ON A PROBLEM BY STEKLOV A. APTEKAREV, S. DENISOV, D. TULYAKOV. n + 1

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1 ON A PROBLEM BY STEKLOV A. APTEKAREV, S. DENISOV, D. TULYAKOV Abstract. Given any δ 0,, we define the Steklov class S δ to be the set of probability measures σ on the unit circle T, such that σ θ δ/2π > 0 at every Lebesgue point of σ. One can define the orthonormal polynomials ϕ nz with respect to σ S δ. In this paper, we obtain the sharp estimates on the uniform norms ϕ n L T as n which settles a question asked by Steklov in 92. As an important intermediate step, we consider the following variational problem. Fix n N define M n,δ = sup ϕ n L T. Then, σ S we prove δ Cδ n + n < M n,δ. δ A new method is developed that can be used to study other important variational problems. For instance, we prove the sharp estimates for the polynomial entropy in the Steklov class. Introduction. One version of the Steklov s problem see [26], [27] is to obtain the bounds on the sequence of polynomials {P n x} n=0, which are orthonormal with respect to the strictly positive weight ρ: P n x P m x ρx dx = δ n,m, n, m = 0,, ρx δ > 0, x [, ]. 0.2 In 92, V.A. Steklov made a conjecture that a sequence {P n x} is bounded at any point x,, i.e., lim sup P n x < 0.3 n provided that the weight ρ does not vanish on [, ]. On page 32, he writes adapted translation from French: I believe that inequality 0.3 is the common property of all polynomials whose orthogonality weight ρ does not vanish inside the given interval, but so far I haven t succeeded in finding either the rigorous proof to that statement or an example when this estimate does not hold at each interior point of the given interval. This problem some related questions gave rise to extensive research, see, e.g., [, 2, 3, 4] the survey [27] for a detailed discussion the list of references. In 979, Rakhmanov [22] disproved this conjecture by constructing a weight from the Steklov class 0.2, for which lim sup P n 0 =. n It is known see, for example [0] that the following bound P n x = o n

2 holds for any x, as long as ρ satisfies 0.2. In his next paper [23], Rakhmanov proved that for every ε > 0 x 0, there is a weight ρx; x 0, ε from the Steklov class such that the corresponding {P n x} grow as where {k n } is some subsequence in N. continuous weight. P kn x 0 k /2 ε n, 0.4 In [], the size of the polynomials was studied for the All Rakhmanov s counterexamples were obtained as corollaries of the corresponding results for the polynomials {ϕ n } orthonormal on the unit circle 2π 0 ϕ n e iθ, σ ϕ m e iθ, σ dσθ = δ n,m, n, m = 0,, 2..., 0.5 ϕ n z, σ = λ n z n +..., λ n > 0 with respect to measures from the Steklov class S δ defined as the class of probability measures σ on the unit circle satisfying σ δ/2π 0.6 at every Lebesgue point. The version of Steklov s conjecture for this situation would be to prove that the sequence {ϕ n z, σ} is bounded in n at every z T provided that σ S δ. This conjecture might be motivated by the following estimate. Consider the Christoffel-Darboux kernel K n ξ, z, µ = ϕ j ξ, µϕ j z, µ for ξ = z as the function of µ. If µ µ 2, then see [0] or [24] j=0 K n z, z, µ 2 K n z, z, µ, z T. 0.7 Here we do not assume µ 2 to be probability measures, of course. Therefore, if σ S δ, we get K n z, z, σ n +, i.e. δ n + ϕ j z, σ 2 δ. 0.8 by taking µ = δ2π dθ in 0.7. So, on average the polynomials ϕ n z, σ are indeed bounded in n one might want to know whether they are bounded for all n. Rakhmanov proved the following Theorem which gave a negative answer to this question. Theorem 0.. [23] Let σ S δ, where δ is sufficiently small. Then, for every sequence {β n } : β n 0, there is σ S δ such that k n ϕ kn z, σ L T > β kn ln k n for some sequence {k n } N. j=0 This estimate is almost sharp due to the following result see, e.g., [9], p. for the real line case; [0], p. 32, theorem 3.5 for the pointwise estimate. Theorem 0.2. For σ S δ, we have ϕ n z, σ L T = o n

3 for completeness, we give the proof in the end of Appendix A. In the proof of the Theorem 0., an important role was played by the following extremal problem. For a fixed n, define M n,δ = sup ϕ n z, σ L T. 0. σ S δ One of the key results in [23] is the following inequality n + C δ ln 3 n M n,δ, C > We recall here a well-known estimate see [0]: Lemma 0.. We have n + M n,δ, n N. 0.3 δ Proof. Indeed, this is immediate from the estimate 0.8. We can also argue differently: = ϕ n 2 dσ δ/2π ϕ n 2 dθ so 0.3 follows from Cauchy-Schwarz T ϕ n 2 L 2 T = c j z j 2 = 2π c j 2 2π L 2 T δ j=0 T j=0 ϕ n L T n + /2 c j 2. Remark. Notice that all we used in the proof is the normalization ϕ n L 2 T,σ = the Steklov s condition on the measure. The problem, though, is whether the orthogonality leads to further restrictions on the size. The purpose of the current paper is to obtain the sharp bounds for the problem of Steklov, i.e., the problem of estimating the growth of ϕ n. We will get rid of the logarithmic factor in the denominator in thus prove the optimal inequalities. The main results are contained in the following two statements: j=0 Theorem 0.3. If δ 0,, then M n,δ > Cδ n. 0.4 Theorem 0.4. Let δ 0,. Then, for every positive sequence {β n } : lim n β n = 0, there is a probability measure σ : dσ = σ dθ, σ S δ such that ϕ kn z, σ L T β kn kn 0.5 for some sequence {k n } N. 3

4 Remark. It will be clear later that both results hold for far more regular weights see Lemma 4. the proof of Theorem 0.4 below. The Steklov condition 0.6 is quite natural for the analysis of the size of the polynomial. Indeed, if µ is a measure not necessarily a probability one then the trivial scaling ϕ n z, mµ = ϕ nz, µ m holds this changes the size of ϕ n accordingly. Let now µ be a probability measure Γ be a small arc which does not support all of µ. Then, we can can take µ m = χ Γ c µ + mχ Γ µ with m very small. So, µ m χ Γ c µ one can expect that ϕ n z, mµ gets large on most of Γ as m 0 in analogy to the case of the whole T. This is indeed true for many measures µ. Thus, if one studies the dependence of ϕ n z, µ L T on µ, then the conditions on the measure which control the size of the polynomial should account for that fact an obvious bound that takes care of this is 0.6 as it does not allow the measure to be scaled on any arc. The problem of estimating the size of ϕ n is one of the most basic most well-studied problems in approximation theory. Nevertheless, the sharp bounds were missing even for the Steklov s class the most natural class of measures for this problem. In the current paper, we not only establish these bounds but also suggest a new method which, we believe, is general enough to be used in the study of other variational problems where the constructive information on the weight is given. For example, one can replace the Steklov s condition by the lower bounds like σ θ wθ, for a.e. θ T where wθ vanishes at a point in a particular way e.g., wθ θ α. The results we obtain are sharp we apply them to estimate polynomial entropies another important quantity to measure the size of the polynomial. Remark. The size, asymptotics, universality of the Christoffel-Darboux kernel were extensively studied, see, e.g. [7, 6, 8, 28, 29]. We, however, will focus on ϕ n itself. Remark. Since S δ is invariant under the rotation {e i jθ 0 ϕ j ze iθ 0, µ} are orthonormal with respect to µθ θ 0, we can always assume that ϕ n is reached at point z =. Therefore, we have M n,δ = sup µ S δ ϕ n, µ. One can consider the monic orthogonal polynomials Φ n z, µ = z n +... the Schur parameters {γ n } so that ϕ n z, µ = Φ nz, µ Φ n L 2 T,µ If ρ n = γ n 2, then see [25] The Szegő formula [25] yields Φ n z, µ = ϕ n z, µλ n, λ n = γ n = Φ n+ 0, µ 0.6 ρ 0... ρ n. 0.7 π exp ln2πµ θdθ = ρ j π π j 0 4

