Christoffel function asymptotics and universality for Szego weights in the complex plane

Transcription

1 University of South Florida Scholar Commons Graduate Theses and Dissertations Graduate School 2009 Christoffel function asymptotics and universality for Szego weights in the complex plane Elliot M. Findley University of South Florida Follow this and additional works at: Part of the American Studies Commons Scholar Commons Citation Findley, Elliot M., "Christoffel function asymptotics and universality for Szego weights in the complex plane" (2009). Graduate Theses and Dissertations. This Dissertation is brought to you for free and open access by the Graduate School at Scholar Commons. It has been accepted for inclusion in Graduate Theses and Dissertations by an authorized administrator of Scholar Commons. For more information, please contact

2 Christoffel Function Asymptotics and Universality for Szegő Weights in the Complex Plane by Elliot M. Findley A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Mathematics College of Arts and Sciences University of South Florida Major Professor: Vilmos Totik, Ph.D. Thomas Bieske, Ph.D. Evguenii Rakhmanov, Ph.D. Boris Schekhtman, Ph.D. Date of Approval: March 3, 2009 Keywords: Faber, Orthogonal Polynomials, Zero-Spacing, Ill-Posed Problems, Potential Theory c Copyright 2009, Elliot M. Findley

3 Acknowledgements I am inexpressibly grateful and deeply humbled to have studied under Vilmos Totik. Among the most insightful men I have ever known, he is certainly the kindest. His fatherly guidance and constant support have enriched my life more than I can articulate and I will forever treasure the time we have spent together. I sincerely thank the mathematics and physics faculty at the University of South Florida for their kind instruction and their many investments in the young minds who depend upon them. Among their rank, I have found at once the most gracious and brilliant people I have ever known. Finally, I pour out my gratitude to the Lord Jesus Christ, the architect of all wisdom and truth and the source of every fortuity. He has heard my prayer and answered.

4 Table of Contents Abstract iii Introduction. Classical Approximation Quadrature Universality Zero-Spacing A Brief History Universality 9 2. Translated Asymptotics Measures on [, ] Asymptotics for Regular Measures Universality Zero Distribution of Orthogonal Polynomials Christoffel Function Asymptotics on General Curves Potential Theory Main Results Applications Orthogonal Polynomials Operator Theoretic Formulation and Ill-Posed Problems Proofs i

5 References 54 About the Author End Page ii

6 Christoffel Function Asymptotics and Universality for Szegő Weights in the Complex Plane Elliot M. Findley Abstract In 99, A. Máté precisely calculated the first-order asymptotic behavior of the sequence of Christoffel functions associated with Szegő measures on the unit circle. Our principal goal is the abstraction of his result in two directions: We compute the translated asymptotics, lim n λ n (µ, x + a/n), and obtain, as a corollary, a universality limit for the fairly broad class of Szegő weights. Finally, we prove Máté s result for measures supported on smooth curves in the plane. Our proof of the latter derives, in part, from a precise estimate of certain weighted means of the Faber polynomials associated with the support of the measure. Finally, we investigate a variety of applications, including two novel applications to ill-posed problems in Hilbert space and the mean ergodic theorem. iii

7 Introduction Let µ be a compactly supported, Borel measure in the complex plane. We are interested in the precise asymptotic behavior of the associated sequence of Christoffel functions, {λ n (µ, z)} n=0. When µ is supported on the real line, the Christoffel functions are defined as follows: λ n (µ, x) = inf P (x) 2 P 2 dµ, (.0.) where the infimum is evaluated over all polynomials P of degree at most n that do not vanish at x. For measures supported on the unit circle or any smooth homeomorphic image thereof, the infimum ranges over the complex polynomials and the definition is slightly modified: λ n (µ, z) = inf P (z) 2 2π P 2 dµ. (.0.2) (In much of the literature, the latter is denoted ω n (µ, e iθ ) when µ is supported on the unit circle. We will adhere to this convention in chapter 2, but revert to the notation of (.0.2) in the final chapter.) From interpolation and quadrature to stochastic processes and statistical inference, this simple sequence has diverse utility. For the simplest example, we look to orthogonal polynomials. Let {p n } n denote the sequence of orthonormal polynomials associated with µ. If we expand P (x) in (.0.) in terms of this sequence, then elemen-

8 tary linear algebra provides the following representation of the Christoffel functions: n λ n (µ, x) = p k (x) 2. (.0.3) k=0 (When µ is supported on a homeomorphic image of the unit circle, as in (.0.2), we will take p n orthonormal to µ/2π, so that this identity will hold for all measures, regardless of the support.) The relationship between Christoffel functions and orthogonal polynomials is deep. Indeed, Paul Nevai, in his very helpful survey of the contributions of Géza Freud ([4]), easily derives Szegő s seminal theory of orthogonal polynomials on the unit circle entirely from a consideration of Christoffel-function asymptotics. For a stellar exploration of further applications of Christoffel function asymptotics, we refer the reader to Grenander s and Szegő s treatise on Toeplitz matrices ([5]). Before proceeding to our main results, we will showcase the versatility of this subject with a few more examples, the first taken from classical approximation theory.. Classical Approximation Consider some orthonormal basis of polynomials {p k } with respect to the measure µ. We may approximate any f L 2 (µ) by the following polynomials: n S n = c k p k, where c k := k=0 f(x)p k (x) dµ(x). S n is a linear function of f, so we may define the linear operators L n (f) := S n and associated norms Ω n (x) := sup L n (f)(x), f L (µ) called the Lebesgue functions. {L n } are integral operators, with associated kernels n K n (x, t) := p k (x)p k (t), k=0 2

9 called the reproducing kernels associated with µ. The reproducing kernels are determined uniquely by their salient property: K n (x, t)p (t) dµ(t) = P (x) for any polynomial P of degree less than n. Let P n denote the best uniform approximant of f by polynomials of degree at most n, with error E n (f). The error of approximation of f by its partial sums is determined by S n f. Since L n (P k ) = P k, S n f (x) S n P n (x) + P n f (x) (Ω n (x) + )E n (f), Now, Ω n (x) = K n (x, t) dµ(t) ( µ(c) ) /2 K n (x, t) 2 dµ(t) ( ) n /2 = µ(c) p k (x) 2 = k=0 ( ) /2 µ(c), (..4) λ n (µ, x) where C is the complex plane. Thus, lower bounds on the series of Christoffel functions imply upper bounds on the rate of convergence of a Fourier series. In fact, the Christoffel and Lebesgue functions sustain a more intimate relationship than the trivial inequality, (..4), suggests: For a class of measures, µ, supported on the interval, [, ], P. Nevai proves in [4, p. 2] that λ n (µ, x) Ω n (x) 2 = o() as n. The asymptotic behavior of the sequence of Christoffel functions has profound implications for approximation theory. For another (perhaps the oldest) example, let us consider integral quadrature ([4])..2 Quadrature Presume that the compact support of µ lies in the interval [, ] so that the zeros of p n (x), denoted x < x 2 < < x n, are real. Let l k denote the k-th Lagrange polynomial: the unique polynomial of degree at most n that vanishes at x j when 3

