Asymptotics of weighted Bergman polynomials Laurent Baratchart - PowerPoint PPT Presentation

The Szeg˝ o function o function of a weight w 1 ∈ L 1 ( ∂ Ω) with • The exterior Szeg˝ log | w 1 | ∈ L 1 ( ∂ Ω): � 1 e i θ + Φ( z ) � � S − e i θ − Φ( z ) log w (Φ − 1 ( e i θ )) d θ w 1 ( z ) = exp , z / ∈ Ω , 4 π T recurs in every asymptotics of orthogonal polynomials (also in non-Hermitian orthogonality). • In fact S − w 1 is the largest (in modulus) nonvanishing analytic function in C \ Ω whose nontangential maximal function lies in L 2 ( ∂ Ω) and whose nontangential limit on ∂ Ω has squared modulus 1 / w 1 a.e.. o function S + • The interior Szeg˝ w 1 ( z ) is defined similarly for z ∈ Ω using the interior conformal map Φ 1 , and this time w 1 | 2 = w 1 on ∂ Ω. | S + • S ± w 1 solve a “Riemann-Hilbert problem”: � − 1 � S − S + w 1 (Φ − 1 w 1 ( ξ ) = ◦ Φ( ξ )) , ξ ∈ ∂ Ω . 1

w �≡ 1: improvements

w �≡ 1: improvements • Smirnov& Lebedev [1964] improved Korovkin’s result by allowing g to have a zero at infinity (of arbitrary multiplicity).

w �≡ 1: improvements • Smirnov& Lebedev [1964] improved Korovkin’s result by allowing g to have a zero at infinity (of arbitrary multiplicity). • Mina-Diaz (2010) has relaxed the zero condition in Korovkin’s result to non-zeroing in a neighborhood of T .

w �≡ 1: improvements • Smirnov& Lebedev [1964] improved Korovkin’s result by allowing g to have a zero at infinity (of arbitrary multiplicity). • Mina-Diaz (2010) has relaxed the zero condition in Korovkin’s result to non-zeroing in a neighborhood of T . • Simanek [2012] obtained ratio asymptotics for large | z | and analytic simply connected Ω, for weights which are conformal images of certain product measures on the unit disk D : � � w = ν ( θ ) × τ ( ρ ) ◦ ϕ, ϕ : Ω → D .

w �≡ 1: improvements • Smirnov& Lebedev [1964] improved Korovkin’s result by allowing g to have a zero at infinity (of arbitrary multiplicity). • Mina-Diaz (2010) has relaxed the zero condition in Korovkin’s result to non-zeroing in a neighborhood of T . • Simanek [2012] obtained ratio asymptotics for large | z | and analytic simply connected Ω, for weights which are conformal images of certain product measures on the unit disk D : � � w = ν ( θ ) × τ ( ρ ) ◦ ϕ, ϕ : Ω → D . • Mina-Diaz and Simanek [2013] gave necessary conditions on w for exterior asymptotics to hold.

w �≡ 1: further remarks

w �≡ 1: further remarks • The results by Korovkin, Suetin, Mina-Diaz and Simanek substantiate the claim that asymptotics of P n depends only on the behavior of w close to ∂ Ω.

w �≡ 1: further remarks • The results by Korovkin, Suetin, Mina-Diaz and Simanek substantiate the claim that asymptotics of P n depends only on the behavior of w close to ∂ Ω. • Saff and Simon speculated that ratio asymptotics exists for | z | large, as soon as w does not vanish too much in a neighborhood of ∂ Ω, at least for reasonably smooth Ω (generalization of a theorem by Rakhmanov on the circle).

w �≡ 1: further remarks • The results by Korovkin, Suetin, Mina-Diaz and Simanek substantiate the claim that asymptotics of P n depends only on the behavior of w close to ∂ Ω. • Saff and Simon speculated that ratio asymptotics exists for | z | large, as soon as w does not vanish too much in a neighborhood of ∂ Ω, at least for reasonably smooth Ω (generalization of a theorem by Rakhmanov on the circle). • Defining what “does not vanish too much” means is part of the question.