5 So, for µ S δ, we have δ ρ n n 0 therefore δ ϕn z, µ Φ n z, µ ϕ n z, µ, z C for any µ S δ. Thus, we have δ /2 M n,δ sup µ S δ Φ n z, µ L T M n,δ 0.9 for fixed δ the variational problems for orthonormal monic orthogonal polynomials are equivalent. The estimate 0.3 can not possibly be sharp for δ very close to. Indeed, if δ = then σ is the Lebesgue measure ϕ n z = z n. We have the following result which provides an effective bound improves 0.3 for δ close to. Lemma 0.2. We have M n,δ δ /2 + n δ Proof. Let σ be one of the maximizers for M n,δ, i.e., ϕ n z, σ L T = M n,δ. The existence of such a maximizer is proved in Theorem. below. Then, d σ = 2π δdθ + d µ where µ = δ. Let Φ n z, σ be the corresponding monic polynomial. We use the variational characterization of Φ n see [24]: Φ n = arg min P z=z n +... P 2 L 2 T, σ. If Φ n = z n + ã n z n ã z + ã 0, then ã = arg D n,δ, D n,δ = min a=a 0,..., a n δ a 2 l 2 + T δ z n + a n z n a z + a 0 2 d µ In particular, upon choosing a = 0, we get D n,δ T d µ = δ so ã 2 l δ/δ. Then, 2 Cauchy-Schwarz inequality gives n δ Φ n L T + δ 0.9 finishes the proof. Now, we would like to comment a little on the methods we use. The proofs by Rakhmanov were based on the following formula for the orthogonal polynomial that one gets after adding several point masses to a background measure at particular locations on the circle see [22]. Lemma 0.3. Let µ be a positive measure on T, Φ n z, µ be the corresponding monic orthogonal polynomials, K n ξ, z, µ = ϕ j ξ, µϕ j z, µ be the Christoffel-Darboux kernel, i.e. l=0 P ξ = P z, K n ξ, z, µ L 2 T,µ, deg P n. Then, if ξ j T, j =,... m, m n are chosen such that then Φ n z, η = Φ n z, µ K n ξ j, ξ l, µ = 0, j l 0.20 m k= m k Φ n ξ k, µ + m k K n ξ k, ξ k, µ K n ξ k, z, µ 0.2 5

6 where η = µ + m m k δ θk, z k = e iθ k, m k 0. k= The limitation that ξ j must be the roots of K is quite restrictive the direct application of this formula with background dµ = dθ yields logarithmic growth at best. In the later paper [23], Rakhmanov again ingeniously used the idea of inserting the point mass but the resulting bound 0.2 contained the logarithm in the denominator the measure of orthogonality was not defined explicitly. We will use a completely different approach. First, we will rewrite the Steklov condition in the convenient form as some estimate that involves Caratheodory function a polynomial see Lemma 3.3 below. This decoupling is basically equivalent to solving the well-known truncated trigonometric moments problem. Then, we will present a particular function a polynomial show that they satisfy the necessary conditions. This allows us to have a good control on the size of the polynomial itself on the structure of the measure of orthogonality. The paper has four parts two Appendixes. The first part contains results on the structure of an optimal measure discussion of the case when δ is n-dependent very small. In the second part, the proof of Theorem 0.3 is given for fixed small δ. We will apply the localization principle to hle every δ 0, prove Theorem 0.4 in the third part. In the last one, two applications are given. First, the lower bounds are obtained for polynomials orthogonal on the real line. Then, we prove the sharp estimates for the polynomial entropies in the Steklov class. The Appendixes contain some auxiliary results we use in the main text. Here are some notation used in the paper: the Cauchy kernel for the unit circle is denoted by Cz, ξ, i.e. Cz, ξ = ξ + z ξ z, ξ T. If the function is analytic in D has a nonnegative real part there, then we will call it Caratheodory function. Given any polynomial P n z = p n z n p z + p 0, we can define its n-th reciprocal or the transform P nz = z n P n /z = p 0 z n + p z n p n. Notice that if z is a root of P n z z 0, then z is a root of P nz. Given two positive functions F F 2 defined on D, we write F F 2 if there is a constant C that might depend only on the fixed parameters such that on D. We write F F 2 if F < CF 2, C > 0 F F 2 F. We use the notation b = O a if b a. The symbol δ a denotes the delta function the point mass supported at a π, π] or at complex point e ia T. If ε is a positive parameter, then ε is the shorth for: ε < ε 0, where ε 0 is sufficiently small. If pz = a n z n a z + a 0, then we define coeffp, j = a j. 6

7 We will use the following stard notation for the norms. If µ is a measure, µ refers to its total variation. For functions f defined on [ π, π], we write f p = f L p [ π,π], p. The symbol f, g σ denotes the following inner product given by f, g σ = fxgxdσ where σ is a measure on M e.g., M = T or M = [ π, π]. M Part. Variational problem: structure of the extremizers. Structure of the extremal measure. In this section, we first address the problem of the existence of maximizers, i.e., µ n S δ for which M n,δ = ϕ n ; µ n.. We will prove that these extremizers exist will study their properties. Theorem.. There are µ n S δ for which. holds. Proof. Suppose µ k S δ is the sequence which yields the sup, i.e. ϕ n, µ k M n,δ, k. Since the unit ball is weak- compact, we can choose µ kj µ this convergence is weak-, i.e. fdµ kj fdµ, j for any f CT. In particular, µ is a probability measure. Moreover, for any interval a, b π, π], we have assuming, e.g., that the endpoints a b are not atoms for µ : dµ δb a/2π [a,b] since each µ kj S δ. This implies µ δ/2π a.e. on T. The moments of µ kj will converge to the moments of µ therefore ϕ n, µ kj ϕ n, µ. Therefore, µ S δ ϕ n, µ = M n,δ. This argument gives existence of an extremizer. Although we do not know whether it is unique, we can prove that every dµ must have a very special form. Theorem.2. If µ is a maximizer then it can be written in the following form dµ = 2π δ dθ + where m j 0 π < θ <... < θ N π. N m j δ θj, N n.2 7