10 j k and equals at x k. Any polynomial of degree at most n can be precisely interpolated at {x k } n k= as follows: p(x) = n p(x k ) l k (x). k= Since {l k } is an orthogonal set with respect to µ, this implies the following quadrature formula: p(x) 2 dµ(x) = n p(x k ) 2 λ nk, where λ nk = lk 2 dµ are the Christoffel-Cotes numbers. The Christoffel-Darboux identity is fundamental for measures supported on the real line: k= p n (x) p n+ (t) p n (t) p n+ (x) K n (x, t) = c n, (.2.5) x t where c n is constant. It implies that K n (x j, x k ) = 0 whenever j k. Therefore, l k (x) = λ n (µ, x k )K n (x, x k ). Squaring both sides and integrating against µ gives λ nk = λ n (µ, x k ). The asymptotic behavior of Christoffel functions is clearly central to the convergence of quadrature schemes..3 Universality The asymptotic behavior of the reproducing kernels, K n, plays a significant role in the theory of random matrices and statistical mechanics. The so-called universality limit is especially important: For certain classes of measures, K n (x + a/n, x + b/n) lim n K n (x, x) = sin ( (a b)/ x 2) (a b)/ x 2, (.3.6) uniformly as a and b range over a compact interval. D.S. Lubinsky recently advanced the frontier of universality limits by means of a very simple relationship between K n and the Christoffel functions, λ n. (See [0].) For two measures µ µ and their 4

11 associated Christoffel functions λ n and λ n, K n (x, y) K n(x, y) K n (x, x) ( ) /2 ( λn (x) λ ) /2 n(x). λ n (y) λ n(x) Universality follows if the translated limit lim n n λ n (µ, x + a/n) can be calculated. Lubinsky does so, and proves (.3.6) for a large class of measures. Among the primary contributions of this dissertation is the extension of Lubinsky s technique to a still broader class of measures at present, the most general class Szegő s class ([3]). Our final application concerns the asymptotic spacing of the zeros of p n..4 Zero-Spacing The spacing of the zeros, x nn < x n,n < < x n, of p n (x) has received much attention in classical analysis. E. Levin and D. Lubinsky ([9]) were the first to compute the precise value of lim n n (x kn x k+,n ), from universality limits. The connection between universality and zero-spacing is fairly transparent: the Christoffel-Darboux identity, (.2.5), implies that K n (x kn, x k+,n ) = 0. Universality suggests that this is possible only if n (x kn x k+,n ) π x 2 as n. In 2.5, we follow Levin s and Lubinsky s technique to prove the following, the most general result presently known: for almost every x [, ], if for some sequence k = k(n), then x kn x = O ( ), (.4.7) n lim (x n kn x k+,n ) n π =. (.4.8) x2 For another application of Lubinsky s technique, see [22]. Such applications of our subject abound. Later, we will furnish a novel application of Christoffel-function asymptotics to ill-posed problems in operator theory. First, let us briefly review the history of this subject. 5

12 .5 A Brief History The asymptotic behavior of {λ n (µ, z)} n differs dramatically according to whether z is contained in the support of µ or not. If, for instance, µ is supported on the unit circle and z <, Szegő proved ([, p. 434]) that lim λ n(µ, z) = ( z 2 ) D(µ, z), n where D(µ, z) is the Szegő function associated with µ, { π } e iθ + z D(µ, z) = exp 4π π e iθ z log µ (θ) dθ, ( z < ). On the support, when z =, the situation is much simpler: λ n (µ, z) µ({z}). What is the rate of this convergence? Szegő again found the correct formula: for absolutely continuous measures with sufficiently smooth, positive weights, µ, lim n λ n(µ, e iθ ) = µ (θ). (.5.9) n As we have seen, this result has profound implications for a variety of subjects. Consequently, many have attempted to extend Szegő s calculation to ever-broader classes of measures. The most precise asymptotic formulas for the sequence of Christoffel functions were given by Freud in his dissertation and reiterated in his treatise on orthogonal polynomials, [4, p. 27]. Let µ be an absolutely continuous measure supported on [, ] whose weight, µ, is in Lip(α) and satisfies µ (x) Q(x) 2 for some polynomial Q(x). He proves the the following representation [4, p. 254]: λ n (µ, x) = n π µ (x) x + 2 O(n α ) (α < ) λ n (µ, x) = n π µ (x) + O(log n) x2 (α = ). 6

13 In particular, lim n λ n(µ, x) = π x 2 µ (x), (.5.0) n the analogue of Szegő s result for measures supported on [, ]. (In fact, the two results are equivalent by virtue of a simple formula we will cite in the next chapter.) After Freud, the next substantial advancement was made by Máté and Nevai in [2]. Szegő s class consists of measures supported on the unit circle whose weights satisfy Szegő s condition, π π log µ (θ) dθ >. (.5.) Máté and Nevai nearly obtained a full generalization of (.5.9) for the Szegő class: they proved, for almost every θ [ π, π], that e π µ (θ) lim inf n n λ n(µ, e iθ ) lim sup n λ n (µ, e iθ ) µ (θ). (.5.2) n Their result constitutes a dramatic improvement. Although they do not obtain the precise value of the limit (.5.9), their hypothesis Szegő s condition is far less restrictive than Freud s: it makes no assumptions on the smoothness of the weight. In particular, (.5.2) applies to every measure whose weight is bounded below by a positive constant. The holy grail was finally recovered in 99 by the circuitous and technical argument of Máté which establishes (.5.9) for Szegő s class in his landmark paper authored jointly with Paul Nevai and Vilmos Totik ([]). Nevai uses Máté s result to prove (.5.0) via the simple formula to which we have already alluded. Totik further shows that Szegő s condition is unnecessarily restrictive: log µ need only be integrable over an open interval containing x. Under this condition, he proves (.5.0) for the much larger class of regular measures, studied by Ullman ([23]). (This class is far broader than Szegő s: µ is regular if its Christoffel functions satisfy the very general condition lim inf n λ n (µ, x) /n.) Since their paper, no further generalizations have been proved. This dissertation comprises two papers by the author ([2, 3]) which substantially 7

14 improve upon the result of Máté ([]) in the following aspects: In [3], we combine Máté s approach with that of D. Lubinsky in [0] to prove the most general universality limit to date. In [2], we adapt Máté s technique to obtain (.5.9) for the class of measures supported on smooth curves satisfying a modified Szegő condition. This is among the first calculation of the precise value of lim n n λ n (µ, z) for measures with general supports. Totik was the first to compute this value for measures supported on disjoint, real intervals ([2]). The only other result of this type was obtained by Golinskii for Chebyshev-type weights on circular arcs ([6]). 8

15 2 Universality In this chapter, µ denotes a finite Borel measure supported on either the unit circle (equivalently on ( π, π)) or the interval [, ]. µ is the weight associated with its absolutely continuous part. If µ is supported on [, ], the Christoffel functions are defined by λ n (µ, x) = inf P n 2 dµ, where the infimum is taken over all complex polynomials of degree at most n which equal unity at x. For measures supported on the unit circle, the integral is evaluated with respect to µ/2π, and the Christoffel functions are denoted by ω n (µ, z). Let {p n } be the orthonormal polynomials associated with the measure µ. For µ supported on the interval, [, ], they are defined (up to a constant multiple of unit modulus) by the conditions p n (t)t k dµ(t) = 0 and p n 2 dµ =. for all 0 k < n. As before, for measures on the unit circle, the integrals are evaluated with respect to µ/2π. The reproducing kernels for µ are defined by n K n (x, t) = p k (x)p k (t), and are related to the Christoffel functions λ n (x) := λ n (µ, x) by k=0 λ n (x) = K n(x, x). (2.0.) 9

