Smallest Eigenvalues of Hankel Matrices for Exponential Weights

Transcription

1 Smallest Eigevalues of Hakel Matrices for Expoetial Weights Y. Che ad D.S. Lubisky Departmet of Mathematics Imperial College 8 Quee s Gate Lodo SW7 B Eglad y.che@ic.ac.uk The School of Mathematics Georgia Istitute of Techology Atlata GA USA lubisky@math.gatech.edu fax: November 3 Abstract We obtai the rate of decay of the smallest eigevalue of the Hakel matrices R I t+k W (t) dt for a geeral class of eve expoetial ;k= weights W = exp ( Q) o a iterval I. More precise asymptotics for more special weights have bee obtaied by may authors. Remark Ruig Title: Smallest Eigevalues of Hakel Matrices

2 The Result Let I = ( d; d) where < d. Let Q : I! [; ) be cotiuous ad W = exp ( Q) be such that all the momets t W (t) dt; = ; ; ; :::; exist. Form the positive de ite Hakel matrix H = t +k W (t) dt I I ;k= ad deote its smallest eigevalue by. The focus of this paper is the rate of decay of the smallest eigevalue of H. May authors have ivestigated the asymptotic behaviour of as!. For example, Widom, ad Wilf ivestigated the behaviour of for weights o a ite iterval satisfyig the Szegö coditio [3]. For the Hermite weight W (x) = exp x, Szegö [] established the asymptotic = 3 3 e exp () = ( + o ()) ; with similar results for Laguerre weights. The rst author, Berg ad Ismail [] showed that remais bouded away from i the momet problem for W is idetermiate. Moreover, the rst author ad Lawrece [3] established asymptotic behaviour of for weights o (; ) such as exp x ; >. Beckerma has explored coditio umbers for Hakel matrices []. It is well kow that is give by the Rayleigh quotiet: ( ) X T H X = mi X T X : X C+ fg : Correspodig to ay of these vectors X = (x ; x ; x ; :::x ) T, we ca de e a polyomial X P (z) = x z : Usig the de itio of H, we see that we ca recast the Rayleigh quotiet i the form ( R I = mi P ) W R P (ei ) d : deg (P ) : () =

3 This extremum property, very similar to the extremal property of Christo el fuctios, is the basis for the aalysis i this paper. Before we de e our class of weights, which is the eve case of the weights i [7], we eed the otio of a quasi-icreasig fuctio. A fuctio g : (; d)! (; ) is said to be quasi-icreasig if there exists C > such that g(x) Cg(y); < x y < d: Of course, ay icreasig fuctio is quasi-icreasig. De itio. Geeral Expoetial Weights Let W = e Q where Q : I! [; ) is eve ad satis es the followig properties: (a) Q is cotiuous i I ad Q() = ; (b) Q exists ad is positive i Ifg; (c) (d) The fuctio lim t!d is quasi-icreasig i (; d), with (e) There exists C > such that The we write W F (C ). Q(t) = ; T (t) := tq (t) Q(t) ; t 6= T (t) > ; t (; d) ; Q (x) Q (x) C Q (x) ; a.e. x (; d) : Q(x) The simplest case of the above de itio is whe I = R ad T is bouded. This is the so called Freud case, for the boudedess of T forces Q to be of at most polyomial growth. A typical example is Q(x) = x ; x R; where >. A more geeral example satisfyig the requiremets of De itio. is Q(x) = exp`(x ) exp`(); 3