Scholium

Scholium • Exterior asymptotics we mentioned are similar to Szeg˝ o asymptotics of orthogonal polynomials on ∂ Ω with respect to � the weight w | ∂ Ω , except for the extra factor ( n + 1) /π .

Scholium • Exterior asymptotics we mentioned are similar to Szeg˝ o asymptotics of orthogonal polynomials on ∂ Ω with respect to � the weight w | ∂ Ω , except for the extra factor ( n + 1) /π . • In fact all these results can be thought of as perturbations of the 1-D case, where the influence of the “germ” of the weight close to the boundary asymptotically dominates all other phenomena.

Scholium • Exterior asymptotics we mentioned are similar to Szeg˝ o asymptotics of orthogonal polynomials on ∂ Ω with respect to � the weight w | ∂ Ω , except for the extra factor ( n + 1) /π . • In fact all these results can be thought of as perturbations of the 1-D case, where the influence of the “germ” of the weight close to the boundary asymptotically dominates all other phenomena. • It is to ensure this dominancy that nonzeroing assumptions on w to the boundary ∂ Ω are made.

Outline

Outline • In this talk we report on fairly weak assumptions on the weight under which exterior asymptotics hold as before.

Outline • In this talk we report on fairly weak assumptions on the weight under which exterior asymptotics hold as before. • We pay a price in that we no longer provide rates of convergence. In fact, with the assumptions we make, convergence can be arbitrarily slow.

Outline • In this talk we report on fairly weak assumptions on the weight under which exterior asymptotics hold as before. • We pay a price in that we no longer provide rates of convergence. In fact, with the assumptions we make, convergence can be arbitrarily slow. • We mainly discuss analytic Jordan domains Ω, meaning that ∂ Ω is the image of the unit circle T under a map analytic and univalent in a neighborhood of T .

Outline • In this talk we report on fairly weak assumptions on the weight under which exterior asymptotics hold as before. • We pay a price in that we no longer provide rates of convergence. In fact, with the assumptions we make, convergence can be arbitrarily slow. • We mainly discuss analytic Jordan domains Ω, meaning that ∂ Ω is the image of the unit circle T under a map analytic and univalent in a neighborhood of T . Results extend to C 1 ,α -domains, as will e stresed later.

Outline • In this talk we report on fairly weak assumptions on the weight under which exterior asymptotics hold as before. • We pay a price in that we no longer provide rates of convergence. In fact, with the assumptions we make, convergence can be arbitrarily slow. • We mainly discuss analytic Jordan domains Ω, meaning that ∂ Ω is the image of the unit circle T under a map analytic and univalent in a neighborhood of T . Results extend to C 1 ,α -domains, as will e stresed later.

Assumptions • Ω is an analytic Jordan domain. In particular, Ψ := Φ − 1 extends conformally into a map from {| z | > 1 − ε } onto C \ Ω 1 , where Ω 1 ⊂ Ω.

Assumptions • Ω is an analytic Jordan domain. In particular, Ψ := Φ − 1 extends conformally into a map from {| z | > 1 − ε } onto C \ Ω 1 , where Ω 1 ⊂ Ω. • Putting Ψ r ( e i θ ) := Ψ( re i θ ), we assume that w ◦ Ψ r converges in L p ( T ) as r → 1, for some p > 1. If F is the limit, we put w 1 := F ◦ Φ.

Assumptions • Ω is an analytic Jordan domain. In particular, Ψ := Φ − 1 extends conformally into a map from {| z | > 1 − ε } onto C \ Ω 1 , where Ω 1 ⊂ Ω. • Putting Ψ r ( e i θ ) := Ψ( re i θ ), we assume that w ◦ Ψ r converges in L p ( T ) as r → 1, for some p > 1. If F is the limit, we put w 1 := F ◦ Φ. • Putting Γ η := Ψ( {| z | = η } ) for 1 − ε < η < 1, we assume that � log − w log + (log − w ) d σ < + ∞ . sup 1 − ε<η< 1 Γ η This last condition expresses that the weight does not vanish too much in the vicinity of ∂ Ω.