16 The reproducing kernels are so named because of their salient feature: For all polynomials, P, with degree at most n, P (x)k n (x, t) dµ(x) = P (t). We will investigate measures whose weights satisfy Szegő s condition locally, that is, log µ (θ) dθ >, I for some interval I. Our results require that the measures also be regular in the sense of Ullman ([23]). For a comprehensive treatment of the theory of regular measures, see the book by Stahl and Totik ([9]). Regularity of a measure µ with compact support K is equivalent to the following condition: For every sequence, {P n } n=, of polynomials whose degrees are not greater than their indices, where P 2 µ = lim sup n ( ) /n Pn K, (2.0.2) P n µ P 2 dµ and P K = sup P (z). z K The class of regular measures is far larger than Szegő s class. D. S. Lubinsky, in [0], established the following inequality relating the reproducing kernels of two measures, µ µ, to their associated Christoffel functions, λ and λ : K n (x, y) K n(x, y) K n (x, x) ( ) /2 ( λn (x) λ ) /2 n(x). λ n (y) λ n(x) He also proves the following asymptotic formula for translated Christoffel functions of regular measures whose weights are positive and continuous on some interval, I: lim n λ n(x + a/n) = π x 2 µ (x), (2.0.3) n uniformly for x I and a in a compact subset of R. With these two formulae, he 0

17 obtains a universality result for the aforementioned class of measures. The goal of this chapter is to prove Lubinsky s result for the broader class of regular measures which satisfy Szegő s condition locally. This is a substantial relaxation of Lubinsky s hypotheses: it obviates the requirement of continuity in favor of a less restrictive local Szegő condition. Theorem 2.0. Let µ be a regular Borel measure on [, ] satisfying Szegő s condition, I log µ (t) dt > on an open interval I [, ]. Fix A > 0. Then, for almost every x I, uniformly for a, b [ A, A]. K n (x + a/n, x + b/n) lim n K n (x, x) = sin ( (a b)/ x 2) (a b)/ x 2, (2.0.4) The vehicle of this extension is the result of Máté, Nevai and Totik ([]) which establishes the limit (2.0.3) (with a = 0) for regular measures satisfying Szegő s condition on an interval I at almost every x I. We will extend the techniques of these authors to obtain (2.0.3) for the broader class of regular, locally Szegő measures and then mimic Lubinsky s procedure to establish universality on I. Finally, we adapt the technique of Levin and Lubinsky ([9]) to prove a result on the distribution of the zeros of orthogonal polynomials associated with locally Szegő weights. 2. Translated Asymptotics In this section we establish the asymptotics (2.0.3) of the translated Christoffel functions for measures on the unit circle which satisfy Szegő s condition. Let dµ(x) = µ (x)dx + dµ s (x), where µ s is the singular part of µ with respect to Lebesgue measure and µ (x)dx is its absolutely continuous part. It is known (see [7, Theorem 8.6]) that µ (t) = lim τ 0 µ([t, t + τ]) τ (2..5)

18 for almost all t (here, for τ < 0, define µ([t, t + τ])/τ by µ([t + τ, t])/ τ ). Recall that t is a Lebesgue point of µ if τ lim µ (t + u) µ (t) du = 0. (2..6) τ 0 τ 0 We use the following terminology: t is a Lebesgue point of µ if the limit in (2..5) exists at t, and with this limit as µ (t), (2..6) is true. Thus, almost all points are Lebesgue points of µ. Theorem 2.. Let µ be a measure on ( π, π) satisfying Szegő s condition. A > 0. Then, for almost every t ( π, π), we have Fix uniformly for a [ A, A]. lim n ω ( ) n µ, e i(t+a/n) = µ (t), (2..7) n Furthermore, (2..7) holds at every t which is a Lebesgue point of µ and for which e it is a Lebesgue point of the Szegő function (see (2..9)) associated with µ. The upper limit actually holds for all finite Borel measures. This follows from the next lemma, an improvement of Lebesgue s result on the convergence of Fejér means (see [7, p. 244]). In what follows, σ n (µ, z) is the n-th Fejér mean of the measure µ, given by with normalized kernels σ n (µ, z) = F n (t) = 2π(n + ) F n (z t) dµ(t) sin 2 ((n + )t/2) sin 2. (t/2) Lemma 2..2 Let µ be an absolutely continuous Borel measure on ( π, π) such that µ µ([ t, t]) (0) := lim, t 0 2t exists. Then the translated Fejér means σ n (µ, e ia/n ) µ (0) as n uniformly for 2

19 a [ A, A]. The proof requires a result concerning Hardy s maximal function which is defined for a measure µ supported on the real line by Mµ(x) = sup t>0 µ([x t, x + t]). 2t Lemma 2..3 Let f be an even, positive function on [ π, π], decreasing away from 0. Then, for any measure µ supported on an interval I = [ t, t] [ π, π], we have Proof. I f dµ Mµ(0) f(t) dt. I Let I = I I 2 I n be a nested sequence of symmetric intervals and choose positive numbers a,..., a n, such that f(t) s(t) := k a kχ Ik (t). Then f dµ k a k µ(i k ) Mµ(0) k a k I k = Mµ(0) s(t) dt, where denotes Lebesgue measure. Since f decreases away from 0 and is even, there is a sequence of simple functions s n of the form s n = k a kχ Ik which dominate f and for which s n (t)dt f(t)dt. This establishes the lemma. Proof. (Lemma 2..2) Without loss of generality assume µ (0) = 0 (subtract a constant if necessary). We begin with the claim that for some constant C > 0, ( F n (t) min n +, ) π 2(n + )t 2 C(n + ) + (n + ) 2 t 2 (2..8) for every t [ π, π] and n 0. To see the first inequality, observe that sin t t 2 π for all t [ π, π]. Thus, F n (t) π 2 2π(n + ) t. 2 But the kernel F n is the average of the first n + Dirichlet kernels, so its maximum is achieved at 0 and is equal to (n + )/2π. This proves the first inequality. The second 3

20 inequality is true for any C > + π/2. Indeed, n + < On the other hand, π C(n + ) C(n + ) > 2(n + )t2 + (n + ) 2 t2 + π/2 > n +. x π 2 (2 + π)x ( + x)π + x π/(2 + π), x and so π 2(n + )t 2 n + (n + )2 t 2 π 2 C(n + ) Cπ/(2 + π) + (n + ) 2 t2 (n + )t 2 π( + π/2) (n + )t 2 (2 + π) = π 2(n + )t 2. This establishes (2..8). Now, choose ɛ > 0 and let I 0 = [ a 0, a 0 ] be an interval centered at 0 such that µ(i) < ɛ I for every symmetric interval I I 0. Define µ 0 = µ I0 and µ = µ µ 0. We show that σ n (µ 0, e ia/n ) and σ n (µ, e ia/n ) converge to zero uniformly for a [ A, A]. Since µ = µ 0 + µ, this will establish the lemma. Using (2..8), we find where the functions σ n (µ, e ia/n ) = F n (t a/n) dµ(t) [ π,π]\i 0 f (a) n (t) = C(n + ) + (n + ) 2 (t a/n) 2 f n (a) t >a 0 (t) dµ(t), tend to zero uniformly for a [ A, A] and t > a 0. So do their integrals, which establishes the convergence for µ. To handle the integral with µ 0, define L n (t) := sup f n (a) (t) = a [ A,A] C(n + ), C(n+), +(n+) 2 ( t A/n) 2 if t A/n, if t > A/n. 4