4 where > ad `. Here we set exp (x) := x ad for `, exp` (x) = exp(exp(exp ::: exp (x)))) {z } ` times is the `th iterated expoetial. A example o the ite iterval I = ( ; ) is Q(x) = exp`(( x ) ) exp`(); x ( ; ) ; where > ad `. I aalysis of expoetial weights, a importat role is played by the Mhaskar-Rakhmaov-Sa umber a u (; d) ; u >, which is the uique root of the equatio u = a u sq (a u s) p ds: s Oe of the features that motivates their importace is the Mhaskar-Sa idetity [9] k P W k L(I)=k P W k L[ ;]; valid for all polyomials P of degree. Throughout, C; C ; C ; ::: deote positive costats idepedet of ; x; t ad polyomials P of degree at most. We write C = C(); C 6= C() to idicate depedece o, or idepedece of, a parameter. The same symbol does ot ecessarily deote the same costat i di eret occurreces. Give sequeces of real umbers (c ) ad (d ) we write c d if there exist positive costats C ad C such that C c =d C for the relevat rage of. Similar otatio is used for fuctios ad sequeces of fuctios. We shall prove: Theorem. Let W be eve ad W F (C ). The for ; r " s exp log + + a s a s # ds! : ()

5 Oe may recast this estimate i umber of other ways: for example, r exp arc sih ds : A itegratio by parts shows that r ( " s # exp log a where b t is the iverse fuctio of a t, that is b at = b (a t ) = t; t > : a s a b t t p + t dt )! ; (3) (For this, oe also eeds lim s!+ s log a s =, which follows from the covergece of R log a s ds, see below). Aother form, which is the iitial form i our proof, is r exp ( [V (i) c ]) ; () where V is a equilibrium potetial, ad c is a equilibrium costat - we shall de e these at the ed of this sectio. Example Let > ad Here where [9] Q (x) = x ; x R: a u = C u = ; u > ;! C = (=) = : () Usig (), the Maclauri series expasio [5, p. 5] log x + p + x = X k= ( ) k (k)! k (k!) (k + ) xk+ ; x ; 5

6 ad some straightforward estimatios, we obtai ( ) B [ X ] ( ) k (k)! k (k!) (k + ) ( ) B k= a k k+ C A ; provided is ot a odd iteger. If is a odd iteger, we obtai istead P [ 3 ] k= ( ) k (k)! a k k (k!) (k+) k+ (log ) ( ) ( )! ( C!) C A : I both estimates, [x] deotes the greatest iteger x. I particular, for the Hermite weight =, this gives exp p ; which accords with Szegö s result, if we recall that Q (x) = x i his formulatio. This paper is orgaised as follows: i Sectio, we establish a geeral lower boud for, usig the same methods that were used i [7] to establish lower bouds for Christo el fuctios. I Sectio 3, we establish upper bouds for by discretizig a potetial. The i Sectio, we complete the proof. Throughout the paper, we assume that W F (C ). (I fact, with more work, our results hold for the class F(Dii) i [7], but i terms of weights de ed by explicit formulas, the di erece is isubstatial). For each t >, it is kow that there is o-egative desity fuctio t o [ a t ; a t ] with total mass t, at a t t (s) ds = t; (5) satisfyig the equilibrium coditio at a t log x s t (s) ds + Q (x) = c t ; x [ a t ; a t ] : (6) We call t the equilibrium desity of mass t, c t the equilibrium costat for t, ad at V t (z) = log a t z s t (s) ds 6

7 the correspodig equilibrium potetial. Oe represetatio for t is t (x) = p a t x at Q (s) a t s Q (x) ds p x a t s ; x ( a t; a t ) : (7) ad oe for c t is See [7, Chapter ]. c t = t log a s ds: Lower Bouds for The result of this sectio is: Lemma. Let < <. The ( C exp V e i c d + sup exp [;] V e i ) c : (8) Here C depeds o, ot o. Throughout we x ad set = [ ; ] : Give x =, we use g (z; x) to deote the Gree s fuctio for C with pole at x, so that g (z; x)+log z x is harmoic as a fuctio of z i C ad vaishes o. Whe x, we set g (z; x), ad whe x =, the Gree s fuctio is deoted by g (z). We also let (z) = z + p z ; z C [ ; ] deote the coformal map of C [ ; ] oto the exterior fw : w > g of the uit ball. The the Gree s fuctio for C with pole at admits the represetatio g (z) = log z : 7