Main result

Main result Theorem Under the previous assumptions it holds that � 1 / 2 � n + 1 Φ n ( z )Φ ′ ( z ) S − P n ( z ) = w 1 ( z )(1 + o (1)) π locally uniformly outside the convex hull of Ω, with S − w 1 the exterior Szeg˝ o function of w 1 .

An example

An example • Let { z k } be a sequence of points in Ω.

An example • Let { z k } be a sequence of points in Ω. • Let { a k } be a summable family of positive numbers.

An example • Let { z k } be a sequence of points in Ω. • Let { a k } be a summable family of positive numbers. • Put �� − 1 � � � � diam Ω + 1 Σ ∞ � � w ( z ) := k =1 a k log log . � � z − z k � �

An example • Let { z k } be a sequence of points in Ω. • Let { a k } be a summable family of positive numbers. • Put �� − 1 � � � � diam Ω + 1 Σ ∞ � � w ( z ) := k =1 a k log log . � � z − z k � � • Then the theorem applies to w on Ω.

An example • Let { z k } be a sequence of points in Ω. • Let { a k } be a summable family of positive numbers. • Put �� − 1 � � � � diam Ω + 1 Σ ∞ � � w ( z ) := k =1 a k log log . � � z − z k � � • Then the theorem applies to w on Ω. • When { z k } is dense in Ω, then w vanishes in the neighborhood of every point.

Structure of the proof

Structure of the proof It has three steps:

Structure of the proof It has three steps: • First we derive an upper bound for κ n .

Structure of the proof It has three steps: • First we derive an upper bound for κ n . This rests on direct estimation of some appropriate integral and requires no assumption on the weight.

Structure of the proof It has three steps: • First we derive an upper bound for κ n . This rests on direct estimation of some appropriate integral and requires no assumption on the weight. • Next we derive an asymptotic lower bound for κ n .

Structure of the proof It has three steps: • First we derive an upper bound for κ n . This rests on direct estimation of some appropriate integral and requires no assumption on the weight. • Next we derive an asymptotic lower bound for κ n . There, we use the characterization: κ n = sup { κ ; ∃ P ( z ) = κ z n + a n − 1 z n − 1 + · · · + a 0 , � P � L 2 ( w ) ≤ 1 } .

Structure of the proof It has three steps: • First we derive an upper bound for κ n . This rests on direct estimation of some appropriate integral and requires no assumption on the weight. • Next we derive an asymptotic lower bound for κ n . There, we use the characterization: κ n = sup { κ ; ∃ P ( z ) = κ z n + a n − 1 z n − 1 + · · · + a 0 , � P � L 2 ( w ) ≤ 1 } . This rests on constructing a sequence of auxiliary polynomials { Q n } whose leading coefficient asymptotically matches the upper bound and whose norm in L 2 ( w ) is asymptotically 1.

Structure of the proof It has three steps: • First we derive an upper bound for κ n . This rests on direct estimation of some appropriate integral and requires no assumption on the weight. • Next we derive an asymptotic lower bound for κ n . There, we use the characterization: κ n = sup { κ ; ∃ P ( z ) = κ z n + a n − 1 z n − 1 + · · · + a 0 , � P � L 2 ( w ) ≤ 1 } . This rests on constructing a sequence of auxiliary polynomials { Q n } whose leading coefficient asymptotically matches the upper bound and whose norm in L 2 ( w ) is asymptotically 1. There assumptions on w are used.