21 The functions L n are even and decreasing away from 0, so we may apply Lemma 2..3 together with the estimate F n (t a/n) L n (t) to obtain σ n (µ 0, e ia/n ) = π π for all a [ A, A]. But, I 0 L n (t) dt π π F n (t a/n) dµ 0 (t) L n (t) dµ(t) ɛ I 0 I 0 L n (t) dt, I 0 L n (t) dt = AC n + n + 2 2AC π A/n C(n + ) + (n + ) 2 (t A/n) 2 dt C + u 2 du which is finite and independent of n. Since ɛ is arbitrary, this completes the proof. Lemma 2..4 Let µ be a finite Borel measure on [ π, π]. Then, at every Lebesgue point t [ π, π] of µ, lim sup n uniformly for all a [ A, A]. n ω n ( µ, e i(t a/n) ) µ (t), Proof. Without loss of generality, we may assume that µ is absolutely continuous. Fix a Lebesgue point of µ, t [ π, π], and define the polynomial P by P (e i(t a/n) ) = and P (e iθ ) 2 = P (ζ) = n e ij(t a/n) ζ j. n j=0 ( ) sin n(t a/n θ) 2 2 = 2π n sin t a/n θ n F n (t a/n θ), 2 5

22 so that ( n ω ) n µ, e i(t a/n) n P ( ) e iθ 2 ( dµ(t) = σ ) n µ, e i(t a/n) 2π By Lemma 2..2, the right hand side converges to µ (t) (uniformly for a [ A, A]), which completes the proof. The proof of the lower bound relies heavily on the Szegő function associated with the measure µ: { π } e iθ + z D(z) = D µ (z) = exp 4π π e iθ z log µ (θ) dθ, ( z < ) (2..9) For measures satisfying Szegő s condition, this function is in Hardy s class, H 2 (see e.g. [7, ], where D 2 is called the outer function associated with µ ). The following properties will be implicitly invoked throughout the proof: D(z) has nontangential limit D(e it ) at almost every point z = e it, which satisfies D(e iθ ) 2 = µ (θ) at almost every θ [ π, π]. In particular, the nontangential limit exists at every z = e it which is a Lebesgue point for D(e iθ ) (see Fatou s theorem [7, p. 34] and apply it to the complex valued harmonic function D). Finally, π π π D(e iθ ) dθ = lim D(re iθ ) dθ. r π Proof. (Theorem 2..) We prove that for almost every e it T, lim inf n n ω n(µ, e i(t a/n) ) µ (t) (2..0) uniformly for a [ A, A]. This together with Lemma 2..4 proves Theorem 2... Without loss of generality we may assume that µ is absolutely continuous, since if not, the monotonicity of the Christoffel functions implies that ω n (µ, z) ω n (µ, z). In this case, (2..0) only increases. We shall show that (2..0) holds at every point t which is a Lebesgue point of µ and for which e it is a Lebesgue point of the Szegő 6

23 function D(e iθ ). Thus, let t be such a point. We may assume that e it = and hence that and τ lim µ (u) µ (0) du = 0 τ 0 τ 0 τ lim D(e iu ) D() du = 0. τ 0 τ 0 These imply that there is a set S with 0 as a density point so that the limit at 0 of µ (u) along S is µ (0), while the limit of D(e iu ) along S is D(). This combined with D(e iu ) 2 = µ (u) a.e. implies that D() 2 = µ (0). Since we want to prove the lower estimate (2..0), we may also assume that for the particular n N appearing in the proof and a [ A, A] the inequality n ω n (µ, e ia/n ) D() 2 = µ (0) holds. For a [ A, A] and n N, let q = q a,n = e ia/n and choose polynomials, P = P a,n, of degree at most n for which ω n (µ, q) = P ( ) e iθ 2 dµ(θ). 2π P (q) 2 Now fix a small ɛ > 0 and α > 4/ɛ 2 with also α > 2A and define K = [ α/n, α/n] and K 2 = [ π, π] \ K. We claim that for sufficiently large n, P (e iθ ) 3 P (q) for all θ K a [ A, A]. If ζ =, and ρ = /n, then for any n and a, P 2 (ρζ)d 2 (ρζ) = 2πi z = P 2 (z)d 2 (z) z ρζ dz 2π π π P D 2 (e iθ ) ρ dθ = π ( ) P e iθ 2 dµ(θ) = 2π( ρ) π ρ P (q) 2 ω n (µ, q) P (q) 2 D() 2. It is known that the zeros of P lie on the unit circle, T, so it is elementary (see [, page 438]) that ( ) n ( + ρ P (ρζ) P (ζ) = ) n P (ζ) P (ζ) (2..) 2 2n 2 and 7

24 Thus, P 2 (ζ)d 2 (ρζ) 4 P (q) 2 D() 2 for all n N, a [ A, A], and ζ =. Now = e i0 is a Lebesgue point of D(e iθ ), so D(z) has a nontangential limit D() at. Hence, for large n, and arg ζ K, D(ρζ) 2 ( ɛ) D() 2 and, therefore, P (ζ) 2 ɛ P (q) 3 P (q), (2..2) which proves the claim. Now let r = + ɛ/n. We show that I /D() Ĩ/D() < 4ɛ P (q) (2..3) where I = π P (e iθ ) r n q n e inθ 2π π r qe iθ D(eiθ ) dθ Ĩ = π P (e 2π iθ ) r n q n e inθ π r qe iθ D(eiθ ) dθ. To this end, define I j = P (e 2π Kj iθ ) r n q n e inθ r qe iθ D(eiθ ) dθ Ĩ j = P (e 2π Kj iθ ) r n q n e inθ r qe iθ D(eiθ ) dθ 8

25 We establish (2..3) by analyzing the following decomposition: I /D() Ĩ/D() = + I 2 D() + ( I D() ) P (e 2π K iθ ) r n q n e inθ r qe iθ dθ ( Ĩ D() + 2π ) P (e K iθ ) r n q n e inθ r qe iθ dθ Ĩ2 D(). (2..4) To establish an upper bound for the first term on the right hand side, observe that 2π P (e K iθ ) r n q n e inθ ( D(e iθ ) D() ) dθ r qe iθ max θ K P (eiθ ) r n q n e inθ r qe iθ D(e iθ ) D() dθ (2..5) 2π K By (2..2), the maximum is max θ K P (eiθ ) r n q n e inθ r n r qe iθ 3 P (q) r 3 P (q) n ɛ, and for large n, by the Lebesgue point property, K D(e iθ ) D() dθ < ɛ 2 so that (2..5) is ɛ P (q). But then, 6α 2α n = ɛ2 3n, I D() P (e 2π K iθ ) r n q n e inθ r qe iθ dθ ɛ P (q) D() (2..6) An analogous argument yields Ĩ D() + P (e 2π K iθ ) r n q n e inθ r qe iθ dθ ɛ P (q) D(), (2..7) for the third term of (2..4). For the second term of (2..4) the Cauchy inequality gives I 2 2 P (e iθ ) 2 D(e iθ ) 2 dθ 2π K 2 2π K 2 r n q n r e iθ q 2 dθ. 9

