EXTREMAL POLYNOMIALS ON DISCRETE SETS

Transcription

1 EXTREMAL POLYNOMIALS ON DISCRETE SETS A. B. J. KUIJLAARS and W. VAN ASSCHE [Received 14 April 1997] 1. Introduction It is well known that orthonormal polynomials p n on the real line can be studied from two different points of view. They can be studied from the orthogonality relation Z 1 1 p n x p m x dr x ˆd m; n ; 1:1 where r is a probability measure on the real line, or from the three-term recurrence relation xp n x ˆa n 1 p n 1 x b n p n x a n p n 1 x ; 1:2 with p 1 x ˆ0, p 0 x ˆ1, where a n > 0 and b n 2 R. Of fundamental interest are the zeros of p n, which are real and simple, and we denote them by x 1; n > x 2; n >...> x n; n. Many authors have contributed to the problem of describing the asymptotic distribution of the zeros. See, for example, [8, 10, 13, 19, 20] for results on zero distributions for orthogonal polynomials given by their orthogonality measure and [4, 9, 15, 21, 22] for polynomials given by the recurrence formula. In this paper, we consider orthogonality measures that have unbounded support, which means that the recurrence coef cients are unbounded. It also means that the zeros of the orthogonal polynomials are unbounded, and to obtain asymptotic results one usually contracts the zeros with a proper scaling factor. In this connection we state the following result. Theorem 1.1 [21, pp. 123±124]. recurrence coef cients in 1:2 satisfy Let a > 1 and A < B. Suppose that the a n f n ˆ 1 4 B A ; f n ˆ 1 2 A B ; where f is increasing and regularly varying with parameter 1=a, that is, for every t > 0, f xt x! 1 f x ˆ t 1 = a : b n The rst author is a postdoctoral fellow of FWO (National Fund for Scienti c Research, Flanders), the second author is a Research Director of FWO. This research was supported by the FWO Research Network WO N and INTAS project ext Mathematics Subject Classi cation: 42C05, 33C25, 31A15. Proc. London Math. Soc. (3) 79 (1999) 191±221.

2 192 a. b. j. kuijlaars and w. van assche Then 1 X n Z x j; n f ˆ f t v t dt 1:3 n f n j ˆ 1 for every bounded continuous function f, where v is the Mellin convolution of the arcsin density on the interval A; BŠ and the beta density at a 1 on 0; 1Š. The density v is known as the Nevai±Ullman density for A; BŠ with parameter a. Explicit formulas are v ˆ b a q A; BŠ with ( b a t :ˆ at a 1 if t 2 0; 1 ; 1:4 0 elsewhere; and 8 < 1 p q A; BŠ t :ˆ p if t 2 A; B ; B t t A 1:5 : 0 elsewhere: The Mellin convolution is de ned by Z v t ˆ b a s q A; BŠ t=s ds Z s ˆ b a t=s q A; BŠ s ds s ; 1:6 and it is the density of a probability measure whose support is the smallest interval containing A, B and 0. The convergence in (1.3) is equivalent to weak convergence of the contracted zero distributions to the Nevai±Ullman measure v t dt. The following problem was posed by one of us in [11, Chapter 13, pp. 203±206]. Problem. Which orthogonality measures r give rise to the Nevai±Ullman density for A; BŠ with parameter a if 0 < A < B? For A < 0 < B, the answer is known and the measures r turn out to be non-symmetric Freud weights [23]. Note that for 0 < A < B, the support of the Nevai±Ullman density is 0; BŠ. The present paper contains the answer to this problem; see Theorem 2.2 below. The measures we nd are discrete measures and they are supported on a sequence of non-negative points tending to in nity. An explicit example is the following (see Theorem 2.2 and (2.9)±(2.10)). Assuming 0 < A < B, we take t k :ˆ bk 1 = a ; w k :ˆ exp 2gtk a ; with b and g given by b :ˆ A 2 F 1 a; 1 A B 1 = a 2 ;1; ; A p p G a g :ˆ G a 1 B a 2 F 1 a; 1 2 ; a 1 2 ; A ; B 2 where 2 F 1 a; b; c; z is the Gauss hypergeometric function. If the polynomials p n,

3 extremal polynomials on discrete sets 193 with deg p n ˆ n, satisfy the discrete orthogonality relation X 1 k ˆ 0 p n t k p m t k w k ˆ d m; n ; then the contracted zero distributions (scaled with factor f n ˆn 1 = a ), converge in the weak sense to the Nevai±Ullman measure for A; BŠ with parameter a. Without any additional effort, our result applies to polynomials that are extremal with respect to an L p -norm. (Recall that orthogonal polynomials are extremal with respect to an L 2 -norm.) The methods we use are based on recent work of Rakhmanov [16] and Dragnev and Saff [6] concerning extremal problems from logarithmic potential theory with a constraint. Recall that in their studies on orthogonal polynomials with respect to a continuous measure on an unbounded set, Gonchar and Rakhmanov [10] and, independently, Mhaskar and Saff [13] introduced the methods of logarithmic potential theory with external elds. Here an important role is played by probability measures that minimize certain logarithmic energies. See the book [18] for a comprehensive account of this theory. Rakhmanov discovered that similar extremal problems govern the asymptotics for polynomials on discrete sets, the only difference is that the class of measures is restricted to those probability measures m that satisfy an upper constraint m < j, where j is a measure determined by the discrete sets. This constraint appears since the zeros of extremal polynomials are separated by the points of the discrete sets. Rakhmanov's paper [16] was directed towards a speci c example on 1; 1Š without a weight (the discrete Chebyshev polynomials), but his methods are useful in a much wider setting. This was further explored by Dragnev and Saff [6] who considered extremal polynomials with a weight and obtained explicit results for Krawtchouk polynomials. See also the recent work of Deift and McLaughlin [5] where the same constrained problem was studied in connection with the continuum it of the Toda lattice. We are grateful to these authors for providing us with copies of their manuscripts. 2. Statement of main results In this paper, E is a countable set of non-negative numbers without nite accumulation points, that is, E :ˆ ft k : k ˆ 0; 1;...g 2:1 with 0 < t 0 < t 1 < t 2 <... and k! 1 t k ˆ 1. Associated with E are the spaces L p E, with 1 < p < 1, with norms k f k p; E given by X 1 = p k f k p; E :ˆ j f x j p ; for f 2 L p E ; 1 < p < 1; x 2 E and k f k 1; E :ˆkfk E :ˆ sup j f x j; for f 2 L 1 E : x 2 E Let w: 0; 1! 0; 1 be a weight on 0; 1 which decreases suf ciently fast at in nity so that the weighted polynomial wp belongs to L p E for every