8 Suppose we have a positive measure µ its moments are given by s j = e ijθ dµ = s R j + is I j, j = 0,,... Then, the following formulas are well-known [25] s 0 s... s n Φ n z, σ = D s s 0... s n n s n s n 2... s z... z n s 0 s... s n s s 0... s n D n = det T n, T n = s n s n 2... s..4 s n s n... s 0 These identities show that Φ n z, σ depends only on the first n moments of the measure σ: Φ n z, σ = Φ n z, s 0,..., s n. Moreover, by definition of the monic orthogonal polynomial, Φ n z, s 0, s,..., s n = Φ n z,, s /s 0,..., s n /s 0, i.e., Φ n z, σ does not depend on the normalization of the measure. The functions F 2, given by F s = Φ n, s 0,..., s n 2, F 2 s = ϕ n, s 0,..., s n 2, s = s 0, s R,..., s I n R 2n+ are the smooth functions of the variables {s 0, s R j, si j }, j =,..., n wherever they are defined. Consider Ω n = {s : T n s > 0}. If s Ω n, then there is a family of measures µ which have s 0,..., s n as the first n moments. That follows from the solution to the truncated trigonometric moment problem. We will need the following Lemma.. The functions F 2 s do not have stationary points on Ω n. Proof. It is known [25] that the map between the first n Schur parameters see 0.6 the first n moments of a probability measure, i.e., {γ j } n j=0 Dn, s,..., s n, is a bijection. The formulas imply that γ n = s n D n 2 D n + fs 0, s, s,..., s n, s n..5 The both polynomials Φ n ϕ n satisfy the recurrences [25], p.57 which shows that Φ n, σ = Φ n, σ γ n Φ n, σ ϕ n, σ = ρ n ϕ n, σ γ n ϕ n, σ, sn Φ n, σ 2 Φn, σ 2 =, Φ n, σ 2 0, s R n because Φ n z, σ Φ n z, σ do not depend on s n, Φ n, σ 0, Φ n, σ 0. Since ρ n = γ n 2, we have ϕ n, σ 2 = C ξ γ n 2 γ n 2, ξ = ϕ n, σ ϕ n, σ, ξ =, C = ϕ n, σ 2. 8 s I n

9 We can rewrite it as which shows that ϕ n, σ 2 = C + γ n 2 2 Reγ n ξ γ n 2, γn ϕ n, σ 2 0. where ϕ n, σ 2 is considered as a function of {γ 0,..., γ n }. Now.5 yields sn ϕ n, σ 2 0 the proof is finished. Remark. The proof actually shows that sn F 2 0. Proof. of the Theorem.2 Our variational problem is an extremal problem for a functional F s 0, s R,..., si n on the finite number of moments {s 0, s R,..., si n} of a measure µ from S δ. We can take F s 0, s R,..., s I n = ϕ n, s 0,..., s n 2. The function F is differentiable. Moreover, s 0 = dµ, s R j = cosjθdµ, s I j = sinjθdµ. Considering the moments as functionals of µ, we compute the derivative of F at the point µ in the direction δµ: F df = s + F s 0 s R s cosθ F s I s sinnθ dδµ. n Consider the trigonometric polynomial: T n θ = F s + F s 0 s R s cosθ F s I s sinnθ. n From the previous Lemma Remark, we know that it has degree n. Let M = max T n θ {θ j ; j =,..., N} are all points where M is achieved. Clearly, N n. Now, if we find a smooth curve µt, t 0, ] such that µt S δ, µ = µ define Ht = F s 0 µt, s R µt,..., s I nµt, then H 0 as follows from the optimality of µ. Now, we will assume that the measure µ is not of the form.2 then will come to a contradiction by choosing the curve µt in a suitable way. We will first prove that the singular part of µ can be supported only at the points {θ j }. Indeed, suppose we have µ = µ + µ 2 where µ 2 is singular supported away from {θ j }. Consider two smooth functions p t p 2 t defined on 0, ] that satisfy µ + p t + p 2 t µ 2 =, p 2 t 0, p = 0, p 2 =. For example, one can take p t = µ 2 t, p 2 t = t. Take µt = µ + p tδ θ + p 2 tµ 2. We have µt S δ H = T n θdµ 2 µ 2 T n θ < 0 since θ is a point of global maximum for T n µ 2 is supported away from {θ j } by assumption. This contradicts optimality of µ so µ 2 = 0. 9

10 We can prove similarly now that µ = 2π δ a.e. Indeed, suppose µ = µ + µ 2, dµ 2 = fθ χ Ωdθ where fθ > 2π δ > 2π δ on Ω, Ω > 0 µ is supported on Ω c. We consider the curve µt = µ + p tδ θ + p 2 tµ 2 t. The choice of p 2 is the same. Then, µt S δ for t ε, provided that εδ is small. The similar calculation yields H < 0 that gives a contradiction. Since the maximizer in the Steklov problem is given by.2, we want to make an observation. The following result is attributed to Geronimus see [0]. Lemma.2. Consider µt = tµ + tδ β where t 0,, β [ π, π. Then, Φ n z, µt = Φ n z, µ t Φ nξ, µk n ξ, z, µ t + tk n ξ, ξ, µ, ξ = eiβ..6 Proof. Notice that the right h side is a monic polynomial of degree n. Then, which yields orthogonality. r.h.s., z j µt = 0, j = 0,..., n The formula.6 expresses monic polynomials obtained by adding one point mass to an arbitrary measure at any location. One can try to iterate it to get the optimal measure dµ. That, however, leads to very complicated analysis. 2. The regime of small n dependent δ. One can make a trivial observation that if µ is any positive measure not necessarily a probability one ϕ n z, µ is the corresponding orthonormal polynomial, then ϕ n z, αµ = α /2 ϕ n z, µ 2. for every α > 0. The monic orthogonal polynomials, though, stay unchanged Φ n z, αµ = Φ n z, µ. Now, consider the modification of the problem: we define M n,δ = sup ϕ n z, µ L T = sup ϕ n, µ, µ δ/2π µ δ/2π i.e., we drop the requirement for the measure µ to be a probability measure. In this case, the upper estimate for M n,δ stays the same the proof of n+ M n,δ. δ is identical. It turns out that the sharp lower bound can be easily obtained in this case. Theorem 2.. We have M n,δ = n+ 0 δ.

11 Proof. Consider dσ = δ 2π dθ + m k δ θk, θ k = k 2π, k =,..., n. 2.2 n+ k= We assume that all m k 0. Consider One gets: Π n z = + z z n, Π n z = n z ε k, ε k = e iθ k. 2.3 k= Π n 2 L 2 T,σ = δn+. We define now Φ n = Π n + Q n where Q n z = q n z n q z + q 0 is chosen to guarantee the orthogonality Φ n, z j σ = 0, j = 0,..., n. Suppose that m k = m for all k. Then, we have the following equations Then, since we get Thus, n δ + δq j + m k=0 ε d k q l l=0 k= ε l j k = 0, j = 0,..., n. = 0, d { n,...,,,..., n} n δ + δq j + mn + q j m q l = 0, Φ n z = Φ n z, σ = z n Now, we have For the orthonormal polynomial, l=0 q j = δ, j = 0,... n. δ + m δ δ + m zn = Φ n L T = Φ n, σ = n + j = 0,... n m δ + m Π nz + δ δ + m n = + mn δ + m Φ n 2 L 2 T,σ = δ + m2 n δ + m 2 + δ2 nm δ + m 2 = δ + mn δ + m ϕ n, σ = Φ n, σ Φ n L 2 T,σ.. δ δ + m zn. For fixed n, this gives lim m ϕ n L T = n + /δ.