8 For further orietatio o the potetial theory we use, see [] or [7]. We also use H [f] to deote the Hilbert trasform of a fuctio f L (R), so that H [f] (z) = f (t) i t z dt; where the itegral must be take i a Cauchy pricipal value sese if z is real. Proof of Lemma. We use the extremal property (), i the form e i d= = sup P I P W ; where the sup is take over all moic polyomials P of degree. Acordigly let P be a moic polyomial of degree m. If deotes a measure of total mass m that places uit mass at each zero of P, the log P admits the represetatio log P (z) = log z t d (t) : Form G (z) = (log z t + g (z; t)) d (t) + V (z) c + ( m) g (z) : Sice log z t+g (z; t) is bouded ad has ite limit as z! t, we see that G is harmoic i C. Moreover, sice as z!, V (z) = log z+o () ad g (z) = log + log z + o (), lim G (z) = g (; t) d (t) c + ( m) log =: G () : z! Thus G is harmoic i C, ad hece has a sigle valued harmoic cougate there, G(z) e say. Hece the fuctio f(z) := exp(g(z) + ig(z))= e z is aalytic i C, with a simple zero at. Cauchy s itegral formula for the exterior of a segmet gives for z =, f (z) = f + (x) f (x) dx = i x z [H [f +] H [f ] (z)] ; (9) 8

9 where f deote boudary values of f o from the upper ad lower half plaes. Note that we set f = outside. Next, f (x) = exp (G (x)) = P W (x) ; x ( ; ) ; () by (6). Moreover, as the Gree s fuctio g is o-egative, f (z) = exp (G (z)) = z P (z) exp (V (z) c ) = z ; z = : The represetatio (9) of f gives for z =, f (z) = (f + (x) f (x)) dx = = P W : Im z Combiig this ad () gives P + e i d + P W si + si I + + = dx x z () exp V e i c P W exp V e i c d: () The rest of the itegral is more di cult. First, ote that sice () holds ad V (z) = V (z) = V ( z) ; e i d + + P e i d sup exp V e i c [;] sup exp V e i c [;] f e i d H [f+ ] e i + H [f ] e i d; 9

10 by (9). Now sice f are cotiuous o, ad are o, they are trivially i L (R), ad the H [f ] belog to the Hardy space of the upper-half plae [6, p. 8]. Next, ormalized Lebesgue measure o the semi-circular arc of the uit circle i the upper-half plae is a Carleso measure with respect to that upper-half plae. So we ca use Carleso s iequality to replace the itegral over the upper ad lower halves of the uit circle, by a itegral alog the real axis: H [f ] e i d C H [f ] (x) dx with C idepedet of f. For a discussio of Carleso s iequality ad Carleso measures, see [] or [6]. The + P + e i d ( C sup [;] exp V e i c ) H [f + ] (x) + H [f ] (x) dx; with C idepedet of f; P (ad ). As the Hilbert trasform is a isometry of L (R), we obtai from () + + P e i d ( C sup exp V e i ) c P W : [;] I Addig this ad () gives, for all moic polyomials of degree ; P e i d= P W I ( C exp V e i c d + sup exp V e i ) c : [;] Now the extremal property () gives the lemma. We ote that this lemma holds more geerally tha for our class of weights: Q does ot eed to be eve or satisfy ay smoothess restrictios. With mior modi catios, the lemma holds for ay expoetial weight W for which the equilibrium measure is supported o a sigle iterval.

11 3 Upper Bouds for I this sectio, we use Totik s method of discretisatio of a potetial [] to obtai a polyomial that gives a upper boud to match the lower boud i the previous sectio. The details are similar to those i [7, Chapter 7]. Theorem 3. There exist C ad C ad for large eough, a polyomial P of degree such that for z = with arg (z) ; 3 ; P (z) C exp ( [V (z) c ]) (3) ad Moreover for such, I P W C : () 3 C 3 exp V e i c d; (5) with C 3 idepedet of. Throughout, we let deote the desity cotracted to [ ; ] so that (s) = ( s) ; s ( ; ) ; ad = : (6) For a xed, we determie poits ad itervals = t < t < t < ::: < t = I = [t ; t + ); ; I = t + t with I = ; :