Structure of the proof It has three steps: • First we derive an upper bound for κ n . This rests on direct estimation of some appropriate integral and requires no assumption on the weight. • Next we derive an asymptotic lower bound for κ n . There, we use the characterization: κ n = sup { κ ; ∃ P ( z ) = κ z n + a n − 1 z n − 1 + · · · + a 0 , � P � L 2 ( w ) ≤ 1 } . This rests on constructing a sequence of auxiliary polynomials { Q n } whose leading coefficient asymptotically matches the upper bound and whose norm in L 2 ( w ) is asymptotically 1. There assumptions on w are used. • At this point, we will know that κ n √ n + 1 = ( π G w 1 ) − 1 / 2 , lim inf n → + ∞ � where G w 1 = exp { T log( w 1 ◦ Ψ) } is the geometric mean.

Structure of the proof cont’d

Structure of the proof cont’d • Having at our disposal a sequence of polynomials Q n with dominant coefficient α n ∼ κ n whose L 2 ( w ) norm is asymptotically 1, we use a technique of Widom: � P n − Q n � 2 L 2 ( w ) = � P n � 2 L 2 ( w ) + � Q n � 2 L 2 ( w ) − 2 ℜ� P n , Q n � w L 2 ( w ) − 2 α n = 1 + � Q n � 2 → 0 . κ n

Structure of the proof cont’d • Having at our disposal a sequence of polynomials Q n with dominant coefficient α n ∼ κ n whose L 2 ( w ) norm is asymptotically 1, we use a technique of Widom: � P n − Q n � 2 L 2 ( w ) = � P n � 2 L 2 ( w ) + � Q n � 2 L 2 ( w ) − 2 ℜ� P n , Q n � w L 2 ( w ) − 2 α n = 1 + � Q n � 2 → 0 . κ n • By [Saff,Stahl,Stylianopoulos, Totik, 2012] [Simanek,2012] | P n / Q n − 1 | ≤ � P n − Q n � L 2 ( w ) d ( z , Conv Ω)+ diam Ω) 2 / d 2 ( z , Conv Ω) , hence P n ∼ Q n outside Conv Ω.

Structure of the proof cont’d • Having at our disposal a sequence of polynomials Q n with dominant coefficient α n ∼ κ n whose L 2 ( w ) norm is asymptotically 1, we use a technique of Widom: � P n − Q n � 2 L 2 ( w ) = � P n � 2 L 2 ( w ) + � Q n � 2 L 2 ( w ) − 2 ℜ� P n , Q n � w L 2 ( w ) − 2 α n = 1 + � Q n � 2 → 0 . κ n • By [Saff,Stahl,Stylianopoulos, Totik, 2012] [Simanek,2012] | P n / Q n − 1 | ≤ � P n − Q n � L 2 ( w ) d ( z , Conv Ω)+ diam Ω) 2 / d 2 ( z , Conv Ω) , hence P n ∼ Q n outside Conv Ω. • Finally one checks by inspection that � 1 / 2 � n + 1 z n S − Q n ( z ) = w 1 ( z ) { 1 + o (1) } , z / ∈ Ω . π

A closer look at the upper bound

A closer look at the upper bound Theorem For Ω an analytic Jordan domain and w ≥ 0 a weight function in L 1 (Ω), it holds that ( cap Ω) n +1 1 √ n + 1 lim sup n →∞ κ n ≤ √ π � � ess sup r → 1 − G 1 / 2 w ◦ Ψ r where cap indicates the logarithmic capacity.

A closer look at the upper bound cont’d

A closer look at the upper bound cont’d Proof: Let A 1 , R to be the annular region between Γ 1 and Γ R , R < 1, and consider the integral: � 1 � 2 π � 2 e − 2 ni θ � P n ◦ Ψ r ( e i θ )Ψ ′ ( re i θ ) / S − w ◦ Ψ r ( e i θ ) J n := rdr d θ. R 0 On the one hand, it holds that � 1 � 2 π | P n (Ψ( re i θ ) | 2 w (Ψ( re i θ )) | Ψ ′ ( re i θ ) | 2 d θ | J n | ≤ rdr 0 R � | P ( ξ ) | 2 w ( ξ ) dm ( ξ ) ≤ 1 . = A 1 , R