4 194 a. b. j. kuijlaars and w. van assche polynomial P. Then, given p and n, there is a monic polynomial T n ˆ T n; p of degree n which minimizes the weighted L p -norm kwp n k p; E among all monic polynomials of degree n. That is, minfkwp n k p; E : P n x ˆx n...gˆkwt n; p k p; E : 2:2 This L p -extremal polynomial T n has n distinct positive zeros, and we are interested in the asymptotic behaviour of these zeros and in the behaviour of the extremal norms kwt n; p k p; E as. We consider the cases where t k, bk 1 = a ; w x, exp gx a ; where a, b and g are xed positive constants. See Theorem 2.2 below for the precise meaning of,. Following the ideas of Rakhmanov [16] and Dragnev and Saff [6], we prove that in these cases the asymptotic behaviour is governed by the probability measure that solves the following constrained extremal problem from potential theory. Minimize the weighted logarithmic energy ZZ Z 1 log dm x dm y 2g x a dm x 2:3 jx yj among all Borel probability measures m on 0; 1 subject to the constraint m < j; where j ˆ j a; b is the measure on 0; 1 with density 2:4 dj a; b :ˆ ab a x a 1 for x > 0: dx The inequality (2.4) means that we have m K < j K for every Borel set K Ì 0; 1. Denoting by U m the logarithmic potential of a measure m, that is, Z U m 1 x :ˆ log jx yj dm y ; we may characterize the extremal measure m of the constrained energy problem (2.3), (2.4) by the variational inequalities U m x gx a > F for x 2 supp j m ; 2:5 U m x gx a < F for x 2 supp m ; where F is some constant; see [6, Theorem 2.1]. For general weights and constraints, such a constrained energy problem is hard to handle. The main problem is to determine the sets supp m and supp j m. However, for the special case given by (2.3)±(2.5) we can obtain explicit formulas for the extremal measure as a Mellin convolution of two standard measures. We use b a, with a > 0, to denote the density of the following beta distribution on 0; 1Š, ( b a t :ˆ at a 1 if t 2 0; 1 ; 0 elsewhere;

5 extremal polynomials on discrete sets 195 and we use q A; BŠ for the density of the arcsine distribution on A; BŠ with 0 < A < B, 8 < 1 p q A; BŠ t :ˆ p if t 2 A; B ; B t t A : 0 elsewhere: Then v ˆ b a q A; BŠ, the Mellin convolution of b a and q A; BŠ, de ned by Z v t :ˆ b a s q A; BŠ t=s ds Z s ˆ b a t=s q A; BŠ s ds s ; 2:6 is the density of a probability measure with support 0; BŠ. Theorem 2.1. Let a; b; g > 0. Then there exist B > A > 0, such that the measure m with density v ˆ b a q A; BŠ solves the constrained energy problem (2.3)±(2.4). Further, the inequalities (2.5) are satis ed with F ˆ 1 a log 1 4 B A : 2:7 Remark. The numbers A and B in Theorem 2.1 depend on a, b and g in the following way. For 0 < a < 1 2 and b a g < p cot pa, we have A ˆ 0 and B is given by p p G a 1 = a B ˆ gg a 1 : 2:8 2 In this case we have supp j m ˆ 0; 1 so that the constraint is not effective. The measure m also minimizes the energy (2.3) among all Borel probability measures on 0; 1. Also note that the number B in (2.8) and the measure m do not depend on b. In the other cases, that is, a > 1 2 or 0 < a < 1 2 and b a g > p cot pa, we have that both B and A are strictly positive, and they are the unique solution of the pair of equations Z 1 B ds p ˆ b a ; p A s a B s s A Z 1 ds p ˆ g: B s a s B s A The integrals can be written in terms of hypergeometric functions, and the corresponding equations are 2 F 1 a; 1 B 2 ;1;A ˆ A a ; 2:9 A b 2 F 1 a; 1 2 ; a 1 2 ; A ˆ gg a 1 2 p B a : 2:10 B p G a In these cases, supp j m ˆ A; 1, so that the constraint is effective on the interval 0; AŠ only. Theorem 2.1 provides the basis for the following asymptotic results for L p - extremal polynomials on a discrete set. In the general form presented below, we

6 196 a. b. j. kuijlaars and w. van assche use the notion of a slowly varying function L x. This is a positive function on 0; 1 such that for every t > 0, L xt x! 1 L x ˆ 1: A De Bruijn conjugate for L is a slowly varying function L such that x! 1 L x L xl x ˆ 1; see [2]. The function L is unique up to asymptotic equivalence. In what follows we will always assume that L and L are continuous functions on 0; 1. Theorem 2.2. Let a, b and g be positive numbers and let A and B be associated with a, b and g as in Theorem 2.1. Let L be a slowly varying function and let L be a De Bruijn conjugate of L. Suppose E ˆft k : k > 0g such that t k ˆ bk 1 = a L k 1 = a 1 o 1 as k! 1; 2:11 and for some constant C > 0, t k 1 t k > Ck 1 = a 1 L k 1 = a for k > 0: 2:12 Let w: 0; 1! 0; 1 be a continuous positive weight such that w x ˆexp gx a L x a for x > 0: 2:13 For 1 < p < 1, let T n; p be the weighted L p -extremal polynomial of degree n with zeros x 1; n > x 2; n >...> x n; n > 0. Then the following hold: (a) `nth root asymptotics of extremal norms': kwt n; p k 1 = n p; E ˆ n 1 = al n 1 = a 1 e 4 B A 1 o 1 as ; (b) `contracted zero distribution': for every bounded continuous function f on 0; 1, 1 X n Z x B j; n f ˆ f t v t dt; n j ˆ 1 n 1 = a L n 1 = a 0 where v ˆ b a q A; BŠ is the Mellin convolution of b a and q A; BŠ (see (2.6)); (c) `largest zero': x 1; n n 1 = a L n 1 = a ˆ B: Remark. An important special case is when x! 1 L x ˆ1. Then we can take L x ˆ1 and the conditions (2.11)±(2.13) simplify to t k ˆ bk 1 = a 1 o 1 as k! 1; and t k 1 t k > Ck 1 = a 1 ; w x ˆexp gx a 1 o 1 as x! 1:

7 extremal polynomials on discrete sets 197 Remark. It is important in Theorem 2.2 that the numbers a in (2.11)±(2.13) are the same. If we are in a case like E ˆfbk 1 = a 0 : k ˆ 0; 1;...g; w x ˆexp gx a ; with a 6ˆ a 0, then the set E and the weight w do not match each other well, since the set E is either too ne or too coarse in relation to w. One can prove the following. (a) If a < a 0, then the discrete set E is too ne for the weight w. As far as nth root asymptotics and zero distributions are concerned, the extremal polynomials behave in the same way as the extremal polynomials for the continuous weight exp gx a on 0; 1. Thus the same relations as in Theorem 2.2(a)±(c) hold with A ˆ 0 and p p G a 1 = a B ˆ gg a 1 2 as in (2.8). (b) If a > a 0, then the discrete set E is too coarse for the weight w. Then roughly speaking, one may think that there is a zero of the extremal polynomial T n in every interval t k 1 ; t k for k ˆ 1;...; n. This gives the it relations kwt n; p k 1 = n p; E ˆ 0; n 1 = a 0 for every bounded continuous function f on 0; 1, 1 X n Z x b j; n f ˆ a 0 b a 0 f t t a 0 1 dt; n n 1 = a 0 0 and j ˆ 1 x 1; n n 1 = a 0 ˆ b: Analogous results hold if a ˆ a 0, but in (2.11)±(2.12) not L, but some other slowly varying function M is involved. If M x =L x!0asx! 1, then the discrete set E is too ne for the weight w, and the same thing happens as under (a) above. If M x =L x!1 as x! 1, then E is too coarse, and we have iting relations analogous to those given in (b). Remark. The condition (2.12) is a separation condition on the sequence ft k g, which is imposed for technical reasons. It would be of interest to nd a weaker condition. Remark. The it relation in part (b) of Theorem 2.2 can be stated in terms of weak convergence of normalized zero distributions. For a polynomial P n of degree n, we denote by n P n the normalized zero distribution n P n :ˆ 1 X n d n xj ; j ˆ 1 where x 1 ;...; x n are the zeros of P n counted according to their multiplicities, and d x is the Dirac point mass at x. We say that a sequence of probability measures