12 Moreover, we use Totik s idea [] of the weight poit or cetre of mass so that We de e ad will prove: = t+ t s (s) ds= t+ t+ t (s) ds (t ; t + ) ; t s (s) ds = : (7) R (z) = Y = z ; Lemma 3. There exists a positive iteger L such that for large eough, ad u with arg (u) ; 3, ad R (u) exp V (u) C ; (8) R (x) exp V (x) C x L ; x ( ; ) : (9) Later o, if I is ubouded, we shall damp dow R o I by multiplyig with aother polyomial so that we obtai (). For the proof of Lemma 3., we eed properties of the discretisatio poits: Lemma 3.3 (a) Uiformly i ad ; (t ) (s) (t + ) I I + ; s [t ; t + ] : () (b) Moreover, if =, (s) C (t + ) I I + ; s [t ; t + ] () with a aalogous assertio if = : (c) There exists C > such that for, ad u; v ( ; ) with u v u 5 ; ()

13 we have (u) = (v) C: Proof (a), (b) These are Lemma 7.6 i [7, p. 9]. (c) Note that the class of weights F (C ) we treat here lies i the class F Lip i [7] (see [7, p. 3]) ad hece we may apply Theorem 6.3(b) i [7, pp. 7-8] with (u) = u =. We obtai for ad u; v ( ; ) ; (u) (v) C p Moreover, from Theorem 6.(b) i [7, p. 6], The subect to (), we obtai = u v : v max fu ; vg (v) C p v : u v u ; so (u) (v) C ( u) 5= u v= C: Proof of (9) of Lemma 3. We see that log R (u) + V (u) = X log u s X = I u ( (s)) ds =: = : (3) Now we proceed i ve steps: Step : A iequality for Fix u [ ; ] ad choose such that u I. Sice I I (by Lemma 3.3), we claim that there exists (; ), idepedet of u; ad, such that for, s I ) s : () u 3

14 To see this, suppose for example that I. The u > ad s t + t t + u t t t, so that I is to the left of I I + I + : (I the third iequality, we use the fact that the ratio decreases as we decrease ). So () holds i this case. The case where + is similar. Next, a Taylor series expasio gives log u u s = log + s u = s u ( + r) s u s ; where r is betwee ad. As r, u log u s u s I u ( ) : dist (u; I ) The the de itio (7) of gives ( r) Step : with I far from I Cosider those with ad I : dist (u; I ) dist (u; I ) u 5 : Let S deote the set of all such idices. Here the rst restrictio o esures that dist (u; I ) C I ad the usig the boud o from Step, X C X I dist (u; I ) S S C fs[;]:s uc ( u ) 5 g C log u : ds s u

15 Step 3: with I close, but ot too close, to I Cosider those with ad dist (u; I ) < u 5 : Let T deote the set of all such idices. Note that from Lemma 3.3(a), (b), ad the (c), uiformly for such, ad some k f; + g ; The X T I I C (u) (t k ) C: C I X T C I fs:s C: I dist (u; I ) uc I g ds s u Step : with I very close to I Now we deal with the at most 3 remaiig terms with. Here we ca apply Lemma 3.3 to obtai, for some costats C ; C ad C 3 (idepedet of ; ; u ad ), = log u s I u ( (s)) ds log u s I C I ( (s)) ds C log u s I I C I ds C3 C log v C 3 C dv C : Thus : X C 5 : Step 5: Fiish the Proof of (9) Combiig (3) ad all the estimates above gives for u ( ; ) ; log R (u) + V (u) C C log u : 5

16 Proof of (8) of Lemma 3. We use the de ed above. For s I ad u C, log u u s = log + s u = log + s s! u + Re u s u + Re u s ; so itegratig over I ad usig (7) gives I u (s) ds + I I : dist (u; I ) Suppose ow that for some C > ; (u) := sup I : dist (u; I ) C Im u : (5) 8 The X :dist(u;i ) 8 (u) C (u) C (u) X :dist(u;i ) 8 X :dist(u;i ) 8 I dist (u; I ) I (Im u) + dist (Re u; I ) ds (Im u) + Re u s C 3; (6) 6