A closer look at the upper bound cont’d

A closer look at the upper bound cont’d Proof cont’d: On the other hand, using the residue formula at infinity for Hardy functions of class H 1 ( C \ D ), we get � 2 d ξ � 1 � r 2 n +1 dr 1 P n (Ψ( ξ )) � J n = 2 π ξ n S − 2 i π ξ w ◦ Ψ r ( ξ ) R T r . � 1 = 2 πκ 2 n ( cap Ω) 2 n +2 r 2 n +1 G w ◦ Ψ r dr . R

A closer look at the upper bound cont’d Proof cont’d: On the other hand, using the residue formula at infinity for Hardy functions of class H 1 ( C \ D ), we get � 2 d ξ � 1 � r 2 n +1 dr 1 P n (Ψ( ξ )) � J n = 2 π ξ n S − 2 i π ξ w ◦ Ψ r ( ξ ) R T r . � 1 = 2 πκ 2 n ( cap Ω) 2 n +2 r 2 n +1 G w ◦ Ψ r dr . R Finally, it is elementary that � 1 r → 1 − G 1 / 2 n →∞ (2 n + 2) − 1 r 2 n +1 G w ◦ Ψ r dr ≤ ess sup lim sup w ◦ Ψ r . R

A closer look at the lower bound

A closer look at the lower bound We first consider the case where Ω = D , the unit disk.

A closer look at the lower bound We first consider the case where Ω = D , the unit disk. Theorem Let w ∈ L 1 ( D ) and assume that r → 1 − w r exists in L p ( T ) , w 1 := lim p > 1 . Then κ n √ n + 1 ≥ ( π G w 1 ) − 1 / 2 , lim inf n → + ∞ where the right-hand side may be finite or infinite depending � � whether T log w 1 > −∞ or T log w 1 = −∞ .

About the proof

About the proof Recall the characterization: κ n = sup { κ ; ∃ P ( z ) = κ z n + a n − 1 z n − 1 + · · · + a 0 , � P � L 2 ( w ) ≤ 1 } .

About the proof Recall the characterization: κ n = sup { κ ; ∃ P ( z ) = κ z n + a n − 1 z n − 1 + · · · + a 0 , � P � L 2 ( w ) ≤ 1 } . The proof rests on the construction of a sequence of auxiliary polynomial whose leading coefficient matches the lower bound and whose norm in L 2 ( w ) is asymptotically 1.

About the proof Recall the characterization: κ n = sup { κ ; ∃ P ( z ) = κ z n + a n − 1 z n − 1 + · · · + a 0 , � P � L 2 ( w ) ≤ 1 } . The proof rests on the construction of a sequence of auxiliary polynomial whose leading coefficient matches the lower bound and whose norm in L 2 ( w ) is asymptotically 1. Such a sequence is given by � 1 / 2 � n + 1 � � Q n ( e i θ ) := e ( n − k n ) i θ P + e ik n θ S − 1 w 1 , + ( e − i θ ) . π Here P + indicates analytic projection that selects Fourier coefficients of non-negative index, and k n → ∞ but k n / n → 0.

About the proof cont’d

About the proof cont’d • In fact, the estimates are obtained first when w ≥ δ > 0.

About the proof cont’d • In fact, the estimates are obtained first when w ≥ δ > 0. This is because the convergence of the Fourier series of S − 1 w 1 , + then takes place n L 2 p ′ , 1 / p + 1 / p ′ = 1.

About the proof cont’d • In fact, the estimates are obtained first when w ≥ δ > 0. This is because the convergence of the Fourier series of S − 1 w 1 , + then takes place n L 2 p ′ , 1 / p + 1 / p ′ = 1. • To remove the assumption that w ≥ δ > 0, we apply the preceding case to w { m } := w + δ m where δ m ∈ (0 , 1) → 0 and we use that κ n increases when the measure decreases.