8 198 a. b. j. kuijlaars and w. van assche fm n g converges in weak sense to m (notation m n! m) if for every bounded continuous function f, Z Z fdm n ˆ fdm: Then part (b) may be stated that for the contracted polynomials P n x :ˆ T n; p n 1 = a L n 1 = a x one has n P n!b a q A; BŠ t dt in weak sense. In 3 we present a number of examples to illustrate Theorem 2.2. Then in 4 we discuss constrained and unconstrained energy problems in order to facilitate future references. We prove Theorem 2.1 in 5. The proof of Theorem 2.2 is substantial and the remaining four sections are devoted to it. It follows from a study of extremal polynomials with respect to uniform norms of the form kw n P n k En where fe n g is a sequence of discrete subsets satisfying a number of conditions. These conditions are discussed in 6. In particular, the sets E n give rise to a constraint j. For Theorem 2.2 we will take w x ˆexp gx a L x a ; E n :ˆ n 1 = a L n 1 = a 1 E; which may be considered as a typical example. Our main intermediate result is Theorem 7.2 which is also of independent interest. Here we compare the uniform norm of polynomials on the discrete sets kw0 n P n k En with the uniform norm on the half-line kw0 n P n k 0; 1. The result is that kw0 n P n k 1 = 0; 1 nˆ kw0 np 1 nk En for every sequence of polynomials fp n g with deg P n < n, provided that the equilibrium measure associated with the weight w 0 (see 4) is bounded by the constraint j. In general, the equilibrium measure associated with the original weight w is not bounded by j, so Theorem 7.2 is not directly applicable to w. However, there is a canonical way of associating with any w a weight w 0 for which Theorem 7.2 holds. Using this, we obtain nth root asymptotics of extremal uniform norms and asymptotic zero distributions of extremal polynomials in Theorem 7.4 below. In 8 we discuss in nite- nite range inequalities which are necessary for L p estimates and for the asymptotics of the largest zeros. Finally, in 9 we complete the proof of Theorem Examples In this section we present some examples to illustrate Theorem 2.2. It is well known that orthonormal polynomials on a discrete set, X 1 k ˆ 0 p n t k p m t k w k ˆ d n; m ; correspond to L 2 -extremal polynomials in the sense of Theorem 2.2 with weight w p satisfying w t k ˆ. The orthonormal polynomials satisfy a three-term w k

9 extremal polynomials on discrete sets 199 recurrence relation (1.2) with certain coef cients a n > 0 and b n 2 R. In a number of cases, both the recurrence coef cients and the orthogonality measure are known explicitly, and then both Theorem 1.1 and Theorem 2.2 might give the asymptotics of the contracted zero distributions. This is the case in the three examples below, Meixner polynomials, Charlier polynomials, and Stieltjes±Carlitz polynomials. We determine for them the constants a, b and g describing the set E and the weight w, and from these we will obtain the interval A; BŠ needed for the contracted zero distributions. The numbers A and B are determined by equations (2.9)±(2.10). We will compare this with the result obtained from Theorem 1.1. Meixner polynomials. Meixner polynomials m n x are orthogonal polynomials satisfying the orthogonality relation X 1 m n k m j k c k b k ˆ 1 c b c n n! b k! n d n; j ; 3:1 k ˆ 0 where 0 < c < 1, b > 0 and b k ˆ G b k =G b is the Pochhammer symbol; see, for example, [3, p. 161]. This means that we have t k ˆ k and w k ˆ w 2 k ˆ c k b k =k! so that the corresponding weight function satis es w x ˆexp 1 2 x log c 1 o 1 : Hence in Theorem 2.2 we have a ˆ b ˆ 1 and g ˆ 1 2 log c. The asymptotic contracted zero distribution is therefore given by the Mellin convolution b 1 q A; BŠ, where A and B are determined from equations (2.9)±(2.10) with a ˆ b ˆ 1 and g ˆ 1 2 log c. The hypergeometric functions involved in this case are explicitly known in terms of elementary functions: 2 F 1 1; 1 2 ;1;z ˆ 1 z 1 = 2 ; 2 F 1 1; 1 2 ; 3 2 ; z ˆ p 1 2 log 1 p z p z 1 : z Hence equation (2.9) reduces to AB ˆ 1, and (2.10) becomes log 1 p A=B p 1 ˆ 1 2 log c: A=B Using AB ˆ 1, we get A ˆ 1 p p c p 1 1 c 2 ˆ ; B ˆ 1 p c p c 1 c 1 ˆ 1 p c 2 : 3:2 c 1 c This corresponds to the result obtained from the recurrence formula, since the orthonormal Meixner polynomials satisfy xp n x ˆa n 1 p n 1 x b n p n x a n p n 1 x with (see [3, p. 218]) so that a 2 n ˆ a n n ˆ c 1 c 2 n n b 1 ; b n ˆ p c 1 c ˆ c n bc ; 1 c b B A ; n n ˆ 1 c 1 c ˆ 1 2 A B ; in accordance with the condition of Theorem 1.1. More precise asymptotics of Meixner polynomials can be found in [12].

10 200 a. b. j. kuijlaars and w. van assche Charlier polynomials. X 1 Charlier polynomials C n x satisfy C n k C m k a k e a ˆ a n n!d k! n; m ; k ˆ 0 where a > 0; see [3, p. 160]. This means that t k ˆ k and w k ˆ w 2 k ˆa k e a =k!. Stirling's formula then gives w 2 x ˆexp x log x 1 o 1 Š: Thus we have (2.13) with slowly varying function L x satisfying L x = log x ˆ1: x! 1 A De Bruijn conjugate of L is L x ˆ1= log x. The discrete set is now too coarse for the weight w and the phenomenon that was noted in the second remark after Theorem 2.2 occurs here. The contracted zero distribution is given by the constraining measure j 1; 1 restricted to 0; 1Š. This agrees with the result obtained from the recurrence coef cients, since for Charlier polynomials we have [3, p. 217] an 2 ˆ an and b n ˆ n a, so that a n n ˆ 0; b n n ˆ 1; which means that A ˆ B ˆ 1. The interval A; BŠ is a singleton so that the Mellin convolution b 1 q A; BŠ gives the beta density b 1. For a detailed asymptotic analysis of Charlier polynomials one can consult [17]. Stieltjes±Carlitz polynomials. Stieltjes, Carlitz and Al-Salam have considered two systems of orthogonal polynomials C n x and D n x with recurrence formulas C n 1 x ˆxC n x a n C n 1 x ; D n 1 x ˆxD n x b n D n 1 x ; and recurrence coef cients a 2n ˆ 2n 2 k 2 ; a 2n 1 ˆ 2n 1 2 ; 3:3 b 2n ˆ 2n 2 ; b 2n 1 ˆ 2n 1 2 k 2 ; with a parameter 0 < k < 1; see [3, pp. 193±195]. We restrict attention to the polynomials C n x but the analysis of the polynomials D n x is similar. The orthogonality for the C n x is X 1 j ˆ 1 q j 1 = 2 1 q 2 j 1 C n s j C m s j ˆK k p n! 2 k 2 n = 2Š d m; n ; 3:4 where 2j 1 p s j :ˆ ; q :ˆ exp p K k 0 2K k K k and K is the complete elliptic integral of the rst kind of the modulus k, p and k 0 ˆ 1 k 2 K k ˆ Z p = 2 0 dv p ; 1 k 2 sin 2 v 3:5 is the complementary modulus. These polynomials are

