s)j C js tj = ( t) 5 Combiig these two estimates i a obvious way gives the result. Lemma 3.8 For some C > 0 (3.) K L, M = O C We estimate K the proof for L M is very similar. By a substitutio t = f + () i (3.3) ad the properties of g + i the previous lemma, we see that K = De e " h (t) = The still " cos g + 8 < t si ( + ) t g+ 0 t dt + O C cos g+ " g 0 + " t [0 " ] cos t g+ g 0 t + t [ " " ] cos g+ " g 0 + " t [ " ] K = 0 h (t) si ( + ) t dt + O C

17 Ideed by Lemma 3.7(f), = " 0 RECURRENCE COEFFICIENTS 7 h (t) si ( + ) t dt cos g + " g 0 + = O " " " si ( + ) t dt 0 ad we assumed 0 < " <. Now we use the orthogoality of 0 si ( + ) t to trigoometric polyomials T of degree to deduce that jk j if T 0 jh T j + O C where the if is take over all T of degree usig Jackso estimates [, Thm..3, p. 05] as jk j sup xy[0]jx yj jh (x) C +5" + O C. We cotiue this h (y)j + O C by the estimates of Lemma 3.7 (f), (g). We assumed i Sectio that " < ad so we are doe. 0 of Theorem.3 I (.), we split the itegrals as i (.), We showed that b a = J + I + J + K K = O C i Lemmas 3. ad 3.3. The remaiig itegrals over I were hadled i Lemma 3.6 ad Lemma 3.8, where we showed for the rst itegral i (.), I = + O C The secod itegral i (.) is similar.

18 8 E. LEVIN, D. S. LUBINSKY. of Corollaries.,.5 ad Theorem. Throughout this sectio, we assume at least the hypotheses of Corollary. - i particular that (a b) is ite. By Theorem.3 ad iteess of (a b), which forces boudedess of f g A = O C ad B = O C so b a b + a (.) A = b a B = b + a + O C + O C b = O a + O C recall a < 0 < b ad a < 0 < a. The rst two coclusios of Corollary. will the follow if we ca show that for each xed " > 0 ad large eough, (.) b a > " We rst gather some techical estimates Lemma. (a) Uiformly for t 6= 0 (.3) a 0 t a t tt (a t ) (b) Uiformly for t 6= 0 (.) Q (a t ) jtj T (a t ) = (c) T (x)! as x! a+ or b. (a) See [, p. 79, Thm. 3.0(a)]. (b) See (3.8) i [, p. 69, Lemma 3.] ad ote that ja j i this case. (c) See Lemma 3.(f) i [, p. 65] ad ote that there the iterval is (c d) rather tha (a b). Lemma.

19 RECURRENCE COEFFICIENTS 9 Assume that W E (C ). The as! (.5) (.6) b a a a dt tt (a t ) dt tt (a t ) Moreover, give " > 0, we have for large eough, dt (.7) tt (a t ) + dt tt (a t ) > " If m > log a m m a 0 m t dt = dt a a t tt (a t ) by Lemma.(a). The costats implicit i are idepedet of m. Sice a m! b as m!, we obtai a b log b a dt tt (a t ) The (.5) follows ad (.6) is similar. Next, we assume (.8), (the case where (.9) holds is similar), T (u) = O(Q (u) " ), u! b The as t!, Lemma.(b) gives (.8) T (a t ) = O (Q (a t ) " ) = O (t " ) This has the cosequece that dt tt (a t ) C as stressed. The (.7) follows. dt t +" C " of Corollary. We add (.5), (.6) ad divide by for give " > 0, as! (.9) b a dt tt (a t ) + Now the result follows immediately from (.). dt tt (a t ) > "

20 0 E. LEVIN, D. S. LUBINSKY Lemma.3 Uder the hypotheses of Corollary.5, (.0) lim! b a b Q [ ] () = ad (.) lim! a a Q [ ] ( ) a = By Lemma.(b), (c), ad (.8), we have for large eough, " Q (a ) ) b Q [ ] " b a b Q [ ] () By hypothesis, give (0 ), there exists " > 0 so that for large eough, b Q[ ] ( " ) + b Q [ ] () The (.0) follows. The other relatio is similar. of Corollary.5 By the lemma, b a b + a (.) = b Q[ ] () + Q [ ] ( ) a ( + o ()) = O b Q [ ] () + Q [ ] ( ) a Now (.) ad the fact that b a decays slower tha ay egative power of (recall Lemma.) give the result. of Theorem. We rst show that these weights satisfy the hypotheses of Corollary. ad.5. Let us assume k (the case where ` is similar). I (.37) of [, p. 9], it is show that as x! T (x) = ( x ) + " k Y exp j x # ( + o ()) j= From this follows that for each " > 0 T (x) = O log Q (x) +" x!