26 The first integral is π P (e iθ ) 2 D(e iθ ) 2 dθ = π P (e iθ ) 2 dµ(θ) = ω n (q) P (q) 2 2π π 2π π The second integral is 2π K 2 r n+ e i(θ+a/n) r 2 dθ 2π K 2 e i(θ+a/n) dθ 2 = 2π K cos(θ + a/n) dθ = 2π since 2 θ + a/n θ on K 2. Thus, K 2 (2 sin((θ + a/n)/2)) 2 dθ ( ) 2 2 (θ + a/n) dθ π 2 2π K 2 π π α/n θ dθ = π n 2 α, I 2 2 π n α ω n(q) P (q) 2 π α D() 2 P (q) 2 ɛ 2 D() 2 P (q) 2, so that I 2 ɛ P (q) D(). (2..8) An analogous argument establishes that Ĩ2 ɛ P (q) D(). (2..9) Equations (2..6), (2..7), (2..8), and (2..9) prove (2..3). If P (ζ) = n k=0 c kζ k, let P (ζ) n k=0 c kζ n k. For ζ = e iθ, P (ζ) = P (ζ)ζ n, so Ĩ = π 2π π P (e iθ ) r n q n e iθ r e iθ q D(eiθ ) dθ = 2πi ( ) n q r ζ = P (ζ)d(ζ) r ζq dζ But r >, so F (ζ) = P (ζ)d(ζ)/(r ζq ) is holomorphic in = { z < } and has 20

27 no singularities there. Also, D H 2 ( ), hence so is F, and therefore, ζ = which, together with (2..3), implies that F (ζ) dζ = lim F (τζ) dζ = 0, τ ζ = I < 4ɛ P (q) D(). Then, Finally, let H n (z) = z n n z = k=0 z k π P (e iθ )H n (rq e iθ )D(e iθ ) dθ = π P (e iθ ) r n q n e inθ 2π π 2π π r qe iθ D(eiθ ) dθ = I π P (e iθ )D(e iθ )e iθ dθ = I 2π π r q e iθ P (ζ)d(ζ) 2πi ζ = r q ζ dζ = I + P (r q)d(r q), so that for large n, But, 2π P (r q)d(r q) 2π π π π π P (e iθ )H n (rq e iθ )D(e iθ ) dθ π P (e iθ )D(e iθ ) 2 dθ = 2π π P (e iθ )H n (rq e iθ )D(e iθ ) dθ 4ɛ P (q)d() 2 ( n k=0 π H n (rq e iθ ) 2 dθ 2π π ) ( + ɛ ) 2k π P (e iθ ) 2 dµ(θ) n 2π π n P (q) 2 ω n (q) 2

28 Thus, P (r q)d(r q) 4ɛ P (q) D() P (q) nω n (q) But, as in (2..), we have ( ) n ( ) n + /r 2 + ɛ/n P (r q) P (q) = P (q) = 2 2( + ɛ/n) ( ) n ɛ/n P (q) ( e ɛ/n) n P (q) e ɛ P (q) 2( + ɛ/n) Also, as n, r q non-tangentially, so that D(r q) D(), and therefore (e ɛ 4ɛ) D() lim inf n nωn (q) This completes the proof of (2..0) since ɛ > 0 is arbitrary. 2.2 Measures on [, ] On [, ] the Szegő class consists of all finite Borel measures µ with support on [, ] for which log µ (x) dx >, (2.2.20) x 2 For measures in this class, the Szegő function is defined on C \ [, ] by ( D(z) = D z2 µ (z) = exp 2π log µ (x) z x ) dx x 2 (2.2.2) with the branch of the square root that is positive for positive z. This has nontangential limit D(x) at almost every x [, ] with D(x) 2 = µ (x). Theorem 2.. translates appropriately for measures on the interval [, ]. Theorem 2.2. Let µ be a measure on [, ] satisfying Szegő s condition. Then, uniformly for a [ A, A], lim n λ n(µ, x + a/n) = πµ (x) x 2 (2.2.22) n 22

29 for almost every x [, ]. Moreover, (2.2.22) holds at every x (, ) which is a Lebesgue point of µ and for D. The proof relies on orthogonal polynomials. Define the integral operators, G n,by G n (f, x) = λ n (µ, x) f(t)k 2 n(µ; x, t)dµ(t), where K n are the reproducing kernels defined in the introduction. The next results are found in [3] as well as [5, p. 230, Corollary 4.3.]. Lemma Let f be continuous on [, ] and let µ be a measure on [, ] which satisfies Szegő s condition. Then, lim n sup G n (f, x) f(x) = 0. x [,] Lemma Let µ be a measure on [, ] satisfying Szegő s condition. Then for any fixed m we have lim sup n x [,] λ n+m (µ, x) λ n (µ, x) = 0. We prove an extension of a corollary given in the same paper. Corollary Let g be a nonnegative function on [, ] and assume that there exists a polynomial P m for which P m g and P m g are continuous on [, ]. Let dµ g (x) = g(x)dµ(x), where µ is a measure on [, ] satisfying Szegő s condition. Then λ n (µ g, x + a/n) lim n λ n (µ, x + a/n) = g(x) uniformly in every compact subset of (, ) devoid of zeros of P m and uniformly for all a [ A, A]. 23

30 Proof. G n (µ; gpm, 2 x) = λ n (µ, x) Pm(t)g(t)K 2 n(µ; 2 x, t)d µ(t) = λ n (µ, x) Pm(t)K 2 n(µ; 2 x, t)d µ g (t) λ n (µ, x)λ n+m (µ g, x)pm(x)k 2 n(µ; 2 x, x) The final equality follows from (2.0.). Thus, uniformly in a [ A, A] = λ n+m(µ g, x) P 2 λ n (µ, x) m(x). lim sup n λ n+m (µ g, x + a/n) λ n (µ, x + a/n) lim n G n (µ; gp 2 m, x + a/n) P 2 m(x + a/n) = g(x) (2.2.23) uniformly on compact subsets of (, ) devoid of zeros of P m. Also, G n (µ g, Pm/g, 2 x) = λ n (µ g, x) P 2 m(t)k 2 n(µ g ; x, t)dµ(t) λ n (µ g, x)λ m+n (µ, x)p 2 m(x)k 2 n(µ g ; x, x) = λ n+m(µ, x) λ n (µ g, x) P 2 m(x) So, as above, lim sup n λ n+m (µ, x + a/n) λ n (µ g, x + a/n) lim n G n (µ g, P 2 m/g, x + a/n) P 2 m(x + a/n) = g(x) (2.2.24) locally uniformly in (, ) (away from zeros of P m ) and uniformly in a [ A, A], since P m /g L and µ g also satisfies Szegő s condition. The result now follows from Lemma and inequalities (2.2.23) and (2.2.24). The proof of Theorem 2.2. relies on an equation which relates the Christoffel function of a measure, µ, supported on the interval [, ] to its projection onto the unit circle. Defined ν by ν(e) := µ({cos θ : θ E}) for E [ π, 0) or E [0, π). Note that ν (t) = µ (cos t) sin t. See [, Lemma 6, p. 446] for a proof of the following: 24

