Polynomial Inequalities in the Complex Plane Vladimir Andrievskii - PowerPoint PPT Presentation

Polynomial Inequalities in the Complex Plane Vladimir Andrievskii Kent State University Crete, 2018 Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities Remez ’36 : � √ � n 2 + √ s � 2 + s � ≤ e c √ sn √ || p n || I ≤ T n ≤ 2 − √ s 2 − s for every real polynomial p n of degree at most n such that |{ x ∈ I : | p n ( x ) | ≤ 1 }| ≥ 2 − s , 0 < s < 2 , where I := [ − 1 , 1 ] and T n is the Chebyshev polynomial of degree n . Set Π( p ) := { z ∈ C : | p ( z ) | > 1 } , p ∈ P n . Let now Γ ⊂ C be an arbitrary bounded Jordan arc or curve. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities Remez ’36 : � √ � n 2 + √ s � 2 + s � ≤ e c √ sn √ || p n || I ≤ T n ≤ 2 − √ s 2 − s for every real polynomial p n of degree at most n such that |{ x ∈ I : | p n ( x ) | ≤ 1 }| ≥ 2 − s , 0 < s < 2 , where I := [ − 1 , 1 ] and T n is the Chebyshev polynomial of degree n . Set Π( p ) := { z ∈ C : | p ( z ) | > 1 } , p ∈ P n . Let now Γ ⊂ C be an arbitrary bounded Jordan arc or curve. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities For V ⊂ Γ we consider its covering U = ∪ m j = 1 U j ⊃ V by a finite number of subarcs U j of Γ . Set m � σ Γ ( V ) := inf diam U j , j = 1 where the infimum is taken over all finite coverings of V . Theorem (A. & Ruscheweyh ’05). Let Γ be an arbitrary bounded Jordan arc or curve. If p ∈ P n and σ Γ (Γ ∩ Π( p )) =: u < 1 4 , diam Γ then � 1 + 2 √ u � n ≤ e c √ un . || p || Γ ≤ 1 − 2 √ u Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities For V ⊂ Γ we consider its covering U = ∪ m j = 1 U j ⊃ V by a finite number of subarcs U j of Γ . Set m � σ Γ ( V ) := inf diam U j , j = 1 where the infimum is taken over all finite coverings of V . Theorem (A. & Ruscheweyh ’05). Let Γ be an arbitrary bounded Jordan arc or curve. If p ∈ P n and σ Γ (Γ ∩ Π( p )) =: u < 1 4 , diam Γ then � 1 + 2 √ u � n ≤ e c √ un . || p || Γ ≤ 1 − 2 √ u Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Weighted Remez-type Inequalities Erdélyi ’92 : Assume that for p n ∈ P n and T := { z : | z | = 1 } we have 0 < s ≤ π |{ z ∈ T : | p n ( z ) | > 1 }| ≤ s , 2 . Then, 0 < s ≤ π || p n || T ≤ e 2 sn , 2 . A & Ruscheweyh ’05: Let Γ be quasismooth (in the sense of Lavrentiev), i.e., | Γ( z 1 , z 2 ) | ≤ Λ Γ | z 1 − z 2 | , z 1 , z 2 ∈ Γ , where Γ( z 1 , z 2 ) is the shorter arc of Γ between z 1 and z 2 and Λ Γ ≥ 1 is a constant. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Weighted Remez-type Inequalities Erdélyi ’92 : Assume that for p n ∈ P n and T := { z : | z | = 1 } we have 0 < s ≤ π |{ z ∈ T : | p n ( z ) | > 1 }| ≤ s , 2 . Then, 0 < s ≤ π || p n || T ≤ e 2 sn , 2 . A & Ruscheweyh ’05: Let Γ be quasismooth (in the sense of Lavrentiev), i.e., | Γ( z 1 , z 2 ) | ≤ Λ Γ | z 1 − z 2 | , z 1 , z 2 ∈ Γ , where Γ( z 1 , z 2 ) is the shorter arc of Γ between z 1 and z 2 and Λ Γ ≥ 1 is a constant. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities Let Ω be the unbounded component of C \ Γ , Φ : Ω → D ∗ the Riemann conformal mapping. For δ > 0, set Γ δ := { ζ ∈ Ω : | Φ( ζ ) | = 1 + δ } . Let the function δ ( t ) = δ ( t , Γ) , t > 0 be defined by dist (Γ , Γ δ ( t ) ) = t . If for p n ∈ P n , |{ z ∈ Γ : | p n ( z ) | > 1 }| ≤ s < 1 2 diam Γ , then || p n || Γ ≤ exp ( c δ ( s ) n ) holds with a constant c = c (Γ) . Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities Let Ω be the unbounded component of C \ Γ , Φ : Ω → D ∗ the Riemann conformal mapping. For δ > 0, set Γ δ := { ζ ∈ Ω : | Φ( ζ ) | = 1 + δ } . Let the function δ ( t ) = δ ( t , Γ) , t > 0 be defined by dist (Γ , Γ δ ( t ) ) = t . If for p n ∈ P n , |{ z ∈ Γ : | p n ( z ) | > 1 }| ≤ s < 1 2 diam Γ , then || p n || Γ ≤ exp ( c δ ( s ) n ) holds with a constant c = c (Γ) . Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities Let Ω be the unbounded component of C \ Γ , Φ : Ω → D ∗ the Riemann conformal mapping. For δ > 0, set Γ δ := { ζ ∈ Ω : | Φ( ζ ) | = 1 + δ } . Let the function δ ( t ) = δ ( t , Γ) , t > 0 be defined by dist (Γ , Γ δ ( t ) ) = t . If for p n ∈ P n , |{ z ∈ Γ : | p n ( z ) | > 1 }| ≤ s < 1 2 diam Γ , then || p n || Γ ≤ exp ( c δ ( s ) n ) holds with a constant c = c (Γ) . Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities A finite Borel measure ν supported on Γ is an A ∞ measure (briefly ν ∈ A ∞ (Γ) ) if there exists a constant λ ν ≥ 1 such that for any arc J ⊂ Γ and a Borel set S ⊂ J satisfying | J | ≤ 2 | S | we have ν ( J ) ≤ λ ν ν ( S ) . The measure defined by the arclength on Γ is the A ∞ measure. Lavrentiev ’36: the equilibrium measure µ Γ ∈ A ∞ (Γ) . Theorem (A ’17) Let ν ∈ A ∞ (Γ) , 1 ≤ p < ∞ , and let E ⊂ Γ be a Borel set. Then for p n ∈ P n , n ∈ N , we have � � | p n | p d ν ≤ c 1 exp ( c 2 δ ( s ) n ) | p n | p d ν Γ Γ \ E provided that 0 < | E | ≤ s < ( diam Γ) / 2 , where the constants c 1 and c 2 depend only on Γ , λ ν , p. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities A finite Borel measure ν supported on Γ is an A ∞ measure (briefly ν ∈ A ∞ (Γ) ) if there exists a constant λ ν ≥ 1 such that for any arc J ⊂ Γ and a Borel set S ⊂ J satisfying | J | ≤ 2 | S | we have ν ( J ) ≤ λ ν ν ( S ) . The measure defined by the arclength on Γ is the A ∞ measure. Lavrentiev ’36: the equilibrium measure µ Γ ∈ A ∞ (Γ) . Theorem (A ’17) Let ν ∈ A ∞ (Γ) , 1 ≤ p < ∞ , and let E ⊂ Γ be a Borel set. Then for p n ∈ P n , n ∈ N , we have � � | p n | p d ν ≤ c 1 exp ( c 2 δ ( s ) n ) | p n | p d ν Γ Γ \ E provided that 0 < | E | ≤ s < ( diam Γ) / 2 , where the constants c 1 and c 2 depend only on Γ , λ ν , p. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities A finite Borel measure ν supported on Γ is an A ∞ measure (briefly ν ∈ A ∞ (Γ) ) if there exists a constant λ ν ≥ 1 such that for any arc J ⊂ Γ and a Borel set S ⊂ J satisfying | J | ≤ 2 | S | we have ν ( J ) ≤ λ ν ν ( S ) . The measure defined by the arclength on Γ is the A ∞ measure. Lavrentiev ’36: the equilibrium measure µ Γ ∈ A ∞ (Γ) . Theorem (A ’17) Let ν ∈ A ∞ (Γ) , 1 ≤ p < ∞ , and let E ⊂ Γ be a Borel set. Then for p n ∈ P n , n ∈ N , we have � � | p n | p d ν ≤ c 1 exp ( c 2 δ ( s ) n ) | p n | p d ν Γ Γ \ E provided that 0 < | E | ≤ s < ( diam Γ) / 2 , where the constants c 1 and c 2 depend only on Γ , λ ν , p. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities A finite Borel measure ν supported on Γ is an A ∞ measure (briefly ν ∈ A ∞ (Γ) ) if there exists a constant λ ν ≥ 1 such that for any arc J ⊂ Γ and a Borel set S ⊂ J satisfying | J | ≤ 2 | S | we have ν ( J ) ≤ λ ν ν ( S ) . The measure defined by the arclength on Γ is the A ∞ measure. Lavrentiev ’36: the equilibrium measure µ Γ ∈ A ∞ (Γ) . Theorem (A ’17) Let ν ∈ A ∞ (Γ) , 1 ≤ p < ∞ , and let E ⊂ Γ be a Borel set. Then for p n ∈ P n , n ∈ N , we have � � | p n | p d ν ≤ c 1 exp ( c 2 δ ( s ) n ) | p n | p d ν Γ Γ \ E provided that 0 < | E | ≤ s < ( diam Γ) / 2 , where the constants c 1 and c 2 depend only on Γ , λ ν , p. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities The sharpness: Theorem (A ’17) Let 0 < s < diam Γ and 1 ≤ p < ∞ . Then there exist an arc E s ⊂ Γ with | E s | = s as well as constants ε = ε (Γ) > 0 and n 0 = n 0 ( s , Γ , p ) ∈ N such that for any n > n 0 there is a polynomial p n , s ∈ P n satisfying � � | p n , s | p ds ≥ exp ( εδ ( s ) n ) | p n , s | p ds . Γ Γ \ E s If in the definition of the A ∞ measure we ask S to be also an arc, then ν is called a doubling measure . Mastroianni & Totik ’00 constructed an example showing that the weighted Remez-type inequality may not be true in the case of doubling measures. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane

Remez-type Inequalities The sharpness: Theorem (A ’17) Let 0 < s < diam Γ and 1 ≤ p < ∞ . Then there exist an arc E s ⊂ Γ with | E s | = s as well as constants ε = ε (Γ) > 0 and n 0 = n 0 ( s , Γ , p ) ∈ N such that for any n > n 0 there is a polynomial p n , s ∈ P n satisfying � � | p n , s | p ds ≥ exp ( εδ ( s ) n ) | p n , s | p ds . Γ Γ \ E s If in the definition of the A ∞ measure we ask S to be also an arc, then ν is called a doubling measure . Mastroianni & Totik ’00 constructed an example showing that the weighted Remez-type inequality may not be true in the case of doubling measures. Vladimir Andrievskii Polynomial Inequalities in the Complex Plane