21 RECURRENCE COEFFICIENTS which is much stroger tha (.8). The remaiig hypotheses to belog to F (C ) ad E (C ) follow easily, ad were outlied i [, p. 9]. We ext show that the hypotheses of Corollary.5 are satis ed with (a b) = ( ). We have log k Q (x) = x ) Q [ ] (u) = (log k u) = (.3) ) Q [ ] (u) = (log k u) = ( + o ()) as u!. The Q [ ] (u " ) Q [ ] (u) = (log k u " ) (log k u) ( + o ()) ( ") ( + o ()) eve if k >. For give > 0 ad correspodigly small ", this is o larger tha + so we ca satisfy (.). If `, we obtai a similar relatio for Q [ ] ( ). The Corollary.5, (.3) ad its aalogue for egative u, give the coclusio of Theorem.. Whe ` = 0, a + decays like a egative power of (cf. [, p. 3]), ad Q [ ] ( ) also decays like a egative power of. The the domiat term i (.7) is that ivolvig (log k ) =, ad the term (log` ) = = = is much smaller, ad ca be absorbed ito the order term. Agai the result follows. I this sectio, we let (a b) = ( 5. A Example ), A > ad Q (x) = exp log x A x ( ) Lemma 5. W = e Q E (C ). We see that (5.) Q 0 (x) = [Q (x) + ] A log x A x x ad hece (5.) T (x) = xq0 (x) Q (x) = + A log x A x Q (x) x

22 E. LEVIN, D. S. LUBINSKY We rst show that (5.3) T (x) A > x (0 ) Now + Q (x) = exp jlog ( x )j A log x A usig the elemetary iequality Hece e u u u 0 T (x) A jlog ( x )j ( x ) Usig the elemetary iequality x log ( t) ( t) t t (0 ) we the obtai (5.3), the most di cult part of De itio.(d). The relatio T (s) C T (t), 0 < s=t < follows for small s t sice A T C there. For s t a little larger, we use the fact that if C (0 ), we have T (x) log x A x i (C ) ad the fuctio o the right-had side is icreasig i x. So we have (d) of De itio.. The requiremet (e) is easy. Fially, we prove (f). Let " > 0 ad K > =". For x close eough to, Q (x) exp K log x = x K The as x! T (x) =Q (x) " = O x K" log x A = o () Lemma 5. (a) The limit (.) of Corollary.5 fails. More precisely, give " (0 ), we have Q [ ] (t " ) lim = t! Q [ ] (t) (b) If A <, the coclusio of Corollary.5 fails. More precisely, lim! a Q [ ] () =

23 RECURRENCE COEFFICIENTS 3 (a) If t = Q (x), the log + t = exp x A ad hece The as x! exp x = exp = exp (log ( + t)) =A = x (log ( + t)) =A ( + o ()) (log t) =A ( + o ()) That is, Q [ ] (t) = exp (log t) =A ( + o ()) The as t! Q [ ] (t " ) Q [ ] (t) = exp (log t) =A ( ") =Ao ( + o ())! (b) We use the relatio (.). The (5.) log Q (a ) = log log T (a ) + O () ) log a A = log (A ) log log a log a +O () Writig log a = (log ) =A we obtai log a A = log A (log ) =A + O (log ) =A Substitutig this i (5.) gives ad hece = A (log ) = O(log log ) =A + O (log ) =A (log )=A + A (log ) =A + (log ) =A +O (log ) =A +O (log ) =A (log log )

24 E. LEVIN, D. S. LUBINSKY Here as =A > =A, we obtai so = A (log ) +=A ( + o ()) log a = (log ) =A A (log ) +=A ( + o ()) ) a = exp (log ) =A + A (log ) +=A ( + o ()) a ) Q [ ] () = exp A (log ) +=A ( + o ())! ( + o ()) as!. The the coclusio of Corollary. does ot traslate ito the coclusio of Corollary.5. Note that for A < we obtai from Corollary., A = ( a ) ( + o ()) = exp (log ) =A + A (log ) +=A ( + o ()) Of course, with a little more work, this may be made more precise. Refereces [] D. Damaik, D. Hudertmark, ad B. Simo, Boud states ad the Szegö coditio for Jacobi matrices ad Schrödiger operators, J. Fuct. Aal. 05 (003), [] R. DeVore, G.G. Loretz, Costructive Approximatio, Spriger, Berli, 993. [3] R. Killip ad B. Simo, Sum Rules ad spectral measures of Schrödiger operators with L potetials, mauscript. [] Eli Levi ad D.S. Lubisky, Orthogoal Polyomials for Expoetial Weights, Spriger, New York, 00. [5] H.N. Mhaskar, Itroductio to the Theory Of Weighted Polyomial Approximatio, World Scieti c, Sigapore, 996. [6] H.N. Mhaskar ad E.B. Sa, Extremal Problems Associated with Expoetial Weights, Tras. Amer. Math. Soc., 85(98), 3-3. [7] E.B. Sa ad V. Totik, Logarithmic Potetials with Exteral Fields, Spriger, New York, 997. [8] B. Simo, Orthogoal Polyomials o the Uit Circle, Parts ad, America Mathemtical Society, Providece, 005. [9] B. Simo ad A. latos, Higher-order Szego theorems with two sigular poits, J. Approx. Theory 3 (005), -9

25 RECURRENCE COEFFICIENTS 5 Mathematics Departmet, The Ope Uiversity of Israel, P.O. Box 808, Raaaa 307, Israel., School of Mathematics, Georgia Istitute of Techology, Atlata, GA , USA., lubisky@math.gatech.edu, elile@opeu.ac.il