31 Lemma Given an arbitrary positive finite Borel measure µ on [, ], for every integer n > and every t [ π, π), where g(x) = x 2. ω 2n (ν, e it ) = π λ n (µ, cos t) + π sin 2 t λ n (µ g, cos t), With these preliminaries, we can proceed with the proof of Theorem Proof. (Theorem 2.2.) As n, cos(t a/n) = cos t + a n sin t + O ( n 2 ) = cos t + a n O(). From Lemma and Corollary 2.2.4, it follows that λ n (µ g, cos(t a/n)) lim n λ n (µ, cos(t a/n)) = sin 2 t uniformly on compact subsets of ( π, π)\{0} and a [ A, A]. Consequently, Lemma gives lim n ω 2n (ν, e i(t a/n) ) λ n (µ, cos(t a/n)) = 2π, (2.2.25) uniformly on compact subsets of ( π, π)\{0} and a [ A, A]. We write cos(t a/n) = cos t + b/n, and note that, while a runs through [ A, A], the b = b a,t covers an interval [ B A,t, B A,t ] (depending on t ( π, π) \ {0} and on A), and here for any t ( π, π) \ {0} and any B > 0 there is an A such that [ B, B] [ B A,t, B A,t ]. The same is true uniformly if t runs through a compact subset of ( π, π) \ {0}. So, since the convergence in (2.2.25) is uniform over any interval [ A, A], we get ω 2n (ν, e i(t a/n) ) lim n λ n (µ, cos t + b/n) = 2π, uniformly for a [ A, A]. Therefore, by Theorem 2.., lim n λ n(µ, cos t + b/n) = πν (t), n 25

32 for almost every t uniformly in b [ B, B]. Substituting cos t with x gives lim n λ n(µ, x + b/n) = π x 2 µ (x), n for almost every x uniformly in b [ B, B], which completes the proof of the first assertion of Theorem To prove the last statement, we only need to observe that under the conformal map w = z z 2, the complement of [, ] is mapped into the unit disk, and the Szegő function D µ is mapped into the Szegő function D ν (see (2..9)) associated with ν. Therefore, x = cos t is a Lebesgue point of µ and for D µ (u) precisely when t is a Lebesgue point of ν and e it is a Lebesgue point of D ν (e iu ). Taking these into account, the last statement follows from Theorem Asymptotics for Regular Measures The assumptions of Theorem 2.2. are unnecessarily restrictive. Regular measures (defined in the introduction by (2.0.2)) that satisfy Szegő s condition locally on I, an interval generate Christoffel functions that exhibit the asymptotic behavior of (2.0.3) when x I. Since we shall now work with a local Szegő condition, we require a local Szegő function. Thus, let us suppose that µ is a finite Borel measure on [, ], and on some open interval I [, ] it satisfies Szegő s condition, i.e. We define I log µ (t)dt >. ( ) i D (z) = Dµ(z) log µ (x) = exp 2π I z x dx. (2.3.26) D (z) has a nontangential limit (from the upper half plane) D (x) at almost every x I and D (x) 2 = µ (x) a.e. (see Lemma below). Theorem 2.3. Let µ be a regular measure on [, ] and let I be an open interval 26

33 in [, ] such that log µ (θ) dθ >. Then, for almost every x I and for every A > 0, I uniformly for a [ A, A]. lim n λ n(µ, x + a/n) = π x 2 µ (x), (2.3.27) n Moreover, (2.3.27) holds at every x I which is a Lebesgue point of µ and of D. Proof. For ɛ > 0, let dν ɛ (x) = dµ(x) + ɛχ [,] (x)dx, where χ is the characteristic function. This measure clearly satisfies Szegő s condition globally and therefore, by Theorem 2.2., lim sup n λ n (ν ɛ, x + a/n) = π x 2 (µ (x) + ɛ). n But µ < ν ɛ for every ɛ and so, by virtue of the monotonicity of the Christoffel functions with respect to measures, lim sup n λ n (µ, x + a/n) π x 2 µ (x), n for almost every x (, ) and uniformly for a [ A, A]. To prove the lower bound, lim inf n n λ n(µ, x + a/n) π x 2 µ (x), (2.3.28) uniformly for a [ A, A] and almost every x I, let dν(x) = dµ(x) + χ [,]\I (x)dx. In Lemma we will prove that x I is a Lebesgue point of Dµ if and only it is a Lebesgue point of the Szegő function D ν associated in (2.2.2) with ν. Therefore, it is enough to show (2.3.28) at every x I which is a Lebesgue point of ν and of 27

34 D ν. Let x be such a point. Assume to the contrary that there are sequences, N N, {a n [ A, A] : n N }, and a real number r < π x 2 for which n λ n (µ, x + a n /n) < rµ (x) for n N. (2.3.29) Let P n represent a polynomial of degree at most n for which P n (x + a n /n) = and λ n (µ, x + a n /n) = Now, for some η > 0, define the polynomials ( ( ) ) 2 [ηn] x + an /n t Q n (t) = P n (t). 4 P (t) 2 dµ(t). (2.3.30) (Here [y] is the integral part of y.) Evidently, Q n (x+a n /n) = and Q n (t) P n (t), t [, ]. Furthermore, since a n /n 0 as n, there is a ρ < such that ( ( ) ) Q n (t) 2 [ηn] x + P n (t) = an /n t < ρ 2n (n ) (2.3.3) 4 on [, ]\I. Now, by (2.0.2), every regular measure σ on [, ] has the property that for any s > and any sequence of polynomials, R n, max R n(t) 2 < s n t [,] for sufficiently large n (see [9, Theorem 3.2.3]). But, R n (t) 2 dσ(t), P n (t) 2 dµ(t) µ([, ]) since λ n (µ, t) µ([, ]). Therefore, for any s > and sufficiently large n N, we have max P n(t) 2 < s n, t [,] 28

35 which, together with (2.3.3) and with the assignment s = /ρ, implies that Q n (t) < ρ n on [, ]\I for sufficiently large n N. This, together with (2.3.29) and (2.3.30) implies that where c = r( + η). Q n (t) 2 dν(t) r n µ (x) + ρ n c < π x 2. In this case [n( + η)] [,]\I dt = c n( + η) ν (x) + o(/n), Since this holds for arbitrary η, we can fix its value so that Q n (t) 2 dν(t) cν (x) + o() for n N, which implies that lim inf n n λ(ν, x + a n/n) < π x 2 ν (x), since Q n (x + a/n) = and Q n has degree [n( + η)]. This contradicts Theorem 2.2. (at x) and this contradiction proves the claim, pending the proof of the next lemma. Lemma With dν(x) = dµ(x) + χ [,]\I (x)dx the nontangential limit (from the upper half plane) at an x I exists for Dµ(z) if and only if it exists for D ν (z). Furthermore, x is a Lebesgue point of Dµ(u) precisely when it is a Lebesgue point of D ν (u). Proof. the function Suppose first that the nontangential limit of D µ(z) exists at x I. Consider ( z2 D µ(z) = exp 2π I log µ (x) z x dx x 2 ). (2.3.32) Since this differs on I R from D ν by an analytic multiplicative factor, it is enough to prove the existence of the nontangential limit for D µ at x. But ( ) i D µ(z)/d µ(z) = exp log µ (t)h(z, t)dt 2π I (2.3.33) 29