11 extremal polynomials on discrete sets 201 symmetric, and if we introduce the polynomials P n x by C 2n x ˆP n x 2, then we obtain orthogonal polynomials on a discrete set in 0; 1 with recurrence relation P n 1 x ˆ x b n P n x an 2 P n 1 x ; where b n ˆ a 2n a 2n 1 ˆ 4n 2 1 k 2 4n 1; 3:6 an 2 ˆ a 2n 1 a 2n ˆ 4n 2 2n 1 2 k 2 : The polynomials P n are orthogonal on the set E ˆft 0 ; t 1 ; t 2 ;...g with t j ˆ sj 2 2j 1 p 2 ˆ 2K k and the corresponding weight is w 2 t j ˆq j 1 = 2 = 1 q 2 j 1, so that w x ˆexp K k p x log q 1 o 1 : 2p This means that we are dealing with the case a ˆ 1 2 ; b ˆ p=k k 2 ; g ˆ K k 2p log q ˆ 1 2 K k 0 : To determine the interval A; BŠ we use the equations (2.9)±(2.10) and the fact that the hypergeometric functions in this case are complete elliptic integrals of the rst kind 2 F ; 1 2 ;1;k2 ˆ 2=p K k ; p so that, using a new variable, ˆ A=B, we have p 2K i, 0 =, ˆp A p p ˆ K k A ; 2K, ˆ1 2 b K k 0 p B : Using the identity K i, 0 =, ˆ,K, 0 p [7, p. 319] and, ˆ A=B, we get p 2K, 0 ˆK k B ; 2K, ˆ1 2 K k 0 p B : 3:7 Taking the ratio of the equations in (3.7), we nd that 2 K k K k 0 ˆ K, 0 K, : 3:8 There are two other useful identities for complete elliptic integrals of the rst kind, namely K k 0 ˆ 2 1 k K 1 k ; K k ˆ 1 1 k 1 k K 2 p! k ; 3:9 1 k see [7, p. 319]. Inserting these in (3.8), we obtain K, 0 K, ˆ K 2 p k = 1 k K 1 k = 1 k ; so that, ˆ 1 k 1 k ;, 0 ˆ 2 p k 1 k :

12 202 a. b. j. kuijlaars and w. van assche Having, in terms of the parameter k, we nd A and B using (3.7) and (3.9). The result is p p A ˆ 2 1 k ; B ˆ 2 1 k : For the contracted zero distribution of P n x we thus nd a Mellin convolution of b 1 = 2 and the equilibrium measure on 4 1 k 2 ; 4 1 k 2 Š. This corresponds with the asymptotic zero distribution found earlier in [22], after applying a quadratic transformation to go from the zeros of C 2n to the zeros of P n. 4. Constrained and unconstrained energy problems In this section we discuss constrained and unconstrained energy problems in order to x notation and terminology. We also include some known results that will be used in the sequel. We restrict ourselves to weights on the half line 0; 1, but this restriction is not essential. De nition 1. A weight w: 0; 1! 0; 1 is called admissible if the following hold: (1) w is continuous; (2) w is not identically zero; and (3) x! 1 xw x ˆ0. Note that this de nition is slightly more restrictive than the one used in [6] or [18], where it is enough that w is upper semi-continuous. It is customary to write Q :ˆ log w so that w ˆ exp Q, and to view Q as an external eld induced by w. Associated with an admissible weight w are a unique Borel probability measure m w on 0; 1 with compact support and a unique constant F w such that U m w Q ˆ F w on supp m w ; 4:1 U m w Q > F w on 0; 1 : 4:2 As for the connection with extremal polynomials, we recall the following result due to Mhaskar and Saff [14], which will be used several times in the sequel. Lemma 4.1. Let w be an admissible weight on 0; 1. Then the following hold. (a) For every monic polynomial P n of degree n, kw n P n k 0; 1 ˆkw n P n k supp m w > exp nf w : (b) If fp n g is a sequence of monic polynomials, deg P n ˆ n, such that kwn P n k 1 = n 0; 1 ˆ exp F w ; then n P n!m w in weak sense. (c) Conversely, if fp n g is a sequence of monic polynomials, deg P n ˆ n, such that n P n!m w and the zeros of P n belong to a xed compact set, then kwn P n k 1 = n 0; 1 ˆ exp F w : The following de nition is also somewhat different from the one in [6].

13 extremal polynomials on discrete sets 203 De nition 2. Let w be an admissible weight on 0; 1. A positive Borel measure j on 0; 1 is called an admissible constraint for w if (1) supp j ˆ 0; 1 ; (2) j fx > 0: w x > 0g > 1; (3) for every compact set K Ì 0; 1, the restricted measure jj K has a potential Z U j j K 1 x :ˆ log K jx yj dj y which is continuous in the complex plane C. Dragnev and Saff [6] require instead of (3) that for every compact K, (4) R R K K log 1=jx yj dj x dj y < 1, which is essentially weaker than (3). The condition (3) was also used in [6] when dealing with asymptotics for extremal polynomials. We note that for (3) it is enough to have (3 0 ) for every a > 0, the restricted measure jj 0; aš has a continuous potential. This follows from the general principle that, whenever 0 < m 1 < m 2 and the potential U m 2 is continuous, then U m 1 is continuous too; see [6, Lemma 5.2]. This principle will be used several times in the sequel to conclude that for a probability measure m with compact support and m < j, with j admissible as in De nition 2, the logarithmic potential U m is continuous. It was proved by Dragnev and Saff [6, Theorem 2.1] that for an admissible weight w and a constraint j which is admissible for w, there is a unique Borel probability measure mw j on 0; 1 satisfying the constraint m j w < j, such that for a constant Fw, j we have U mj w Q < Fw j on supp m j w ; 4:3 U mj w Q > Fw j on supp j m j w : 4:4 We note that the constant Fw j is uniquely determined by (4.3), (4.4). This is so because we are working on the set 0; 1 which is connected. Therefore, the closed sets supp m j w and supp j m j w have non-empty intersection, and Fw j is the value of U mj w Q on this intersection. Working on a disconnected set, we nd that the constant may not be unique; see [6, Example 2.4]. 5. Proof of Theorem 2.1 Let a > 0 and B > A > 0. For simplicity we write b :ˆ b a and q :ˆ q A; BŠ.We let v ˆ b q and we denote by m the probability measure with density v. Note that supp m ˆ 0; BŠ. We have by the de nition of Mellin convolution v x ˆ Z min 1; x = A x = B b t q x=t dt t ˆ Z B max x; A b x=t q t dt t : 5:1 Lemma 5.1. (a) The density v satis es 8 >< ˆ Cax a 1 if x 2 0; AŠ; v x < Cax a 1 if x 2 A; B ; >: ˆ 0 if x 2 B; 1 ; 5:2

