Interpolating real polynomials Joaquim Ortega-Cerd` a Universitat - PowerPoint PPT Presentation

Interpolating real polynomials Joaquim Ortega-Cerd` a Universitat de Barcelona, BGSMath Providence, June 6, 2018

Interpolating sequences Let X be a set and H a reproducing kernel Hilbert space of real functions defined on X , i.e. for all x ∈ X , there is a K x ∈ H such that f ( x ) = � f , K x � . We normalize the reproducing kernel and denote κ x = K x / � K x � .

Interpolating sequences Let X be a set and H a reproducing kernel Hilbert space of real functions defined on X , i.e. for all x ∈ X , there is a K x ∈ H such that f ( x ) = � f , K x � . We normalize the reproducing kernel and denote κ x = K x / � K x � . Definition A sequence Λ ⊂ X is an interpolating sequence for H whenever | c λ | 2 ≃ � � � c λ κ λ � 2 . λ ∈ Λ λ ∈ Λ

Riesz sequences and Interpolating sequences in PW Let Λ ⊂ R , then Definition A sequence of functions { f λ ( z ) = sin π ( z − λ ) π ( z − λ ) } λ ∈ Λ is a Riesz sequence for the Paley Wiener space whenever, 2 � � � � | c λ | 2 � � � � | c λ | 2 . c λ f λ � � � � � � � λ ∈ Λ λ ∈ Λ λ ∈ Λ This implies that Λ is uniformly separated.

The density of a interpolating sequences There is a density that almost describes interpolating sequences Definition The upper Beurling-Landau density of a sequence Λ ⊂ R is # { Λ ∩ ( x − r , x + r ) } D + (Λ) = lim r →∞ sup . 2 r x ∈ R

The density of a interpolating sequences There is a density that almost describes interpolating sequences Definition The upper Beurling-Landau density of a sequence Λ ⊂ R is # { Λ ∩ ( x − r , x + r ) } D + (Λ) = lim r →∞ sup . 2 r x ∈ R Theorem (Beurling) A separated sequence Λ ⊂ R is interpolating for PW if D + (Λ) < 1 . Moreover if Λ is interpolating then D + (Λ) ≤ 1 .

Our setting Let Ω be a smooth bounded strictly convex domain in R d .

Our setting Let Ω be a smooth bounded strictly convex domain in R d . Let P n be the real polynomials of degree n .

Our setting Let Ω be a smooth bounded strictly convex domain in R d . Let P n be the real polynomials of degree n . Let dV be the normalized Lebesgue measure restricted to Ω . We denote by N n = dim ( P n ) .

Our setting Let Ω be a smooth bounded strictly convex domain in R d . Let P n be the real polynomials of degree n . Let dV be the normalized Lebesgue measure restricted to Ω . We denote by N n = dim ( P n ) . We endow P n with the norm given by L 2 ( V ) . � � p � 2 = | f ( x ) | 2 dV ( x ) . Ω

Interpolating sequences Let Λ = { Λ n } n ⊂ Ω be a sequence of finite sets of points of Ω ⊂ R d . Definition We say that Λ is an interpolating sequence if there is a constant C > 0 such that 2 � � | c λ | 2 ≤ � � C − 1 � � � c λ κ n | c λ | 2 , ≤ C � � λ � � λ ∈ Λ n � λ ∈ Λ � λ ∈ Λ n were κ n λ is the normalized reproducing kernel. We are interested in the geometric distribution of points in Λ .

Alternative definition Λ is an interpolating is equivalent to the two following properties. | p ( λ ) | 2 K n ( λ, λ ) ≤ C � p � 2 , � ∀ p ∈ P n λ ∈ Λ n and for any sequence of sets of values { v λ } λ ∈ Λ v there are polynomials p n ∈ P n such that p n ( λ ) = v λ with | v λ | 2 � p n � 2 ≤ C � K n ( λ, λ ) . λ ∈ Λ n

The “natural” normalization The natural normalization is | p ( λ ) | 2 . c λ, n = sup p ∈P n , � p � = 1

The “natural” normalization The natural normalization is | p ( λ ) | 2 . c λ, n = sup p ∈P n , � p � = 1 This can be computed as follows. Take p 1 , . . . , p N n an orthonormal basis of P n and construct: � K n ( z , w ) = p j ( z ) p j ( w ) , j

The “natural” normalization The natural normalization is | p ( λ ) | 2 . c λ, n = sup p ∈P n , � p � = 1 This can be computed as follows. Take p 1 , . . . , p N n an orthonormal basis of P n and construct: � K n ( z , w ) = p j ( z ) p j ( w ) , j � � n d , n d + 1 c λ, n = K n ( λ, λ ) ≃ min . � d ( λ )

The “natural” normalization The natural normalization is | p ( λ ) | 2 . c λ, n = sup p ∈P n , � p � = 1 This can be computed as follows. Take p 1 , . . . , p N n an orthonormal basis of P n and construct: � K n ( z , w ) = p j ( z ) p j ( w ) , j � � n d , n d + 1 c λ, n = K n ( λ, λ ) ≃ min . � d ( λ ) Moreover K n is the reproducing kernel: � p ( z ) = K n ( z , w ) p ( w ) dV ( w ) , ∀ p ∈ P n Ω

Carleson mesures The Plancherel-Polya sequences are a particular case of Carleson measures. Definition A sequence of measures in Ω , µ k is Carleson if there is a constant C > 0 such that � | p | 2 d µ k ≤ C � p � 2 , ∀ p ∈ P k . Ω We have a geometric characterization of Carleson measures.