36 with h(z, t) = ( ) z 2 z x t + z = t 2 t2 ( z 2 + t 2 ), which is an analytic function (in z) on I, so the nontangential limit ( ) i D µ(u)/d µ(u) = exp log µ (t)h(u, t)dt 2π I (2.3.34) of (2.3.33) certainly exists at any u I. This shows that, indeed, the nontangential limit of D µ = Dµ ( D µ/d µ) also exists at x. It is a simple exercise to show that if f is a nonzero C -function, then x is a Lebesgue point of Dµ(u) if and only if it is a Lebesgue point of f(u)dµ(u). With f(u) = D µ(u)/d µ(u) this is the same as x being a Lebesgue point of D µ(u). Applying the same argument once more with f(u) = D ν (u)/ D µ(u) we find that x is a Lebesgue point of Dµ(u) if and only if it is a Lebesgue point of D ν (u). The proof of the converse implication (i.e. going from D µ(u) to Dµ(u)) is very similar. Since the real part of i/(z t) for z = x + iy is the Poisson kernel y/((x t) 2 + y 2 ) of the upper half plane, it is a standard exercise to show that Dµ(z) 2 tends nontangentially to µ (x) at every Lebesgue point of µ. 2.4 Universality We will now apply Lubinsky s technique ([0]) to prove the universality result, Theorem The proof follows directly from the following: Lemma 2.4. Let µ and µ satisfy the conditions of the hypothesis in Theorem 2.0. and assume further that µ (x 0 ) = (µ ) (x 0 ) > 0 for some x 0 I which is a Lebesgue point of µ, µ, D µ and D µ (see (2.3.26)). Then lim n n K n Kn (x 0 + a/n, x 0 + b/n) = 0, uniformly for a, b [ A, A], where K n and K n are the reproducing kernels associated respectively with µ and µ. 30

37 Proof. First assume that µ µ on [, ]. It was proven in [0, (3.5)] that K n (x, y) K n(x, y) K n (x, x) λ n (x) λ n (y) λ n(x) λ n(x). (2.4.35) Here, the λ s are the associated Christoffel functions. By Theorem 2.3. lim n(x 0 + a/n) = π n x 2 0µ (x 0 ) (2.4.36) and lim n λ n(x 0 + a/n) = π x 2 0(µ ) (x 0 ), (2.4.37) uniformly for all a [ A, A]. Now replace x with x 0 + a/n and y with x 0 + b/n in (2.4.35). Then, since µ (x 0 ) = (µ ) (x 0 ), it follows that lim n λ n(x 0 + a/n) K n K n (x 0 + a/n, x 0 + b/n) = 0, uniformly for a, b [ A, A], which implies, again because of (2.4.36), that lim n n K n Kn (x 0 + a/n, x 0 + b/n) = 0 uniformly for a, b [ A, A]. (2.4.38) For arbitrary µ and µ satisfying the conditions of the lemma, define the measure dν(x) = max(dist(x, I), µ (x), (µ ) (x))dx + dµ s (x) + dµ s(x), where µ and µ s denote, respectively, the absolutely continuous and singular components of the measure µ. Clearly, ν µ, µ so ν satisfies Szegő s condition locally on I and is a regular measure on [, ]. Hence (2.4.38) holds for the pairs (ν, µ) and (ν, µ ) and, consequently, for (µ, µ ). This completes the proof. 3

38 Proof. (Theorem 2.0.) Let x be a Lebesgue point of µ and of D µ (see (2.3.26)) and assume µ (x) > 0. Define dµ (u) = µ (x)du, u [, ]. By Lemma 2.4. lim n n K n Kn (x + a/n, x + b/n) = 0. (2.4.39) Applying Lubinsky s original theorem [0, Theorem.] to K n we find that, as n, K n (x + α/µ (x)k n(x, x), x + β/µ (x)k n(x, x)) K n(x, x) sin π(α β), (2.4.40) π(α β) uniformly for any fixed B and α, β [ B, B]. Now, choose α = α n and β = β n so that α/µ (x)kn(x, x) = a/n and β/µ (x)kn(x, x) = b/n. Then, because of (2.4.37), as n, α a/π x 2 and β b/π x 2, hence the statement in Theorem 2.0. follows from (2.4.40) and (2.4.39) (see also (2.4.37)). 2.5 Zero Distribution of Orthogonal Polynomials Finally, we apply the techniques of Levin and Lubinsky to extend their Theorem. in [9]. The application of universality to study zero spacing is also found in Freud s text, [4] as well as in [22]. In what follows, x kn denotes the k-th zero of the orthogonal polynomial p n associated with a given measure µ, defined on the interval [, ]. Let the zeros be ordered according to x nn < x n,n < x n 2,n < < x n. (2.5.4) Theorem 2.5. Let µ be a finite regular Borel measure on [, ] which satisfies Szegö s condition locally in some interval I. Fix an x I for which (2.0.4) holds and for which µ (x) > 0. If for some sequence k = k(n) x kn x = O ( ), n then lim (x n kn x k+,n ) n π x =. 2 32

39 Proof. Let l kn be the Lagrange interpolation polynomial associated with the point x kn which vanishes at every other zero of p n and satisfies l kn (x kn ) =. l kn has the representation l kn (z) = K n(x kn, z) K n (x kn, x kn ). There is a bounded sequence, a n such that x kn = x + a n n. Since µ (x) > 0, Theorem 2.3. implies that K n (x kn, x kn )/K n (x, x) as n, so that Theorem 2.0. applied to the Lagrange polynomials gives ( l kn x + b ) = sin((a n b)/ x 2 ) n (a n b)/ + o(), (2.5.42) x 2 uniformly for bounded b. The first term on the right hand side (taken as a function of b), changes sign when a n b = π x 2 and therefore, by (2.5.4), the zero x k+,n has the representation, for some bounded sequence b n < 0 with x k+,n = x kn + b n n, (2.5.43) lim inf n b n π x 2. So, by (2.5.42), 0 = l kn (x k+,n ) = sin(b n/ x 2 ) b n / x 2 + o(). (2.5.44) We claim that lim n b n = π x 2. To this end choose any subsequence of {b n } with limit point b. Equation (2.5.44) gives, upon passing through this subsequence, sin(b/ x 2 ) b/ x 2 = 0. 33

40 Since π x 2 b 0, we must have b = π x 2, which proves the claim. This together with equation (2.5.43) gives (x k+,n x kn ) n = b n π x 2, as n, which completes the proof. 34

41 3 Christoffel Function Asymptotics on General Curves We now turn our discussion to measures with general supports in the plane. How does the asymptotic behavior of the sequence of Christoffel functions depends on the geometry of the support, Γ := supp(µ), when the point of evaluation, z Γ? It is easy to see, for instance, that λ n (µ, z) µ({z}) as n if Γ is sufficiently regular (e.g. smooth) and z lies on the outer boundary of Γ. What is the rate of this convergence for general measures, and how is this rate determined by Γ? For measures supported on a circle or a union of intervals, we already know the answer: lim n λ n(µ, e iθ ) = n 2π µ (e iθ ) or lim n λ n (µ, x) = π x 2 µ (x) (3.0.) n (almost everywhere on the supports). These equations hint at the answer to our question, but to understand the hint we require some elementary potential theory. 3. Potential Theory Every measure, µ, with compact support, Γ, has associated logarithmic energy defined as follows: I(µ) := log z w dµ(z)dµ(w). If Γ is sufficiently dense, then I(ν) > ɛ > 0 for all probability measures ν supported on Γ. The fundamental theorem of potential theory guarantees a unique ν, denoted ν Γ, whose energy is least among all such measures. It is known as the (logarithmic) equilibrium measure, supported on the outer boundary of Γ and determined entirely by the geometry of this boundary. Let Ω be the unbounded component of C \ Γ 35