14 204 a. b. j. kuijlaars and w. van assche where C ˆ C A; B :ˆ 1 p Z B A s a p ds: 5:3 B s s A (b) The logarithmic potential of m satis es 8 < < F Dx a if x 2 0; A ; U m x ˆ F Dx a if x 2 A; BŠ; 5:4 : > F Dx a if x 2 B; 1 ; where F ˆ 1 a log 1 4 B A 5:5 and D ˆ D A; B ˆ Z 1 B s a p ds: 5:6 s B s A Proof. Using the second integral in (5.1) we obtain 8 Z ax a 1 1 B s a p ds if x 2 0; AŠ; >< p A B s s A Z v x ˆ ax a 1 1 B s a p ds if x 2 A; BŠ; p x B s s A >: 0 if x 62 0; BŠ; which proves part (a). To evaluate the logarithmic potential of m, we use the rst integral in (5.1) and we get Z U m x ˆ log jx tj v t dt ˆ ˆ Z B 0 Z 1 0 log jx tj b s Z Bs As Z min 1; t = A t = B b s q t=s ds s dt log jx tj q t=s dt ds s ; where the last equality follows by interchanging the order of integration, which can easily be justi ed using Tonelli's theorem. The change of variables t 7! st gives Z 1 Z B U m x ˆ b s log s log x s t q t dt ds ˆ 0 Z 1 0 Z x = B 0 Z 1 x = B A b s log sds Z B as a 1 log x A s t q t dt ds Z B as a 1 log x s t q t dt ds A ˆ: I 0 I 1 x I 2 x : 5:7

15 extremal polynomials on discrete sets 205 An integration by parts easily gives Z 1 I 0 ˆ b s log sdsˆ 1 0 a : To evaluate I 1 x we make the change of variables s ˆ x=y to nd Z 1 Z B I 1 x ˆ x a ay a 1 log jy tjq t dt dy: B A 5:8 An integration by parts leads to I 1 x ˆ x a B a log Z 1 Z B 1 4 B A y a 1 q t dt dy : B A y t p The inner integral in the last term is equal to 1= y B y A and we arrive at I 1 x ˆ x a B a log Z! B A dy p : 5:9 B y a y B y A For I 2 x, we consider the three cases x 2 A; BŠ, x 2 0; A and x 2 B; 1 separately. If x 2 A; BŠ, then x=s 2 A; BŠ for every s 2 x=b; 1Š. Since q t dt is the equilibrium measure of A; BŠ, we have Z B log x s t q t dt ˆ log 1 B A for every such s. Hence for x 2 A; BŠ, I 2 x ˆ log 1 4 B A Z 1 A x = B 4 as a 1 ds ˆ log 1 4 B A 1 x=b a : 5:10 For x 2 0; A, we use the fact that the potential of the equilibrium measure q t dt attains its maximum on A; BŠ so that Z B log x s t q t dt < log 1 B A A for all s 2 x=b; 1Š with strict inequality if s 2 x=a; 1Š. It follows that for x 2 0; A, I 2 x < log 1 4 B A Z 1 x = B Similarly, if x > B then Z B log A 4 as a 1 ds ˆ log 1 4 B A 1 x=b a : 5:11 x s t q t dt > log 1 B A for every s 2 1; x=bš with equality only for s ˆ x=b. Then Z x = B Z B I 2 x ˆ as a 1 log x s t q t dt ds 1 > log Z x = B 1 4 B A as a 1 ds A 1 ˆ log 1 4 B A 1 x=b a ; for x 2 B; 1 : 5:12 Combining (5.7)±(5.12) we obtain part (b) of the lemma. 4

16 206 a. b. j. kuijlaars and w. van assche Proof of Theorem 2.1. In view of the previous lemma it suf ces to show that, for any given g > 0, there exists a (necessarily unique) pair A, B with B > A > 0, such that either A ˆ 0; C 0; B < b a ; D 0; B ˆg; or A > 0; C A; B ˆb a ; D A; B ˆg: In both cases it then follows from (5.2) and (5.4) that m < j a; b and that the inequalities (2.5) are satis ed. Consider rst C A; B as given by (5.3). We have C A; B ˆ Z B A s a q s ds; and since q s ds is a probability measure on A; BŠ, we obtain the estimates B a < C A; B < A a : 5:13 From this it follows that in order to nd solutions to C A; B ˆb a we can restrict ourselves to 0 < A < b and B > b. Introducing in (5.3) the change of variables s ˆ A t B A, we get Z 1 C A; B ˆ1 A t B A a dt p ; p 0 t 1 t from which we see that C = B < 0. It is also clear that for any given A < b, C A; B ˆ0: B! 1 Since C A; b > b a by (5.13), there exists a unique B ˆ B A > b such that C A; B ˆb a. Similarly, we have that C = A < 0 and for any given B > b, that C b; B < b a and 8 >< 1 if a > 1 2 ; C A; B ˆC 0; B ˆ A! 0 B a G 1 2 G 1 a >: G 1 2 a if a < 1 2 : It now follows that the mapping A 7! B A is strictly decreasing for A 2 0; b and that 8 >< 1 if a > 1 2 ; B A ˆb; A! b B A ˆ A! 0 >: b G 1 2 a 1 = a if a < 1 G 1 a 2 : G 1 2 Then we get from (5.6) that D A; B A ˆ 1; A! b and after some calculations for the case a < 1 2 (, that D A; B A ˆ 0 if a > 1 2 ; A! 0 b a p cot pa if a < 1 2 : Hence, by continuity, there is an A 2 0; b such that D A; B A ˆ g in case

17 extremal polynomials on discrete sets 207 a > 1 2 or in case a < 1 2 and g > b a p cot pa. This completes the proof of Theorem 2.1 in those cases. What remains is the case a < 1 2 and g < b a p cot pa. In this case we take A ˆ 0 and we note that for B > B 0, where G 1 2 a 1 = a B 0 ˆb G ; 1 2 G 1 a we have C 0; B < b a. Also it follows from the above that D 0; B 0 ˆ b a p cot pa, and that B! 1 D 0; B ˆ0. Hence, since g < b a p cot pa, we can nd B > B 0 such that D 0; B ˆg, which proves Theorem 2.1 in this case also. 6. Conditions on the sets E n In the next sections we will study the asymptotics for polynomials that are extremal with respect to a uniform norm of the form kw n P n k En, where w is an admissible weight on 0; 1 and fe n g is a sequence of discrete subsets (that is, at most countable and no nite accumulation points) of 0; 1. A typical example will be E n ˆ n 1 = a L n 1 = a 1 E, where E is as in Theorem 2.2. In this short section we state a number of conditions that will be imposed on the sequence fe n g. These conditions are analogous to the notion of an admissible triangular scheme used in [6]. For each n, we denote by j n the counting measure of E n, normalized by the factor 1=n, that is, j n A :ˆ 1 n card A Ç E n ; A Ì 0; 1 : 6:1 We will always assume that the following conditions hold. Condition 1. For each n and each compact set A, j n A is nite. Condition 2. There is an admissible constraint j for w, such that Z Z fdj n ˆ fdj for every continuous function f on 0; 1 with compact support. It follows from De nition 2(3) that j has no mass points. Therefore, Condition 2 is equivalent to the condition that j n a; bš ˆ j a; bš 6:2 for every b > a > 0; see, for example, Theorem in [1]. More generally, we have j n K ˆj K ; 6:3 for every compact K Ì 0; 1 such that j K ˆ0, where K denotes the topological boundary of K. For the next condition, we rst need a de nition. De nition 3. We call a point x > 0anormal point for the sequence fe n g if there exist a constant r x > 0, a neighbourhood O x of x and an index n x, such that