12 Remark. This Theorem has the following implication for our original problem. Suppose we consider the class S δ but δ is small in n. Then, 2. gives n + M n,δn = + o δ n where δ n = C, m n +, as n. nm n Thus, for δ small in n, the upper bound for M n,δ is sharp. If one takes m n = /n in the proof above to make the total mass finite, the polynomials ϕ n are bounded in n as δ. Part 2. The proof of Theorem 0.3: the case of small fixed δ 3. Lower bounds: fixed δ n. In this section, we prove the sharp lower bound for small fixed δ. The main result is the following Theorem. Theorem 3.. There is δ 0 0, such that M n,δ0 n. 3. Remark. In this section, we are not trying to control the size of δ 0. The full range δ 0, will be covered in part 3 by using certain localization technique. 3.. Notation Basics from the theory of polynomials orthogonal on the circle. We start by introducing some notation recalling the relevant facts from the theory of polynomials orthogonal on the unit circle. The following trivial Lemma will be needed later see, e.g., [2], p. 08 Lemma 3.. Let n. If a polynomial P n of degree at most n has all zeroes outside D, then D n z = P n z + P nz has all exactly n zeroes on the unit circle. Proof. We have The first factor has no zeroes in D. P n/p n = on T. So, + P n P n D n z = P n + P n, z D. P n In the second one, P n/p n is a Blaschke product indeed, is holomorphic in D, continuous up to the boundary, its boundary values belong to the circle with center at z = radius. Thus, Re + P n 0. P n Therefore, either Re + P nz P n z > 0 z D or + P nz P n z 0. 2

13 The last condition implies P n = P n, so P n 0 on D by assumption of the Lemma P n 0 on C \ D because it is equal to P n. Then, P n 0 on C so P n const. Therefore P n = P n const. This is possible for n = 0 only. Finally, + P n P n does not have zeroes in D. Since D n is invariant under the transform, it has the following property: D n w = 0 implies D n w = 0. Therefore, D n has no zeroes in z > as well. One can actually show that D n has the degree n under the assumptions of the Lemma. Indeed, if D n z 0, z D, then Var arg D rt n =0, r <. Therefore Var arg Dn r T =2πn = degd n n. But Dn = D n. We will be mostly working with the orthonormal polynomials ϕ n the corresponding ϕ n. It is well known [25] that all zeroes of ϕ n are inside D thus ϕ n has no zeroes in D. However, we also need to introduce the second kind polynomials ψ n along with the corresponding ψn. Let us recall [25], p. 57 that { ϕn+ = ρ n zϕ n γ n ϕ n, ϕ 0 = / µ = ϕ n+ = ρ n ϕ n γ n zϕ n, ϕ 0 = probability case 3.2 / µ = the second kind polynomials satisfy the recursion with Schur parameters γ n, i.e., { ψn+ = ρ n zψ n + γ n ψn, ψ 0 = µ = ψn+ = ρ n ψn + γ n zψ n, ψ0 = µ = The following Bernstein-Szegő approximation result is valid: Theorem 3.2. Suppose dµ is a probability measure {ϕ j } {ψ j } are the corresponding orthonormal polynomials of the first/second kind, respectively. Then, for any N, the function F N z = ψ N z ϕ N z = Cz, e iθ dθ dµ N θ, dµ N θ = 2π ϕ N e iθ 2 = dθ 2π ϕ N eiθ 2 T has the first N Taylor coefficients identical to the Taylor coefficients of the function F z = Cz, e iθ dµθ. T In particular, the polynomials {ϕ j } {ψ j }, j N are the orthonormal polynomials of the first/second kind for the measure dµ N. We also need the following Lemma: Lemma 3.2. The polynomial P n z of degree n is the orthonormal polynomial for a probability measure with infinitely many growth points if only if. P n z has all n zeroes inside D counting the multiplicities. 2. The normalization conditions dθ 2π P n e iθ 2 =, coeffp n, n > 0 are satisfied. T Now, we are ready to formulate the main result of this section

14 3.2. The reduction of the problem: Decoupling Lemma. The proof of the Theorem 3. will be based on the following result. Lemma 3.3. The Decoupling Lemma To prove 3., it is sufficient to find a polynomial ϕ n a Caratheodory function F which satisfy the following properties:. ϕ nz has no roots in D. 2. Normalization on the size rotation ϕ nz 2 dθ = 2π, ϕ n0 > Large uniform norm, i.e., 4. F C T, Re F > 0 on T, 5. Moreover, uniformly in z T. T 2π ϕ n n. T Re F e iθ dθ =. 3.5 ϕ nz + F zϕ n z ϕ nz < C δ Re F /2 z 3.6 Proof. By Lemma 3.2, the first two conditions guarantee that ϕ n z is an orthonormal polynomial of some probability measure. It also determines the first n Schur parameters: γ 0,..., γ n. The third one gives the necessary growth. Next, let us show that the fourth fifth conditions are sufficient for the existence of a measure σ S δ for which ϕ n is the n th orthonormal polynomial. By the fourth condition, F defines the probability measure σ which is purely absolutely continuous has positive smooth density σ given by σ θ = Re F e iθ 2π. 3.7 Denote its Schur parameters by { γ j }, j = 0,,... the orthonormal polynomials of the first second kind by { ϕ j } { ψ j }, j = 0,,..., respectively. Notice that the normalization condition for σ implies ϕ 0 = ψ 0 =. By Baxter s Theorem [25] we have γ j l in fact, the decay is much stronger but l is enough for our purposes. Then, let us consider the probability measure σ which has the following Schur parameters γ 0,..., γ n, γ 0, γ,... We will show that this measure satisfies the Steklov s condition. Denote γ n = γ 0, γ n+ = γ, The Baxter Theorem implies that σ is purely a.c., σ belongs to Wiener s class W T, σ is positive on T. The first n orthonormal polynomials corresponding to the measure σ will be {ϕ j }, j = 0,..., n. Let us compute the polynomials ϕ j ψ j, orthonormal with respect to σ, for the indexes j > n. Since the second kind polynomials correspond to the Schur parameters { γ j } see 3.3, the recursion can be rewritten in the following matrix form ϕn+m ψ n+m Am B = m ϕn ψ n C m D m ϕ n ψn 3.9 ϕ n+m ψ n+m 4

15 where A m, B m, C m, D m satisfy A0 B 0 0 = C 0 D 0 0 Am B m = C m D m ρ 0... ρ m, z z γ m γ m z γ0... z γ 0 thus depend only on γ n,..., γ n+m i.e., γ 0,..., γ m by 3.8. Moreover, we have ϕm ψm Am B ϕ m ψ m = m. C m D m Thus, A m = ϕ m + ψ m /2, B m = ϕ m ψ m /2, C m = ϕ m ψ m/2, D m = ϕ m + ψ m/2 their substitution into 3.9 yields 2ϕ n+m = ϕ n ϕ m ψ m + ϕ n ϕ m + ψ m = ϕ m ϕ n + ϕ n + F m ϕ n ϕ n 3.0 where F m z = ψ mz ϕ mz. Since { γ n } l {γ n } l, we have [25], p. 225 F m F as m ϕ n Π, ϕ n Π as n. uniformly on D. The functions Π Π are the Szegő functions of σ σ, respectively, i.e., they are the outer functions in D that give the factorizations Π 2 = 2πσ, Π 2 = 2π σ. 3. In 3.0, send m to get 2Π = Π ϕ n + ϕ n + F ϕ n ϕ n. 3.2 Thus, the first formula in 3. shows that for the sufficiently regular measures, Steklov s condition σ δ/2π is equivalent to Π ϕ n + ϕ n + F ϕ 2 n ϕ n δ, z = e iθ T. 3.3 Since ϕ n = ϕ n on T, we have Π ϕ n + ϕ n + F ϕ n ϕ n 2 Π ϕ n + F ϕ n ϕ n < 2C δ Π Re F /2 = 2C δ due to 3.6, 3.7, the second formula in 3.. Thus, to guarantee 3.3, we only need to take C δ = δ /2 in 3.6. In this section, we assume δ to be fixed so the exact formulas for Cδ C δ will not be needed. Proof. of Theorem 3.. The proof will be based on the Decoupling Lemma will contain two parts. In the first one, we will make the choice for ϕ n, F study their basic properties to check conditions,3-5 of the Decoupling Lemma. In the second part, we will verify the normalization condition, i.e., condition 2. 5