17 by (5). Moreover, X :dist(u;i )> 8 C X I C 5 : Combiig this, (3) ad (6) gives :dist(u;i )> 8 R (u) exp V (u) C6 ; provided (5) holds. Now we show that (5) does hold if u ad arg (u) ; 3 : (7) We cosider two subcases: (I) I is a ite iterval I this case! d < as! : The the coditio (7) esures that Im u C, with C idepedet of u ad. Hece (5) is immediate. (II) I = ( ; ) I this case! ;!, ad (7) implies that u for large eough 8. The for ; dist (u; I ) 8 ) I 3 ; : 3 (The threshhold does ot deped o u; ; ; ). (7.8) i [7, pp ]), I 3 ; ) I 3 ; Sice (see (7.89) ad (with costats i the relatio idepedet of ), ad sice Im u, we see that (5) reduces to C ; which is true as = o () : 7

18 (See (3.3) i [7, p. 7] ad ote that i the eve case = ). From this we deduce: Lemma 3. Let L be as i Lemma 3.. There exist polyomials R of degree + L such that for z with arg (z) ; 3, R (z) exp (V (z) c ) C (8) ad Proof Observe that V ( u) = R W C i I: (9) = a log u t (t) dt log u s (s) ds = log + V (u) : We set R (z) := ( a z ) L R a z exp c log a ; where L is as i Lemma 3.. We see that R ( u) exp (V ( u) c ) = u L R (u) exp V (u) ad (8) follows from (8), o settig z = u. (Note that below). Next, for x [ ; ], from (6) u is bouded R W ( x) = R ( x) exp (V ( x) c ) = x L R (x) exp V (x) C; 8

19 by (9). The k R W k L(I)=k R W k L[ ;] C: Although the sup-orm of RW is bouded, all we ca deduce from this last lemma is that the L orm over I is O ( ). This is a problem if! ;!. To x this, we multiply R by a polyomial of degree O ( ) that behaves like ( + x ) o [ ; ]. But that would give a polyomial of degree + O ( ), rather tha. To avoid this, we show that the polyomials Rm with m = O ( ) still satisfy the coclusios of the previous lemma, ad for this we eed: Lemma 3.5 Let K >. Assume that lim = :! Assume that for, we are give a iteger m = m () with The for u K; Proof We use [7, p. 6, eq. (.3)] ad [7, p. 6, eq. (.35)] m = O ( ) ;! : (V (u) c ) (V m (u) c m ) C: c = (t) = log a s ds s (t) ds; where s is the equilibrium desity for the iterval s = [ a s ; a s ], so that s (t) = ( pa ; t ( a s; a s ) ; s t ; otherwise : 9

20 The Gree s fuctio for C s with pole at has the represetatios g s (u) = log u t s (t) dt + log a s s = log u u + a s a s : The we see that So, = = V (u) + c log u t s (t) ds dt + log a s ds g s (u) ds: (3) (V (u) c ) (V m (u) c m ) = Here for s [m; ] ; s g s (u) = log u u + a s a s s u + u + A Thus m a s a s m log + u K : a m a s g s (u) ds: g s (u) ds C m a m C a m C : The last relatio follows as m ) a m (see (3.7) i [7, p. 7]). We tur to the Proof of Theorem 3.

21 If ( ) is bouded, the we ca ust choose P = R ad the assertios (3) ad () of Theorem 3. follow from the correspodig oes i Lemma 3.. Now we cosider the case where ( ) is ubouded. For, let ` = ` () deote the greatest iteger L. By Corollary i [8], there exist for large eough, polyomials S` of degree ` with ad The we set S` (x) + x ; x [ ; ] S` (z) C; z = : P (z) = R a polyomial of degree. The i [ so P (x) W (x) a ` (z) S` z ; ; ], (9) gives C + (x=) ; P W C: Restricted rage iequalities (see Theorem. i [7, p. 96]) the give (). Moreover, (8) ad Lemma 3.5 with m = ` give for z = with arg (z) ; 3 ; P (z) C R ` (z) C exp ( [V ` (z) c `]) C exp ( [V (z) c ]) : So we have (3). Fially, the extremal property () of gives (5). Proof of Theorem. If we combie Lemma. ad Theorem 3., we see that the followig three assertios together give Theorem.: (I) 3 exp V e i c d r a exp ( [V (i) c ]) : (3)