18 208 a. b. j. kuijlaars and w. van assche for n > n x and for every pair of different points x 1 ; x 2 2 E n Ç O x,wehave jx 1 x 2 j > r x =n: Condition 3. The set of points that are not normal for the sequence fe n g has logarithmic capacity zero. The above three conditions are suf cient when dealing with extremal uniform norms. We need one more condition that we will use when discussing extremal L p -norms with p < 1. Condition 4. for every b > 0, and There exist two positive functions f and w on 0; 1 such that log f x x! 1 x b ˆ 0; 6:4 log w n ˆ 0; n 6:5 card E n Ç 0; xš < f x w n ; for x > 1; n > 1: 6:6 Example 6.1. If E n ˆ n 1 = a L n 1 = a 1 E, where E satis es the conditions of Theorem 2.2, then the sequence fe n g satis es all Conditions 1±4. To prove this we put c n :ˆ n 1 = a L n 1 = a ; and we write f x, g x if f x =g x!1 for x! 1. Now Condition 1 is clear and for Condition 2 we note that from t k, bk 1 = a L k 1 = a it follows that card E Ç 0; xš, x a L x a : b 6:7 Thus for each xed b > 0, j n 0; bš ˆ 1 n card E Ç 0; bc nš, 1 bc a n L c n b n a, b a L n 1 = a L n 1 = a L n 1 = a a, b a ; b b where the last relation holds by the de nition of the De Bruijn conjugate. Hence j n 0; bš ˆ b aˆ j b a; b 0; bš; where j a; b is the measure with density ab a t a 1 as in 2. Thus Condition 2 holds with j ˆ j a; b ; see (6.2). To get Condition 3, we show that every x > 0 is a normal point. Given x > 0, we get from (6.7) that t k =c n is close to x if the index k is approximately k < c n x a L c b n a ˆ n x a L n 1 = a L n 1 = a L n 1 = a a, n x a : 6:8 b b

19 extremal polynomials on discrete sets 209 Further, because of (2.12), t k 1 t k c n > C c k kc n and it is easy to show from (6.8) that c k =c n is bounded away from 0, with a bound depending on x. Thus t k 1 t k > r x c n n ; which shows that x is a normal point. Finally, we get from (6.7), card E n Ç 0; xš ˆ card E Ç 0; c n xš, c n x a L c b n x a < c n x a c b n x «; where «> 0 is an arbitrary positive number, and c n x is suf ciently large. Thus Condition 4 is satis ed with f x, x a «b a ; w n, c n a «: 7. L 1 -estimates In this section we discuss the asymptotics for weighted extremal polynomials in the uniform norm on varying sets. We suppose that w: 0; 1! 0; 1 is an admissible weight and that j is an admissible constraint for w. Further, we let fe n g be a sequence of discrete subsets of 0; 1. The following lemma is the analogue of [6, Lemma 5.3]. In it we use the measure mw j and the constant Fw j from the constrained energy problem as described in 4. Lemma 7.1. Suppose that the sequence fe n g satis es the Conditions 1 and 2. Then there exists a sequence of monic polynomials P n, with deg P n ˆ n, such that sup kw n P n k 1 = n E n < exp Fw : j 7:1 Proof. In this proof we put m :ˆ mw j and F :ˆ Fw.Ford j > 0 we de ne K d :ˆfx > 0: U m x Q x < F dg; 7:2 where, as usual, we write Q ˆ log w. Then K d is a compact set, since both U m and Q are continuous functions (see 3). We observe that j K d > 0 for an at most denumerable collection of d. Therefore we may assume, by taking a smaller d if necessary, that j K d ˆ0. Then by (6.3), j n K d ˆj K d 7:3 and also j nj Kd ˆ jj Kd : 7:4

20 210 a. b. j. kuijlaars and w. van assche Next, we see from (3.3), (3.4) and (7.2) that K d is a proper subset of supp m. Therefore m K d < 1. It also follows from (3.3) and (7.2) that K d Ç supp j m ˆ0=, so that j and m coincide on K d. In particular, we have j K d < 1 and so it follows from (7.3) that j n K d < 1 for n suf ciently large, say for n > n 0. Now we construct the monic polynomials P n for n > n 0 by specifying their zeros. In K d the set of zeros of P n is E n Ç K d. That gives nj n K d < n zeros in K d. In supp m nk d we choose n 1 j n K d zeros by discretizing the measure m restricted to the complement in K d. In this way we obtain monic polynomials P n, with deg P n ˆ n, such that E n Ç K d Ì fx: P n x ˆ0g Ì supp m ; 7:5 and n P n ˆm: 7:6 Then we have kw n P n k En ˆkw n P n k En n K d, since P n vanishes on E n Ç K d. Outside K d we have from (7.2) the inequality w < exp U m F d. This gives kw n P n k En ˆkw n P n k En n K d < e F d n kexp nu m P n k En n K d Next, it follows from (7.5), (7.6) and Lemma 4.1(c) that kexp nu m P n k 1 = n supp m ˆ 1; so that (7.7) gives sup kw n P n k 1 = n E n < e F d : Now (7.1) follows, since d can be chosen arbitrarily small. < e F d n kexp nu m P n k supp m : 7:7 In order to get an estimate in the other direction we will use Condition 3 as well. A major tool will be the following comparison theorem, which says that the uniform norms of weighted polynomials on the sets E n are in nth root sense comparable to the uniform norms on 0; 1, provided that the extremal measure associated with the weight is dominated by the constraint j. Theorem 7.2. Let w 0 be an admissible weight on 0; 1, and j an admissible constraint for w 0 such that m w0 < j. Let fe n g be a sequence satisfying Conditions 1±3 of the previous section and let V Ì 0; 1 be an open neighbourhood of supp m w0 in 0; 1. Then kw n 0 P sup n k 1 = V nˆ deg P n < n; P n 6 0 kw0 np 1: 7:8 nk En Ç V Theorem 7.2 is actually a basic result in the sense that all later results depend on it, rather than on Condition 3. In future research one might look for generalizations of Condition 3 so that (7.8) continues to hold. The proof of Theorem 7.2 requires some preinary work and is postponed to the end of this section. First we show how to get the correct estimates from it.