16 3.3. The choice of parameters. In what follows, we take ε n = n.. The choice of F. Consider two parameters: α /2, ρ 0, ρ0 where ρ 0 is sufficiently small. Let us emphasize that these parameters are fixed will not be changed in the estimates below, however many constants in these inequalities will actually depend on them. We do not trace this dependence here. Take F z = C n ρ + εn z + + ε n z α, 3.4 where the positive normalization constant C n will be chosen later. We have two terms inside the brackets. The first one gives the right growth at point z = : + ε n = n this choice is motivated by conditions 3 5 of the Decoupling Lemma. The role of the second term will be explained later. We will need more information on F. Clearly F is smooth has a positive real part in D. Notice that for z = e iθ T θ, we have +ε n z = ε n + θ2 2 iθ + O θ 3, +ε n z 2 = ε n + θ2 2 + O θ θ + O θ 3 2 = = ε 2 n + θ 2 + ε n θ 2 + O θ 3 Since w = w w 2, we obtain +ε n z = ε n + θ2 2 + iθ + O θ 3 ε 2 n + θ 2 + ε n θ 2 + O θ 3 = ε n + iθ ε 2 n + θ 2 + O θ 2 ε 2 n + θ The last equality can be verified directly by subtraction. So, F e iθ = C ρεn n ε 2 n + θ 2 + i ρθ ρθ ε 2 n + θ ε n z α 2 + O ε 2 n + θ We have + ε n e iθ α = + ε n cos θ 2 + sin 2 θ α/2 exp iαγ n θ. 3.7 sin θ Γ n θ = arctan π/2, π/ ε n cos θ The following bound is true ρε n ρθ ε 2 n + θ ε n e iθ α 2 L + O ε 2 n + θ 2 [ π,π] uniformly in n. Then, for every fixed υ > 0, we have ρ + ε n z + + ε n z α ρ z + z α, n 3.9 uniformly in {z = e iθ, θ > υ}. Since Re ρ e iθ + e iθ α dθ θ >υ we can choose C n to guarantee 3.5 then C n uniformly in n. Consider the formulas They yield F ε n = n for θ < ε n 3.20 F θ for θ > ε n. 3.2 Indeed, in the last inequality, the upper bound F θ 6

17 is immediate. For the lower bound, F Im F = ρ sin θ + ε n cos θ 2 + sin 2 θ + + ε n cos θ 2 + sin 2 θ α/2 sin α arctan sin θ. + ε n cos θ These terms have the same signs, so Im F ρ sin θ + ε n cos θ 2 + sin 2 θ = ρ sin θ 2 + ε n cos θ + ε 2 θ, ε n < θ < π/2. n For θ : θ > π/2, the estimate is a trivial corollary of 3.9. F θ 2. The choice of ϕ n. Let ϕ n be chosen as follows ϕ nz = C n f n z, f n z = P m z + Q m z + Q mz 3.22 where P m Q m are certain polynomials of degree m = [δ n] 3.23 where δ is small will be specified later. Notice here that Q m is defined by applying the n th order star operation. The constant C n will be chosen in such a way that π π ϕ n 2 dθ = 2π i.e., 3.4 is satisfied. To prove the Theorem, we only need to show that π /2 C n = f n dθ π uniformly in n that f n satisfies the other conditions of the Decoupling Lemma. The choice of ϕ n is motivated by the following observation. The estimate 3.6 requires F /2, zϕ n z ϕ nz < C δ Re F z z T. Since F z is much larger than Re F z around z =, the point of growth, the factor ϕ n ϕ n should provide some cancelation. The sum of the second the third terms in 3.22, the polynomial Q m + Q m has degree n is symmetric so it drops out in ϕ n ϕ n. However, it has zeroes on T due to Lemma 3. thus can not be a good choice for ϕ n due to violation of conditions 2 in Decoupling Lemma. P m, the first term in 3.22, will be be chosen to achieve a certain balance. It will be small around z = it will push the zeroes of Q m + Q m away from D to guarantee 3.4. Consider the Fejer kernel F m θ = sin 2 mθ/2 m sin 2 = cosmθ θ/2 m cosθ, F m0 = m 3.25 the Taylor approximation to the function z α, i.e., k R k,α z = + d j z j 7

18 see Appendixes for the detailed discussion. We define Q m as an analytic polynomial without zeroes in D which gives Fejer-Riesz factorization Q m z 2 = G m θ + R m,α/2 e iθ G m θ = F m θ + 2 F m θ π + m 2 F m θ + π m Clearly, the right h side of 3.26 is a positive trigonometric polynomial of degree m so this factorization is possible Q m is unique up to a unimodular factor. We choose this factor in such a way that Q m 0 > 0. Since Q m is an outer function, we have the following canonical representation see [9], page 24 π Q m z = exp Cz, e iθ ln Q m e iθ dθ 2π π, z < Notice that Q m is a polynomial of degree n with positive leading coefficient. Since Q m e iθ is even in θ, this representation shows that Hz = ln Q m z is analytic in D has real Taylor coefficients indeed, Hz = Hz. That, on the other h, implies that Q m z = e Hz has real coefficients as well. For P m, we take P m z = Q m z z 0.R m, α z 3.29 deg P m = 2m + < n by the choice of small δ. Consequently, deg ϕ n = n. Now that we have chosen F ϕ n, it is left to show that they satisfy the conditions of the Decoupling Lemma. The second term in 3.4 the structure of 3.29 will become important in what follows.. f n has no zeroes in D. For f n, we can write f n = Q m z z 0.R m, α z + + z n e 2iϕ, z T 3.30 where e 2iϕθ = Q me iθ Q m e iθ so ϕ is an argument of Q m. The polynomial Q m has no zeroes in D is analytic in D has positive real part. Indeed, Re + Q m 0, Q m z 0.R m, α z + + Q m Q m 3.3 since Q m = Q m. Since Q m has real coefficients, Q m is real. Furthermore, since Q m 0 > 0, Q m is real at the real line Q m 0 in D, then Q m >0. So ϕ0 = 0 Re + Q m = 2, z =. Q m For the first term in 3.3, we have where z T Re [ z 0.R m, α z ] = cos θ 0.X 0.Y sin θ 3.32 X = Re R m, α, Y = Im R m, α. 8

19 Notice that R m, α < 2 for z D So X < 2 in D as well. The function Y is odd in θ Y θ < 0 for θ 0, π as follows from the Lemma 8.5 in Appendix A for small θ; see [4], Theorem for the general case. Thus, the function f n z has a positive real part on T, in particular, has no zeroes in D. We conclude then that ϕ n has no zeroes in D. 2. The growth at z =. The formula 3.30 implies Then, yield due to f n = 2Q m. f n m n 3. Steklov s condition. We need to check 3.6 with ϕ n replaced by f n. From , we get f n 2 Q m 2 = G m θ + R m,α/2 e iθ 2. Lemma 8.3 implies R m,α/2 e iθ 2 ε m + θ α ε n + θ α. The exact form of the Fejer s kernel 3.25 gives m G m θ m 2 θ 2 + We have Re F = ρ + ε n cos θ + ε n cos θ 2 + sin 2 θ n n 2 θ 2 + = + + ε n cos θ 2 + sin 2 θ α/2 cos α arctan Since α < arctan < π/2, we get The estimate uniform in θ, implies Therefore, ε n ε 2 n + θ 2. Re F ρε n + θ 2 ε 2 n + θ 2 + ε 2 n + θ 2 α/2. Re F Then, for the second term in 3.6, we get The uniform bounds θ 2 ε 2 n + θ 2 ε 2 n + θ 2 α/2, ε n ε 2 n + θ 2 + ε2 n + θ 2 α/2. sin θ + ε n cos θ f n Q m Re F / F f n f n 2 = F P m P m 2 F z 2 Q m 2. F z, Q m 2 Re F together with 3.34, imply 3.6 with ϕ n replaced by f n.. 9