42 and g Ω (z, ), its Green s function with pole at infinity. If Γ is smooth then we may exploit the following useful representation of equilibrium measure: dν Γ (z) = g Ω (z, ) ds (z Γ). (3..2) 2π n Here ds denotes arc length measure and n is the outward normal along Γ. Recall that g Ω (z, ) is the unique harmonic function on Ω which vanishes on the boundary Γ and satisfies g Ω (z, ) log z as z. If Φ is a conformal mapping of Ω onto the exterior of the unit disk, then by the uniqueness of Green s function, g Ω (z, ) = log Φ(z). Thus, by equation (3..2), dν Γ = (2π) Φ ds. For example, if Γ = [, ] then dν Γ (x) = (π x 2 ) dx. (See [6, ch. 3] for a stellar introduction.) Equations (3.0.) can now be consolidated: lim n λ n(µ, z) = dµ (z), (3..3) n dν Γ where Γ is the unit circle or interval. Since the Christoffel functions are infima, it is easy to show that the upper bound, lim sup n λ n (µ, z) dµ (z), (3..4) n dν Γ holds for all finite Borel measures (a.e. on the support). (See [, p. 435].) Until fairly recently, however, (3..3) was known only for continuous weights bounded away from zero. That it holds almost everywhere (with respect to ν Γ ) for the more general class of measures satisfying Szegö s condition, ( ) dµ log dν Γ >, (3..5) dν Γ was a long-standing conjecture, finally proved by Máté, Nevai, and Totik in 99 ([]). In this chapter, we aim to prove (3..3) for measures supported on smooth curves, Γ C,α, whose weights satisfy Szegő s condition, (3..5). We accomplish this by an 36

43 abstraction of Máté s technique in [] and investigate some applications. 3.2 Main Results In what follows, C denote the complex plane, U, a bounded, simply connected domain with boundary Γ := U in the class C,α ; and Ω := C \ U. denotes the closed unit disk and R := {z : z R} with boundaries γ and γ R, respectively. Φ is the outer mapping function of Γ, a conformal mapping of Ω onto C \ with Φ( ) = and Ψ := Φ. Φ maps U conformally and Ψ := (Φ ). Γ R := Ψ(γ R ) if R and Γ R := Ψ (γ R ) if R <. U R is the interior of Γ R and Ω R := C \ U R. Let ds denote arc length measure along Γ and let dµ = W Φ ds + dµ s be a measure supported on Γ with singular part µ s. Note that dµ/dν Γ = 2π W almost everywhere, so the following, our main result, is the correct abstraction of equation (3..3): Theorem 3.2. Let dµ = W Φ ds+dµ s be a positive, Borel measure supported on a closed curve Γ C,α for some α > 0 and assume that W satisfies Szegő s condition, (3..5). If W is bounded and dµ s 0, or α > /2, then for ν Γ -almost every ζ 0 Γ. lim n λ n(µ, ζ 0 ) = W (ζ 0 ), (3.2.6) n Theorem 3.2. is really the intersection of two broader results. Unfortunately, our proof of the upper bound, (3..4), requires a more stringent restriction on the smoothness of Γ than that of the lower bound. This inadequacy appears to be intrinsic to our method, as will become transparent in the proof. Theorem Let dµ = W Φ ds + dµ s be a positive, Borel measure supported on a closed curve Γ C,α. If W is bounded and dµ s 0, or α > /2, then for ν Γ -almost every ζ 0 Γ. lim sup n n λ n (µ, ζ 0 ) W (ζ 0 ), (3.2.7) 37

44 Theorem Let dµ = W Φ ds + dµ s be a positive, Borel measure supported on a closed curve Γ C,α where α > 0. If W satisfies (3..5) then for ν Γ -almost every ζ 0 Γ. lim inf n n λ n(µ, ζ 0 ) W (ζ 0 ), (3.2.8) Theorem admits the simpler proof, since it is an upper bound on the infimum, λ n. We simply need a sequence of polynomials, Q n (z), whose L 2 (µ)-norms converge at the optimal rate dictated by Theorem This is easy if Γ is the unit circle: We proved it already in Theorem 2..4 using the sequence Q n (z) = n (ζ 0 z) k (3.2.9) n For general Γ, the upper bound can be deduced from this case by substituting z k in Q n (z) by the (k-th order) -Faber polynomial associated with Γ, k=0 F k (z) = Φ k (ζ) 2πi Γ R ζ z Φ (ζ)dζ, (z U R ). (3.2.0) Let Q n (z) := n F k (z) and S n (z) := n Φ (z)φ k (z). (3.2.) n n k=0 k=0 With d µ(θ) := dµ(ψ(e iθ )), we have S n 2 dµ = n n = k=0 π π Φ(ζ) k 2 Φ (ζ) 2 dµ(ζ) n n k=0 e ikθ 2 Ψ (e iθ ) 2 d µ(θ) = 2π n σ n(d µ/ Ψ 2, ), (3.2.2) and so the revised sequence, {Q n }, will serve our purpose if we can show that Q n S n 0 with sufficient rapidity. We can, at least if α > /2 or W is bounded. 38

45 Lemma For any α > 0 and Γ C,α, Q n S n 0 pointwise and Q n S n 2 ds = o(n ). (3.2.3) If α > /2 then the limit holds uniformly for z Γ: Γ sup Q n (z) S n (z) = o(n /2 ). (3.2.4) z Γ Our choice of Q n in (3.2.) is a natural generalization of (3.2.9): The Faber polynomials generalize the monomials, z k, in a variety of contexts, e.g. locally uniform (taylor-series type) expansions of analytic functions in U. Let us quickly verify Theorem using these estimates. Proof. (Theorem 3.2.2) Minkowskii gives n Qn L2 (µ) n Q n S n L2 (µ) + n S n L2 (µ). (3.2.5) Under the conditions of Theorem 3.2.2, Lemma implies that Q n S n L2 (µ) = o(/ n) so the first term on the right-hand side vanishes as n. Our choice of the particular conformal mapping in (3.2.) is arbitrary, so we may presume that the Fejér means in (3.2.2) converge at z =. Thus, (3.2.5) implies that lim sup n n 2π Q n 2 n dµ lim n 2π S n 2 dµ = W (ζ 0 ) Φ (ζ 0 ) 2, where ζ 0 = Ψ(). Finally, since Q n (ζ 0 ) S n (ζ 0 ) 0 and S n (ζ 0 ) = Φ (ζ 0 ), lim sup n 2π n Q n (ζ 0 ) 2 Q n 2 dµ W (ζ 0 ). (3.2.6) The more difficult proof of the lower bound, (3.2.8), will employ Hardy space methods. D(z) and Q(z) represent, respectively, the outer functions associated with log W in U and Ω. They are analytic and non-vanishing in their respective domains 39