21 extremal polynomials on discrete sets 211 Corollary 7.3. Let w be an admissible weight on 0; 1, and j an admissible constraint for w. Let fe n g be a sequence satisfying Conditions 1±3. Then for every sequence of monic polynomials P n, deg P n ˆ n, we have inf kwn P n k 1 = n E n > exp Fw : j 7:9 Proof. We write m :ˆ mw j and F :ˆ Fw j and de ne a second weight w 0 by w 0 :ˆ min w; exp F U m ˆ exp F U m on supp m ; w elsewhere on 0; 1 : Then it is easy to see that w 0 is an admissible weight on 0; 1 with m w0 ˆ m and F w0 ˆ F. Since m w0 ˆ m j w < j, we obtain, from Theorem 7.2 with V ˆ 0; 1, kw n 0 P n k 1 = 0; 1 nˆ kw0 np 1: 7:10 nk En Because F w0 ˆ F, we obtain by Lemma 4.1, inf kwn 0 P n k 1 = n 0; 1 > exp F : 7:11 We also have w 0 < w, so that kw n P n k En > kw0 n P n k En : 7:12 Combining (7.10)±(7.12), we obtain (7.9). From Lemma 7.1 and Corollary 7.3 we obtain the following result, which may be viewed as the constrained analogue of Lemma 4.1. Theorem 7.4. Let w be an admissible weight on 0; 1 and j an admissible constraint for w. Let fe n g be a sequence of subsets of 0; 1 satisfying Conditions 1±3. Then the following hold. (a) If T n is a monic polynomial of degree n minimizing kw n P n k En among all monic polynomials P n, then kwn T n k 1 = n E n ˆ exp Fw : j 7:13 (b) For every sequence of monic polynomials fp n g, with deg P n ˆ n, such that kwn P n k 1 = n E n ˆ exp Fw ; j 7:14 we have n P n ˆm j w: 7:15 Proof. Part (a) follows immediately from Lemma 7.1 and Corollary 7.3. For part (b), we use the weight w 0 introduced in the proof of Corollary 7.3, and note that the relations (7.10)±(7.12) hold for every sequence of monic polynomials fp n g. If (7.14) holds as well, we obtain from this kwn 0 P n k 1 = n 0; 1 ˆ exp F w : j Then (7.13) follows by Lemma 4.1, since Fw j ˆ F w0 and m j w ˆ m w0. We assume that the weight w 0, the measure j and the sequence of sets fe n g are such that the conditions of Theorem 7.2 are satis ed. For the proof of Theorem 7.2 we rst need a lemma.

22 212 a. b. j. kuijlaars and w. van assche Lemma 7.5. For each n, let P n be a monic polynomial of degree n with real simple zeros separated by E n, that is, between two consecutive zeros of P n, there is at least one point of E n. Suppose also that the zeros are uniformly bounded. Let V Ì 0; 1 be a neighbourhood of supp m w0 in 0; 1. Then kw n 0 P n k 1 = V nˆ kw0 np 1: 7:16 nk En Ç V Proof. Passing to a subsequence if necessary, we may assume that the sequence of normalized zero distributions fn P n g converges in weak sense to a probability measure n with compact support. Observe that n < j, since the zeros of P n are separated by E n. Therefore the potential U n is continuous. Then we note that kw0 n P n k V ˆkw0 n P n k 0; 1 < kw0 n exp nu n k 0; 1 k exp nu n P n k 0; 1 : By Lemma 4.1, we have k exp nu n P n k 1 = n 0; 1 ˆ 1, so that sup kw0 n P n k 1 = n V < kw 0 exp U n k 0; 1 : Hence to establish (7.16) it is enough to prove that inf kwn 0 P n k 1 = n E n Ç V > kw 0 exp U n k 0; 1 : 7:17 To prove (7.17), we rst show that the maximum of w 0 exp U n is attained on the set K n :ˆ supp j n Ç supp m w0 : To show this, we use (3.1), (3.2) to nd that ( log w 0 U n ˆ U m w 0 n F w0 on supp m w0 ; < U m w 0 n 7:18 F w0 on 0; 1 : Since m w0 < j by assumption, it follows that m w0 < n outside the set K n. Thus U m w 0 n is subharmonic in CnKn Å, and then the maximum principle for subharmonic functions implies that U m w 0 n attains its maximum on K n. Then by (7.18) we see that also w 0 exp U n attains its maximum on K n. So we can nd x 0 2 K n ˆ supp j n Ç supp m w0 such that w 0 exp U n x 0 ˆkw 0 exp U n k 0; 1 : It could be that x 0 is a non-normal point for fe n g. Therefore we rst take a normal point which is close to x 0 as follows. Let h > 0. Since w 0 and U n are continuous and since x 0 2 supp m w0 Ì V with V open, there is a neighbourhood V 0 Ì V of x 0 such that for every x 0 2 V 0, w 0 exp U n x 0 > kw 0 exp U n k 0; 1 h: 7:19 Furthermore, the set supp j n Ç V 0 is non-empty, and therefore it has positive capacity. Since the set of non-normal points has zero capacity (by Condition 3) there is a normal point x 0 in supp j n Ç V 0. The result is that we have a normal point x 0 for which (7.19) and x 0 2 V Ç supp j n 7:20 hold.

23 extremal polynomials on discrete sets 213 For «> 0, we use D «to denote the interval x 0 «; x 0 «. By de nition of a normal point, there exist a number r > 0 and an «0 > 0 such that for all x 1 ; x 2 2 E n Ç D «0, with x 1 6ˆ x 2, we have jx 1 x 2 j > r=n. We may assume that «0 < 1 2. Let «and d be such that 0 < d < «< «0. In order to estimate kw n 0 P n k En Ç V, we essentially follow a part of the proof of Lemma 2 in [16]; see also [6, Lemma 5.5]. For the convenience of the reader, we include all the arguments. We write P n ˆ Q n R n such that Q n is a monic polynomial whose zeros coincide with the zeros of P n in D «and the zeros of R n are the zeros of P n in RnD «. Let n 1 be the restriction of n to RnD «. Then n R n!n 1, and it follows that uniformly on D d. Then jr n x j 1 = n ˆ exp U n 1 x inf jr n x j 1 = n ˆ inf exp U n 1 x > inf exp U n x : x 2 D d x 2 D d x 2 D d 7:21 The second inequality in (7.21) holds since U n x > U n 1 x for x 2 D d, and this is a consequence of the fact that d < 1 2. Next, we have by (7.20) that n D d < j D d. Therefore, for n large enough, we have n Q n D d ˆn P n D d < j n D d. Then there is an x n 2 D d Ç E n such that all zeros of Q n have distance at least r= 2n to x n. Now, we divide the zeros of Q n into two groups: those that are bigger than x n and those that are smaller than x n. The rst group we arrange in increasing order p 1 < p 2 <...< p k ; for k ˆ k n ; and the second group in decreasing order q 1 > q 2 >...> q l ; for l ˆ l n : Here k l ˆ deg Q n ˆ nn P n D «. We have seen that jx n p 1 j > r= 2n ; jx n q 1 j > r= 2n : It follows from the assumption that the zeros of P n are separated by E n and the fact that jx 1 x 2 j > r=n for all x 1 ; x 2 2 E n Ç D «, with x 1 6ˆ x 2, that Thus jx n p j j > j 1 r=n; for j ˆ 2;...; k; jx n q j j > j 1 r=n; for j ˆ 2;...; l: jq n x n j ˆ Yk j ˆ 1 jx n p j j Yl j ˆ 1 jx n q j j > 1 r k k 1 r l l 1! 1! 2 n 2 n ˆ 1 r k l k 1! l 1!: 4 n Since k l ˆ nn P n D «, we easily get k 1! l 1! > 1 2 nn P n D «1! 2 : Then using also the fact that n P n D «!n D «as and Stirling's formula,