20 3.4. Normalization: checking condition 2 of the Decoupling Lemma. We only need to check now that f n 2 dθ T see It is sufficient to consider θ [0, π] as f n e iθ is even. Our first goal is to obtain a convenient lower bound on f n 2. Notice that 3.26 yields Q m e iθ 2 R m,α/2 e iθ 2 the Lemma 8.3 from Appendix A gives we should use notation β = α > 0 here f n 2 m + θ α z 0.Rm, β z + + z n e 2iϕ 2. Then, the representation 3.32 leads to z 0.R m, β z + + z n e 2iϕ 2 = 3.35 For the first term, we have cos θ 0.X 0.Y sin θ + + cosnθ 2ϕ 2+ 0.Y cos θ 0.X sin θ + sinnθ 2ϕ 2. cos θ 0.X 0.Y sin θ + + cosnθ 2ϕ 0.Y sin θ θ 2 α 3.36 where the last inequality follows from Lemma 8.5. Thus, where f n 2 m + θ α [ θ 2 α 2 + Ψθ + sinnθ 2ϕ 2 ] 3.37 Ψ = Im z 0.R m, β z = 0.Y cos θ 0.X sin θ. In what follows, we will control Ψ ϕ to analyze Ψθ + sinnθ 2ϕ in We will locate the zeroes {θ j } of this highly oscillatory function will show that away from these points the normalization condition is easily satisfied. More delicate analysis will be needed to integrate f n 2 over small neighborhoods of {θ j }. To bound Ψ θ, we use Lemma Indeed, Ψ θ = Im d [ z 0.R m, β z] dθ 0.R m, β e iθ + 0. eiθ d dθ R m, βe iθ β θ O max n, θ = O. Then, uniformly in n. This estimate Ψ0 = 0 imply by integration. For the phase ϕ, we have ϕ0 = 0 Ψ θ 3.38 Ψθ θ 3.39 ϕ θ m 20

21 where the last inequality is proved in Appendix B. Since we have the derivative of ϕ under control, nθ 2ϕθ = n + Oδ n By making δ small, we can make sure that the function nθ 2ϕθ is monotonically increasing n/2 < nθ 2ϕθ < 2n. 3.4 To study the zeroes {θ j }, we first introduce auxiliary points { θ j }. The monotonicity of nθ 2ϕθ allows us to uniquely define { θ j } as solutions to the equation: n θ j 2ϕ θ j = πj 2, j = 0,..., [cn] Then, sin n θ j 2ϕ θ j = j. On the other h, from 3.4, one has 2j π < 4n θ j < 2πj n, 3.43 By the estimate 3.39, we can choose some small positive constant c>0 so that for j = 0,,, [cn] the expression Ψθ + sinnθ 2ϕθ changes the sign on [ θ j, θ j+ ]. Indeed, j[ Ψ θ j + sin n θ j 2ϕ θ j ] = j[ Ψ θ j + j ] < Ψ θj 2πj = O < 0 ; n j[ Ψ θ j+ + sin n θ j+ 2ϕ θ j+ ] > Ψ θj+ 2πj = O > 0. n Moreover, we require that c is chosen such that Ψ θ j <, j = 0,,..., [cn] 2 that υ, defined as υ = θ [cn], is smaller than the parameter υ from the Lemmas 8.3, 8.4, 8.5 in the Appendix A. Now, let us show that there is the unique point θ j such that θ j [ θ j, θ j+ ] : Ψθ j + sin nθ j 2ϕθ j = The existence of such θ j is a simple corollary of continuity sign change. Note, that the function Ξθ= nθ 2ϕθ πj, restricted to the segment [ θ j, θ j+ ], takes values from [ π 2, π 2 ]. So, for each θ j that satisfies 3.44, we have sin Ξθ j = j Ψθ j nθ j 2ϕθ j + j arcsin Ψθ j = πj If, for fixed j, there are several solutions θ j to 3.44, then the derivative of the function nθ 2ϕθ + j arcsin Ψθ in the left h side of 3.45 is non-positive at at least one of these θ j. However the lower estimate on the derivative reads n 2ϕ θ j + j Ψ θ j Ψ 2 θ j > n Om O > n 2 2 which shows that θ j is unique. Since Ψ θ j < 2, we have arcsin Ψ θ j < π 4 θ j θ j > 2n π 2 π 4, θ j+ θ j > 2n π 2 π 4. Now, from 3.40, , one gets [, i.e., I j = θ j 0.0 n, θ j ] [ θj, n θ ] j+. 2

22 If θ [ θ j, θ j+ ]\I j, then nθ 2ϕθ nθ j 2ϕθ j > n n > The estimate for the integral over I 0 = [0, 0.0n ] is easy since on that arc we have Q m m by 3.26 f n 2 m +cosnθ 2ϕθ 2 by 3.30, The bound on the derivative of ϕ implies that f n m on I 0 so f n 2 dθ n 2. I 0 Let θ I j assume that some ξ is located between θ j θ. From the definition of I j, we get ξ θ j < 00n. Therefore, nξ 2ϕξ πj nθ j 2ϕθ j πj + nξ 2ϕξ nθ j 2ϕθ j arcsin Ψθ j + 2n ξ θ j π ; π cosnξ 2ϕξ cos > So, for θ I j, j, we have θ Ψθ + sinnθ 2ϕθ = Ψ ξ + cosnξ 2ϕξn 2ϕ ξ dξ n θ θ j. For θ [ θ j, θ j+ ]\I j, j, one gets θ j sinnθ 2ϕθ+Ψθ sinnθ 2ϕθ sinnθ j 2ϕθ j Ψθ Ψθ j by the triangle inequality. Then, for the last term, we apply 3.38 to get Ψθ Ψθ j θ θ j θ j+ θ j = On. For the first one, x x 2 x + x 2 sin x sin x 2 = 2 sin cos 2 2 where x = nθ 2ϕθ x 2 = nθ j 2ϕθ j. Next, we use 3.46 to write π 2 x x Consider x + x 2 /2. For x 2, we have x 2 = πj j arcsin Ψθ j with arcsin Ψθ j π/4. Then, for every x [πj π/2, πj + π/2], we get cos x + x 2 > cos3π/8. 2 Therefore, outside j I j. Ψθ + sin nθ 2ϕθ Now, let us obtain the estimates outside the small fixed arc {θ : θ > υ}. The bound 3.30 implies Q m 2[ 2 Re z 0.R m, α z] fn 2 Q m z 0.R m, α z 2. 22

23 We have the uniform convergence R m, α z z α, m, z T Q m z z α, m, z = e iθ, θ > υ. The direct calculation shows see, e.g., 3.36 that Re z 0.R m, α z, θ > υ. Consequently, From 3.37, we have fn 2 2 [0,υ] j I j c θ α dθ + + n θ >υ [cn] f n 2 dθ Cυ, α n θj α 0.0n [cn] θ 2+2α j + dθ θj 2 α 2 + n 2 θ + 2 n θ 2α 2 dθ θ >υ where we used 3.43, 3.44, α /2,. Together with 3.48, that implies f n 2 dθ the proof of Theorem 3. is finished. T f n 2 dθ Remark. It is immediate from the proof that the constructed polynomial satisfies the following bound: ϕ n e iθ, σ 2 n Cα + nθ 2 + θ α, α 0.5, Measure of orthogonality Our method allows one to compute a measure of orthogonality σ for which the orthonormal polynomial has the required size it is interesting to compare it to the results on the maximizers we obtained before. The calculations given below will show that σ is purely absolutely continuous. Its density can be represented as a sum of background Bθ, 0 < C Bθ C 2, a combination of peaks positioned at θ j to be defined later. Qualitatively, each peak resembles the mollification of the point mass by the Poisson kernel. Our analysis can establish the parameters of mollification a mass assigned to each peak. The formulas yield σ = 4 σ ϕ n + ϕ n + F ϕ n ϕ n 2 = 2 Re F π ϕ n + ϕ n + F ϕ n ϕ n 2. This expression is explicit as we know the formulas for all functions involved. We have σ = C n Q m 2 Re F 2 + e inθ 2ϕ + H n e iθ + e inθ 2ϕ H n e iθ + F H n e iθ e inθ 2ϕ H n e iθ 2 where 4. H n z = z 0.R m, α z, C n