An anisotropic metric In the ball there is an anisotpric distance given by � � � � 1 − | x | 2 + 1 − | y | 2 d ( x , y ) = arccos � x , y � + . This is the geodesic distance of the points in the sphere S d defined as x ′ = ( x , 1 − | x | 2 ) and y ′ = ( x , � � 1 − | x | 2 ) .

An anisotropic metric In the ball there is an anisotpric distance given by � � � � 1 − | x | 2 + 1 − | y | 2 d ( x , y ) = arccos � x , y � + . This is the geodesic distance of the points in the sphere S d defined as x ′ = ( x , 1 − | x | 2 ) and y ′ = ( x , � � 1 − | x | 2 ) . If we consider balls B ( x , r ) in this distance they are comparable to a box (a product of intervals) which is of size R in the tangent directions and R 2 + R 1 − | x | 2 in the normal direction. �

Geometric characterization The geometric characterization of the Carleson measures is the following: Theorem Let Ω be a ball. A sequence of measures µ n is Carleson if there is a constant C such that for all points z ∈ Ω µ n ( B ( z , 1 / n )) ≤ CV ( B ( z , 1 / n )) .

Bochner-Riesz type kernels Proof. The main ingredient in the proof is the existence of well localized kernels (the needlets of Petrushev and Xu), i.e. kernels L n ( x , y ) such that for an arbitrary k there is a constant C k such that: � K n ( x , x ) K n ( y , y ) | L n ( x , y ) | ≤ C k , ( 1 + nd ( x , y )) k and moreover L n ( x , x ) ≃ K n ( x , x ) and L n ∈ P 2 n and reproduce the polynomials of degree n .

The Nyquist density We try to identify which is the critical density. We will use the following result: Theorem (Berman, Boucksom, Witt-Nystr¨ om) If µ is a Bernstein-Markov measure then K n ( x , x ) d µ ( x ) ∗ ⇀ µ eq . N n The Bernstein-Markov condition is technical and it is satisfied when µ = χ Ω dV . The measure µ eq is the equilibrium measure.

The equilibrium potential Definition Given a compact K = Ω ⊂ R d and any z ∈ C d one defines the Siciak-Zaharjuta equilibrium potential as � log | p ( z ) | � u K ( z ) = sup : sup | p | ≤ 1 . deg( p ) K

The equilibrium potential Definition Given a compact K = Ω ⊂ R d and any z ∈ C d one defines the Siciak-Zaharjuta equilibrium potential as � log | p ( z ) | � u K ( z ) = sup : sup | p | ≤ 1 . deg( p ) K Then the equilibrium measure is defined as the Monge-Ampere of u K µ eq = ( i ∂ ¯ ∂ u K ) d . The equilibrium measure is a positive measure supported on K .

What does µ eq look like? The measure µ eq is a well-known object in pluripotential theory. In the examples we mentioned before it is well understood. Theorem (Bedford-Taylor) If Ω is an open bounded convex set in R d then d µ eq ( x ) ≃ d euc ( x , ∂ Ω) − 1 / 2 dV ( x ) .

Main result Theorem If Λ is an interpolating sequence for the polynomials in a bounded smooth strictly convex domain then 1 � δ λ ≤ µ eq . lim sup N n n →∞ λ ∈ Λ n

Main result Theorem If Λ is an interpolating sequence for the polynomials in a bounded smooth strictly convex domain then 1 � δ λ ≤ µ eq . lim sup N n n →∞ λ ∈ Λ n In particular, given any ball B in Ω we have #(Λ n ∩ B ) ≤ µ eq ( B ) , lim sup N n n →∞ thus µ eq is the Nyquist density.

The Kantorovich-Wasserstein distance Given a compact metric space K we defines the K-W distance between two measures µ and ν supported in K as �� KW ( µ, ν ) = inf d ( x , y ) d ρ ( x , y ) , ρ K × K where ρ is an admissible measure, i.e. the marginals of ρ are µ and ν respectively.

The Kantorovich-Wasserstein distance Given a compact metric space K we defines the K-W distance between two measures µ and ν supported in K as �� KW ( µ, ν ) = inf d ( x , y ) d ρ ( x , y ) , ρ K × K where ρ is an admissible measure, i.e. the marginals of ρ are µ and ν respectively. Alternatively: �� KW ( µ, ν ) = inf d ( x , y ) d | ρ | ( x , y ) , ρ K × K where ρ is an admissible complex measure, i.e. the marginals of ρ are µ and ν respectively

The complex transport plan The K-W distance metrizes the weak- ∗ convergence. We want to prove that KW ( b n , σ n ) → 0 , where b n ≤ K n ( x , x ) dV ( x ) / N n is smaller than the Bergman measure and σ n = 1 � δ λ N n λ ∈ Λ n