24 214 a. b. j. kuijlaars and w. van assche we obtain inf jq n x n j 1 = n r n D«nn D > «n 2e n D«ˆ rn D n D««: 7:22 2e Combining (7.21) and (7.22) and using x n 2 E n Ç D d, we obtain the following estimate on the polynomials P n, inf kp nk 1 = n E n Ç D d > rn D n D««inf exp U n x : 7:23 2e x 2 D d Letting d! 0 in (7.23), we note that D d Ì V for d suf ciently small, and we have inf kp nk 1 = n E n Ç D «Ç V > rn D n D««exp U n x 2e 0 : Hence inf kwn 0 P n k 1 = n E n Ç D «Ç V > inf w 0 x rn D n D««exp U n x x 2 D «2e 0 : 7:24 Then, letting «! 0 in (7.24), we have n D «!0, and we obtain from (7.19), inf kwn 0 P n k 1 = n E n Ç V > w 0 x 0 exp U n x 0 > kw 0 exp U n k 0; 1 h: Finally, we let h! 0 and (7.17) follows. This completes the proof of the lemma. Having the lemma, we are ready for the proof of Theorem 7.2. Proof of Theorem 7.2. Without loss of generality, we may assume that V is a bounded set. Let Pn be a polynomial of degree at most n such that kw0 n Pn k V kw n 0 P kw0 np n ˆ sup n k V k En Ç V kw0 np : deg P n < n; P n 6 0 : 7:25 nk En Ç V Multiplying Pn by a suitable constant, we may assume, without loss of generality, that kw0 n Pn k V ˆkw0 n Pn k supp mw0 ˆ 1. Let x 0 2 supp m w0 Ì V such that j w0 n Pn x 0 j ˆ 1. Then x 0 62 E n and Pn minimizes the norm kw0 n P n k En Ç V among all polynomials P n of degree less than or equal to n satisfying j w0 n P n x 0 j ˆ 1. Then Q n x :ˆ x n Pn x 0 P n x 1 x 0 7:26 is a monic polynomial of degree n which minimizes k Äw 0 n Q n ke Ä n among all monic polynomials Q n of degree n, where we have put ÄE n :ˆfx: x 1 x 0 2 E n Ç V g; Äw 0 x :ˆ x 1 w 0 x 1 x 0 ; for x 2 ÄE n : 7:27 From this we obtain, by a standard argument, that Q n has n simple zeros in the convex hull of ÄE n. Moreover, the zeros are separated by the points of ÄE n. By (7.26) and (7.27) this implies that Pn has at least n 1 real simple zeros in the convex hull of E n Ç V, which are separated by E n. (It could be that one zero of Pn is outside the convex hull of E n Ç V; it could also be that Pn has degree n 1, namely if 0 is a zero of Q n.) Now we distinguish two cases. If Pn has degree n for all n, and all zeros lie

25 extremal polynomials on discrete sets 215 within a xed compact set, then we can apply Lemma 7.5 to the monic polynomials 1=g n Pn, where g n is the leading coef cient of Pn. Then Theorem 7.2 follows in view of (7.25). In the other case, we have that either Pn has degree n 1, in which case we put d n ˆ 0, or there is one zero of Pn, which we denote by 1=d n, with d n! 0. Then it is easy to nd a bounded sequence fa n g such that for each n, the zeros of x a n Pn x are separated by E n. Since V is bounded, we can take a n such that a n > 2 sup V. Then we de ne a monic polynomial P n of degree n by x a P n x :ˆ c n n 1 d n x P n x 7:28 where c n is some constant. The zeros of P n are uniformly bounded and separated by E n. Then by Lemma 7.5 we have kw0 n P n k 1 = V nˆ kw0 np 1: 7:29 nk En Ç V Since d n! 0 and a n > 2 sup V with a n bounded, we see that the function x 7! j 1 d n x = x a n j is uniformly bounded on V and uniformly bounded away from 0. Hence it follows from (7.28) and (7.29) that kw0 n P n k 1 = V nˆ kw0 np n 1 k En Ç V and the theorem follows by virtue of (7.25). 8. In nite- nite range inequality and L p -estimates We assume again that w ˆ exp Q is an admissible weight on 0; 1, that j is an admissible constraint for w, and that the sequence fe n g satis es Conditions 1±3. An in nite- nite range inequality says that the norm of a weighted polynomial w n P n is attained essentially on some bounded set. For example, one has, for the uniform norm, j w n P n x j < kw n P n k supp mw exp n Q x U m w x F w for x > 0; 8:1 see [13, 18]. In this section we generalize this to discrete sets. We de ne d x :ˆ Q x U mj w x Fw j 8:2 and for «>0, S «:ˆfx > 0: d x <«g: 8:3 Each S «is a compact set which contains supp m j w by (3.3). The sets S «increase with «and if «1 < «2 then S «1 is contained in the interior of S «2. Note that it is possible that supp m j w is strictly contained in S 0. The next lemma gives the analogue of (8.1) for the discrete case. Lemma 8.1. For every «> 0, there exists an n 0 ˆ n 0 «such that for every n > n 0 and every polynomial P n of degree less than or equal to n, we have j w n P n x j < kw n P n k En exp n d x «; for x 2 0; 1 ns «: 8:4

26 216 a. b. j. kuijlaars and w. van assche Proof. De ne, as in the proof of Corollary 7.3, w 0 :ˆ min w; exp Fw j U mj w. Then m w0 ˆ m j w < j, so that by Theorem 7.2 there is an n 0 such that for n > n 0, kw n 0 P n k 1 = n 0; 1 sup deg P n < n kw0 np < e «: 8:5 nk En Let n > n 0, let P n be a polynomial of degree at most n and let x > 0 with x 62 S «. Then w 0 x ˆw x and we get j w n P n x j ˆ j w0 n P n x j < kw n 0 P n k 0; 1 exp nd x < kw n 0 P n k En e n«exp nd x < kw n P n k En exp n d x «: For the rst inequality we used the in nite- nite range inequality (8.1) applied to w 0, and for the second inequality we used (8.5). The last inequality holds because w 0 < w. Corollary 8.2. For every «> 0, there is an n 0 such that for n > n 0 and for every polynomial P n of degree less than or equal to n, kw n P n k En ˆkw n P n k En Ç S «: Proof. This follows immediately from Lemma 8.1. To obtain similar in nite- nite range inequalities in L p with p < 1 we also need Condition 4 on E n. In addition, we will assume for the L p -estimates that Q has at least polynomial growth at 1. Lemma 8.3. Let w ˆ exp Q be an admissible weight on 0; 1 such that Q x > cx b for some c; b > 0 and for x large enough. Let j be an admissible constraint for w and let fe n g be a sequence of sets satisfying Conditions 1±4. Then the following hold for every p 2 1; 1. (a) For every «> 0, there exist d > 0 and n 0 such that for n > n 0 and every polynomial P n of degree less than or equal to n, we have kw n P n k p; En n S «< kw n P n k p; En e d n : 8:6 (b) For every sequence of polynomials P n, with deg P n < n and P n 6 0, we have kw n P n k 1 = p; En nˆ kw n 1: 8:7 P n k En Proof. (a) Let «> 0 and set d :ˆ 1 6 «.Forx 62 S «,wehaved x > «and we can nd c; b > 0 (not necessarily the same as in the statement of the lemma), such that d x > cx b 2 3 «; for x 2 0; 1 ns «: Then by (8.4), with «replaced by 1 3 «, we have for x 62 S «and every polynomial P n of degree less than or equal to n, with n large enough, j w n P n x j < kw n P n k En exp cnx b 2dn :