24 as follows from 3.24, Consider the first factor. We can apply 3.20, 3.2, Lemma 8.3 to get Recall that F is given by Q m 2 Re F. F z = C n ρ + ε n z + + ε n z α, C n = ρ/ + ε n + + ε n α 4.3 where the last formula for C n comes from the normalization 3.5 Re 2π F e iθ dθ = Re F 0 the Mean Value Formula for a harmonic function continuous in D. Substitution into the second factor in 4. gives σ 2 + H n F e inθ 2ϕ H n + F 2 + H n F T For z = e iθ small positive θ the negative values can be hled similarly, we have H n e iθ = iθ + 0.θiR m, α e iθ + O θ 2 Lemma 8.5 can be used for R m, α. Next, consider F. It can be written as F e iθ = C ρ n ε n iθ + ε n iθ α + O Therefore, for the first factor in 4.4, we have when ρ 0, ρ 0 α ρ 0 α is small. Consider J = 2 + H n + F 2 + H n F. Notice first that J > for θ 0. Indeed, J 2 = 2 + H n + F H n F 2 4 Re 2 + H n F = H n F 4.5 H n + H n + H n 2 F 2 + H n F = 2 = 4Re F H n Re H n 2 + H n F 2 the last expression is positive by the choice of F H n. For θ = 0, we have J0 =. For small θ, the following asymptotics holds J 2 θ 2 α ρεn ε 2 n + θ 2 + θ α, θ > 0.n 4.6 If we write J 2 ρεn ε 2 n + θ 2 + ε α n θ n α, θ < 0.n. J = rθe iυθ, Υ0 = 0 24

25 then σ e inθ 2ϕθ Υθ +rθ 2 = cosnθ 2ϕθ Υθ+rθ 2 +sin 2 nθ 2ϕθ Υθ. 4.7 Consider { θ j }, the solutions to nθ 2ϕθ Υθ = π + 2πj, j < N 3 that belong to some small fixed arc θ < υ. We have Υ0 = 0 the direct estimation gives Υ θ < 0.n uniformly for all θ. Indeed, it is sufficient to show that 2 + H n + F θ 2 + H n + F < 0.0n H n F θ 2 + H n F < 0.0n. 4.9 The both inequalities are proved in Lemma 8.8 from Appendix B. Now we can argue that the distance between the consecutive { θ j } is of size n σ r θ j 2 + n 2 θ θ j = m 2 j ỹ j ỹ 2 j + θ θ j 2 on θ : θ θ j < Cn. In the Poisson kernel, the mollification parameter ỹ j is the mass m j is given by Notice, that N 3 m j j= N 3 ỹ j = r θ j n m j = nr θ j 0.n < θ <υ dθ rθ < as follows from 4.6 α /2,. Away from these { θ j } the density is. Remark. In the estimates above, we assumed that ρ is small: ρ 0, ρ 0. The choice of ρ 0 is made in Lemma 8.8 see the Remark after it in 4.5. Thus, we first fix a parameter α then fix ρ. In fact, we need ρ to be small only to control σ it is irrelevant for the proof of the main Theorem. The measure σ constructed in the proof has no singular part. Its regularity can be summarized in the following Lemma. For δ 0,, p [, ], C > 0, let us introduce the following class of measures given by a weight let S p,c δ = {σ : dσ = wθdθ, w δ/2π, w =, w p C} M p,c n,δ = sup ϕ n z, σ L T σ S p,c δ 25

26 Lemma 4.. For every p [, there is δ 0 0, C > 0 such that M p,c n,δ 0 n Proof. The upper estimate is obvious since M p,c n,δ M n,δ. For the lower one, we only need to show that for every large p one can take α in the proof of Theorem 3. so close to that w p < C where w = σ C is independent of n. The estimate 4.7 the bounds on the derivatives of ϕ Υ yield T N 3 w p dθ + Cn 0 dθ r θ j 2 + n 2 θ 2 p + N 3 dθ + 0 θ 2p 2 2α < if α 4p 3/4p 2,. Here we used 4.6 to estimate rθ. nr θ j 2p Part 3. Bernstein s method localization. The proofs of Theorem 0.3 Theorem 0.4 In this part, we will use the localization principle to first prove the lower bounds on M n,δ in the full range of δ Theorem 0.3 then iterate this construction prove Theorem The method by Bernstein localization principle. Given a weight w on [ π, π], we define λw = exp ln2πwθdθ, Λw = w. 4π We have 0.7, 0.8 yield exp T ϕ n z, w = w /2 ϕ n z, w/ w ln2πwθ/ w dθ 4π T Φ n z, w ϕ n z, w/ w. So, λw Φ n z, w ϕ n z, w Λw. 5. In [6], S. Bernstein studied the asymptotics of the polynomials when the weight of orthogonality is regular introduced a method which we will use when proving the following Theorem. Theorem 5.. Let w 2 be two weights on [ π, π] so that Then for all n. w θ = w 2 θ, θ [ ϵ, ϵ]. 5.2 ϕ n, w ϕ n, w 2 Λw 2 λw + 4Λw ϕ n e iθ, w ϕ n e iθ, w 2 w + w 2 dθ ϵλw θ >ϵ

27 Proof. Following Bernstein, we write n π Φ n z, w = ϱ n ϕ n z, w 2 + ϕ j z, w 2 Φ n e iθ, w ϕ j e iθ, w 2 w 2 θdθ 5.4 π j=0 with some coefficient ϱ n. By orthogonality, n Φ n z, w = ϱ n ϕ n z, w 2 + ϕ j z, w 2 j=0 π π Φ n e iθ, w ϕ j e iθ, w 2 w 2 θ w θ dθ = π ϱ n ϕ n z, w 2 + Φ n e iθ, w K n e iθ, z, w 2 w 2 θ w θ dθ. π The Christoffel-Darboux kernel K n admits a representation see, e.g., [5], p. 225, formula 8.2.: Then, K n ξ, z, µ = ϕ nz, µϕ nξ, µ ϕ n z, µϕ n ξ, µ zξ Φ n, w ϕ n, w 2 ϱ n + 4ϵ Φ n e iθ, w ϕ n e iθ, w 2 w + w 2 dθ. θ >ϵ We will now use 5.. Comparing the coefficients in front of z n in 5.4, we get so λw 2 ϱ n Λw 2 ϕ n, w ϕ n, w 2 Λw 2 λw + 4Λw ϕ n e iθ, w ϕ n e iθ, w 2 w + w 2 dθ ϵλw θ >ϵ by the repetitive application of The proofs of Theorem 0.3 Theorem 0.4. We start with a Lemma which will immediately imply Theorem 0.3. It allows to perturb very general measures have the orthogonal polynomial grow. Lemma 6.. Assume δ 0, ], p [2, the weight w satisfies the following properties: w =, w δ/2π, w L p [ π, π]. Then, for arbitrary ϵ > 0 n N, there is a weight w such that w =, w δ/2π ϵ, w w p ϵ 6. ϕ n, w > Cϵ, δ, p, w n. Proof. Take any δ 0, ]. For every p p,, Lemma 4. yields σ : dσ = σ dθ so that σ δ 0 /2π > 0, σ L p T < Cp, σ L T =, ϕ n, σ > Cp n. The constants Cp above are n independent. Consider an interval I τ = τ, τ. Then, I τ dσ τ /p. 27