22 (II) Give < <, there exists C > such that exp V e i c = exp ( [V (i) c ]) exp C ; (3) uiformly for ad [ ; ] [ [ ; + ] : (III) s! V (i) c = log + + ds: (33) a s a s (Recall that =! as! ). Proof of (I), (II) Observe that as is eve, V e i V (i) = Here for all ad t; = a a log log t + e i (t) dt + t cos t e i (t) dt: t cos t e i t cos (t ( ) + ) while for ; 3, we have uiformly i ; t, istead of ust. The we obtai for all [ ; ] ; ad for ; 3, V e i V (i) C cos V e i V (i) cos t t (t + ) (t) dt (3) t (t + ) (t) dt: (35) I all cases, the costats are idepedet of ;. Now we eed the estimates (t) C p a t ; t ( ; )

23 ad (t) p a t ; t ; : These estimates follow from Theorem. i [7, pp. 7-8]. Let us substitute these bouds i (3) ad (35). Some straightforward estimatio gives for all [ ; ] ; V e i V (i) C cos : (36) ad for ; 3, V e i V (i) cos (37) (For =, we iterpret = as ). Now (36) directly gives (3). Moreover, this last relatio gives for some C ; C ; C 3 ; exp V e i c d= exp ( [V (i) c ]) exp exp C cos d C r a d C 3 : Similarly (37) gives a matchig upper boud, ad so we have (I) also. Proof of (III) From (3), Sice g s c V (i) = g s (i) ds: admits the represetatio s g s (z) = log z z + a s a s ; we obtai (33). Ackowledgemet We thak a referee for a careful readig of the mauscript, ad for poitig out a error i the proof of Lemma 3.. 3

24 Refereces [] B. Beckerma, The Coditio Number of Real Vadermode, Krylov ad Positive De ite Hakel Matrices, Numerische Mathematik, 85(), [] C. Berg, Y. Che ad M.E.H. Ismail, Small Eigevalues of large Hakel matrices: the idetermiate case, Math. Scadiavica, 9(), [3] Y. Che ad N. Lawrece, Small Eigevalues of large Hakel Matrices, J. Physics A, 3(999), [] J. B. Garett, Bouded Aalytic Fuctios, Academic Press, Orlado, 98. [5] I.S. Gradshtey ad I.M. Ryzhik, Tables of Itegrals, Series ad Products, Academic Press, th Editio, Sa Diego, 98. [6] P. Koosis, Itroductio to H p Spaces, Cambridge Uiversity Press, Cambridge, 998. [7] E. Levi ad D.S. Lubisky, Orthogoal Polyomials for Expoetial Weights, Spriger, New York,. [8] D.S. Lubisky, Best Approximatio ad Iterpolatio of ( + (ax )) ad its Trasforms, to appear i J. Approx. Theory. [9] H.N. Mhaskar ad E.B. Sa, Where does the sup Norm of a Weighted Polyomial Live?, Costr. Approx., (985), 7-9. [] E.B. Sa ad V. Totik, Logarithmic Potetials with Exteral Fields, Spriger, Berli, 997. [] G. Szego, O some Hermitia Forms associated with two give curves of the complex plae, (i) Gabor Szegö: Collected Papers, Vol., P. 666, Birkhäuser, Basel, 98. [] V. Totik, Weighted Approximatio with Varyig Weights, Spriger Lecture Notes i Maths., Vol. 3, Spriger, Berli, 99. [3] H. Widom ad H. S. Wilf, Small Eigevalues of large Hakel matrices, Proc. Amer. Math. Soc., 7(966),