About the proof cont’d • In fact, the estimates are obtained first when w ≥ δ > 0. This is because the convergence of the Fourier series of S − 1 w 1 , + then takes place n L 2 p ′ , 1 / p + 1 / p ′ = 1. • To remove the assumption that w ≥ δ > 0, we apply the preceding case to w { m } := w + δ m where δ m ∈ (0 , 1) → 0 and we use that κ n increases when the measure decreases. • Besides, the needed convergence m →∞ G w { m } lim = G w 1 1 follows easily from dominated and monotone convergence applied to the positive and negative parts of the functions.

About the proof cont’d

About the proof cont’d To pass to analytic Ω, we use the Faber polynomials of the second kind F n , defined as the singular part at infinity of Φ n Φ ′ :

About the proof cont’d To pass to analytic Ω, we use the Faber polynomials of the second kind F n , defined as the singular part at infinity of Φ n Φ ′ : ∞ Φ n ( z )Φ ′ ( z ) = cap n +1 Ω z n + α n − 1 z n − 1 + · · · + α 0 + � β j z − j j =1 ∞ � β j z − j . = F n ( z ) + j =1

About the proof cont’d To pass to analytic Ω, we use the Faber polynomials of the second kind F n , defined as the singular part at infinity of Φ n Φ ′ : ∞ Φ n ( z )Φ ′ ( z ) = cap n +1 Ω z n + α n − 1 z n − 1 + · · · + α 0 + � β j z − j j =1 ∞ � β j z − j . = F n ( z ) + j =1 If we let V R := Ψ( { z : | z | > R } ) , R > R 0 .

About the proof cont’d To pass to analytic Ω, we use the Faber polynomials of the second kind F n , defined as the singular part at infinity of Φ n Φ ′ : ∞ Φ n ( z )Φ ′ ( z ) = cap n +1 Ω z n + α n − 1 z n − 1 + · · · + α 0 + � β j z − j j =1 ∞ � β j z − j . = F n ( z ) + j =1 If we let V R := Ψ( { z : | z | > R } ) , R > R 0 . we get by Cauchy’s theorem: Φ n ( ξ )Φ ′ ( ξ ) 1 � F n ( z ) = Φ n ( z )Φ ′ ( z ) + d ξ, z ∈ V R . 2 i π ξ − z Γ R Then, a straightforward majorization gives us � ≤ CR n , � � F n ( z ) − Φ n ( z )Φ ′ ( z ) � z ∈ V R . R > R 0 ,

About the proof cont’d

About the proof cont’d • Consider the test polynomial Q n associated with the weight w ◦ Ψ on D : Q n ( z ) = α n z n + γ n − 1 z n − 1 + · · · + γ n − k n z n − k n

About the proof cont’d • Consider the test polynomial Q n associated with the weight w ◦ Ψ on D : Q n ( z ) = α n z n + γ n − 1 z n − 1 + · · · + γ n − k n z n − k n • On Ω, we pick our test polynomial to be Q n ( z ) = α n F n + γ n − 1 F n − 1 + · · · + γ n − k n F n − k n .

About the proof cont’d • Consider the test polynomial Q n associated with the weight w ◦ Ψ on D : Q n ( z ) = α n z n + γ n − 1 z n − 1 + · · · + γ n − k n z n − k n • On Ω, we pick our test polynomial to be Q n ( z ) = α n F n + γ n − 1 F n − 1 + · · · + γ n − k n F n − k n . • Q n is a polynomial of degree n with dominant coefficient � 1 / 2 � n + 1 ( cap (Ω)) − ( n +1) . G − 1 / 2 w 1 π

About the proof cont’d • Consider the test polynomial Q n associated with the weight w ◦ Ψ on D : Q n ( z ) = α n z n + γ n − 1 z n − 1 + · · · + γ n − k n z n − k n • On Ω, we pick our test polynomial to be Q n ( z ) = α n F n + γ n − 1 F n − 1 + · · · + γ n − k n F n − k n . • Q n is a polynomial of degree n with dominant coefficient � 1 / 2 � n + 1 ( cap (Ω)) − ( n +1) . G − 1 / 2 w 1 π • Previous estimates on F n and our choice of k n make Q n → 0 locally uniformly in Ω.