28 We now introduce two new weights w 2, w given by: { σ w 2 = /δ 0, θ I τ, w = w w, θ / I 2 / w 2. τ We have w 2 2π δ a.e. on T w w 2 p w L p [ τ,τ] + w 2 L p [ τ,τ] o + Cp τ p p by Hölder s inequality. Here o 0 as τ 0. Therefore, w w 2 o + Cp τ p p. The triangle inequality normalization w = give w 2 = + o + Oτ p p. We can choose τ small enough that the last two conditions in 6. are satisfied for w. For the corresponding polynomials, we have ϕ n, σ /δ 0 = δ 0 ϕ n, σ so ϕ n, σ /δ 0 n. Apply Lemma 5. with w = σ /δ 0 w 2 defined above. Notice that 0 < C δ, p λw 2 C 2 δ, p, C δ, p Λw 2 C 2 δ, p w = w 2 on I τ by construction. We have ϕ n e iθ, w ϕ n e iθ, w 2 w 2 dθ max ϕ n e iθ, w w 2 L Iτ c 2 T ϕ n e iθ, w 2 L 2 T θ >τ θ >τ ϕ n e iθ, w ϕ n e iθ, w 2 w dθ max ϕ n e iθ, w w L Iτ c 2 T ϕ n e iθ, w 2 L 2 T. The polynomials orthonormal with respect to a measure in Steklov class have uniformly bounded L 2 T norm see the proof of Lemma 0.. For the estimation of max I c ϵ ϕ n e iθ, w, we use 3.49 to get max ϕ n e iθ, w Iτ c where α <. Thus, the localization principle 5.3 gives the proof is finished. n + n 2 τ 2 + τ α ϕ n, w Cϵ, δ, p, w n Now the proof of the Theorem 0.3 is immediate. /2 τ /2 Proof. of Theorem 0.3 It is sufficient to take w = 2π δ =. Remark. Notice that this proof allows to improve Lemma 4. to cover the full range of δ : δ 0,. This statement is much stronger than the Theorem 0.3 itself: it shows that n growth can be achieved on far more regular weights. Now, we can iterate this construction to prove Theorem 0.4. Proof. of Theorem 0.4. Fix any δ 0, a sequence {β n } : lim n β n = 0. We can assume without loss of generality that β <. Choose any p [2, parameter C >. We construct the sequence of weights {w n } through the following induction: First step: We let w = 2π k =. Then, ϕ k, w = > β. 28

29 Inductive assumption: We assume that the weight w n the natural numbers k <... < k n are given so that ϕ kj, w n > β kj kj, j =,..., n w n =, w n p < C, w n δ + δ2 n /2π. 6.2 Inductive step: For every ϵ > 0 N we can use the perturbation Lemma 6. to get w n+ so that w n+ =, w n+ δ + δ2 n /2π ϵ, w n+ w n p ϵ ϕ N, w n+ > Cϵ, δ, p N. Notice that for fixed j the functional ϕ j, σ is continuous in σ in weak then in L p T topology. The second inequality in 6.2 is strict. So, we first choose ϵ so small that:. w n p + ϵ < C. 2. ϕ kj, ν > β kj kj, j =... n, as long as ν : ν w n p < ϵ. 3. δ + δ2 n /2π ϵ > δ + δ2 n /2π. Then, with fixed ϵ, take N large so that N > k n Cϵ, δ, p N > β N N. We can always achieve that since lim l β l = 0. Now, let k n+ = N. Thus, we constructed the new weight w n+ k n+ that satisfy all induction assumptions. At each step when going from n to n + we choose new ϵ that depends on n, the step of induction. By construction, w n+ w n p 2 n so w n converges to some w in L p T norm. Moreover, w δ/2π a.e. on T. We use the continuity of ϕ j, σ in σ again to get that finishes the proof. ϕ kj, w β kj kj, j Remark. It is clear that our construction allows to have the polynomials grow simultaneously at any finite number of points on the circle. We also can make the measure of orthogonality symmetric with respect to both axis OX OY. Indeed, the measure we constructed in the Theorem is given by the even weight w. Now, for every N N, we can take w N x = wnx then ϕ Nj z, w N = ϕ j z N, w To make the measure symmetric with respect to both OX OY, it is sufficient to take N = 2. Remark. The conjecture of Steklov its solution can be interpreted as follows. It is known that {Φ n z} satisfy the recursion Therefore, Φ n+z = Φ nz γ n zφ n z, Φ 0z =. n Φ nz = z γ j Φ j z. Recall that σ S δ implies {γ j } l 2 one can define a maximal function in analogy to the Carleson maximal function in Fourier series, i.e., Mθ = sup γ j Φ j e iθ n, Mθ sup Φ ne iθ. n j=0 29 j=0

30 Then, for the example we constructed, γ j Φ j e iθ L T but Me iθ / L T. j=0 Part 4. Applications In this part, we apply the obtained results to hle the case of the orthogonality on the segment on the real line. We also prove the sharp bounds for the polynomial entropy in the Steklov class. 7. Back to the real line. In the case when the measure σ is symmetric on T with respect to the real line, one can relate {ϕ n z, σ} to polynomials orthogonal on the real line through the following stard procedure. Let ψ, x [, ], ψ = 0 be a non-decreasing bounded function with an infinite number of growth points. Consider the system of polynomials {P k }, k = 0,,... orthonormal with respect to the measure ψ supported on the segment [, ]. Introduce the function { ψcos θ, 0 θ π, σθ = 7. ψcos θ, π θ 2π, which is bounded non-decreasing on [0, 2π]. Consider the polynomials ϕ k z, σ = λ k z k +, λ k = ρ 0... ρ k > 0 orthonormal with respect to σ. Then, ϕ n is related to P k by the formula P k x, ψ = ϕ 2kz, σ + ϕ 2k z, σ 2π [ + λ 2k ϕ 2k0, σ ] z k, k = 0,,..., 7.2 where x=z+z /2 [0, 24]. This reduction also works in the opposite direction: given the symmetric measure σ we can map it to the measure on the real line the corresponding polynomials will be related by 7.2. We are ready to formulate the Theorem. Theorem 7.. Let δ >0. Then, for every positive sequence {β n } : lim n β n = 0, there is a measure ψ : dψ = ψ dx supported on [, ] such that ψ x δ for a.e. x [, ] for some sequence {k n } N. P kn 0, ψ β kn kn 7.3 Proof. Indeed, in the Theorem 0.4 we can take σ : dσ = w dθ to be symmetric with respect to both axis, i.e., w 2π θ = w θ symmetry with respect to OX w θ = w π θ symmetry with respect to OY. Moreover, we can always arrange for all {k n } to be divisible by 4 ϕ kn, σ β kn kn Now, we take σ : dσ = w θ π/2dθ, i.e., the rotation of σ by π/2 apply 7. to it. The symmetries of σ yield the symmetry of σ with respect to OX so this transform is applicable. Notice that ϕ n z, σ = e inπ/2 ϕ n ze iπ/2, σ where the first factor is introduced to make the leading coefficient positive. Also, notice that ϕ n, σ is real-valued so ϕ n, σ = ϕ n, σ. We have ϕ kn i, σ + ϕ k n i, σ = ϕ kn i, σ + ϕ kn i, σ = ϕ kn, σ + ϕ kn, σ = 2ϕ kn, σ 2β kn kn 30