Interpolation by multivariate polynomials in convex domains
IINTERPOLATION BY MULTIVARIATE POLYNOMIALSIN CONVEX DOMAINS
JORGE ANTEZANA, JORDI MARZO, AND JOAQUIM ORTEGA-CERD `A
Abstract.
Let Ω be a convex open set in R n and let Λ k be a finitesubset of Ω. We find necessary geometric conditions for Λ k to beinterpolating for the space of multivariate polynomials of degree atmost k . Our results are asymptotic in k . The density conditionsobtained match precisely the necessary geometric conditions thatsampling sets are known to satisfy and they are expressed in termsof the equilibrium potential of the convex set. Moreover we provethat in the particular case of the unit ball, for k large enough,there is no family of orthogonal reproducing kernels in the spaceof polynomials of degree at most k . Introduction
Given a measure µ in R n we consider the space P k of polynomials oftotal degree at most k in n -variables endowed with the natural scalarproduct in L ( µ ). We assume that L ( µ ) is a norm for P k , i.e. thesupport of µ is not contained in the zero set of any p ∈ P k , p (cid:54) = 0. Inthis case the point evaluation at any given point x ∈ R n is a boundedlinear functional and ( P k , L ( µ )) becomes a reproducing kernel Hilbertspace, i.e for any x ∈ R n , there is a unique function K k ( µ, x, · ) ∈ P k such that p ( x ) = (cid:104) p, K k ( µ, x, · ) (cid:105) = ˆ p ( y ) K k ( µ, x, y ) dµ ( y ) . Given a point x ∈ R n the normalized reproducing kernel is denoted by κ k,y , i.e. κ k,y ( µ, x ) = K k ( µ, x, y ) (cid:107) K k ( µ, x, · ) (cid:107) L ( µ ) = K k ( µ, x, y ) (cid:112) K k ( µ, x, x ) . We will denote by β k ( µ, x ) the value of the reproducing kernel in thediagonal β k ( µ, x ) = K k ( µ, x, x ) . The function 1 /β k ( µ, x ) is the so called Christoffel function. For brevitywe may omit sometimes the dependence on µ . Date : January 21, 2021.Supported by the the Spanish Ministerio de Econom´ıa y Competividad (grantMTM2017-83499-P) and the Generalitat de Catalunya (grant 2017 SGR 358). a r X i v : . [ m a t h . C A ] J a n JORGE ANTEZANA, JORDI MARZO, AND JOAQUIM ORTEGA-CERD `A
Following Shapiro and Shields in [15] we define sampling and inter-polating sets:
Definition 1.
A sequence Λ = { Λ k } of finite sets of points on R n is said to be interpolating for ( P k , L ( µ )) if the associated family ofnormalized reproducing kernels at the points λ ∈ Λ k , i.e. κ k,λ , is aRiesz sequence in the Hilbert space P k , uniformly in k , i.e there is aconstant C > k such that for any linear combinationof the normalized reproducing kernels we have:(1.1) 1 C (cid:88) λ ∈ Λ k | c λ | ≤ (cid:13)(cid:13) (cid:88) λ ∈ Λ k c λ κ k,λ (cid:13)(cid:13) ≤ C (cid:88) λ ∈ Λ k | c λ | , ∀{ c λ } λ ∈ Λ k . The definition above is usually decoupled in two separate conditions.The left hand side inequality in (1.1) is usually called the
Riesz-Fischer property for the reproducing kernels and it is equivalent to the fact thatthe following moment problem is solvable: for arbitrary values { v λ } λ ∈ Λ k there exists a polynomial p ∈ P k such that p ( λ ) / (cid:112) β k ( λ ) = (cid:104) p, κ k,λ (cid:105) = v λ for all λ ∈ Λ k and (cid:107) p (cid:107) ≤ C (cid:88) λ ∈ Λ k | v λ | = (cid:88) λ ∈ Λ k | p ( λ ) | β k ( λ ) . This is the reason Λ is called an interpolating family.The right hand side inequality in (1.1) is called the Bessel propertyfor the normalized reproducing kernels { κ k,λ } λ ∈ Λ k . The Bessel propertyis equivalent to have(1.2) (cid:88) λ ∈ Λ k | p ( λ ) | β k ( λ ) ≤ C (cid:107) p (cid:107) for all p ∈ P k . That is, if we denote µ k := (cid:80) λ ∈ Λ k δ λ β k ( λ ) , we are re-quiring that the identity is a continuous embedding of ( P k , L ( µ )) into( P k , L ( µ k )).The notion of sampling play a similar but opposed role. Definition 2.
A sequence Λ = { Λ k } of finite sets of points on R n is said to be sampling or Marcinkiewicz-Zygmund for ( P k , L ( µ )) ifthe associated family of normalized reproducing kernels at the points λ ∈ Λ k , κ k,λ ( x ) is a frame in the Hilbert space P k , uniformly in k , i.ethere is a constant C > k such that for any polynomial p ∈ P k :(1.3) 1 C (cid:88) λ ∈ Λ k |(cid:104) p, κ k,λ (cid:105)| ≤ (cid:107) p (cid:107) ≤ C (cid:88) λ ∈ Λ k |(cid:104) p, κ k,λ (cid:105)| , ∀ p ∈ P k . Observe that the left hand side inequality in (1.3) is the Bessel condi-tion mentioned above. If we were considering a single space of polyno-mials P k then the notion of interpolating family amounts to say that NTERPOLATION BY POLYNOMIALS IN CONVEX DOMAINS 3 the corresponding reproducing kernels are independent. On the otherhand, the notion of sampling family corresponds to the reproducingkernels span the whole space P k .In this work we will restrict our attention to two classes of measures: • The first is dµ ( x ) = χ Ω ( x ) dV ( x ) where Ω is a smooth boundedconvex domain and dV is the Lebesgue measure. • The second is of the form dµ ( x ) = (1 − | x | ) a − / χ B ( x ) dV ( x )where a ≥ B is the unit ball B = { x ∈ R n : | x | ≤ } .In these two cases there are good explicit estimates for the size ofthe reproducing kernel on the diagonal K k ( µ, x, x ) , and therefore bothnotions, interpolation and sampling families, become more tangible.In [2] the authors obtained necessary geometric conditions for sam-pling families in bounded smooth convex sets with weights when theweights satisfy two technical conditions: Bernstein-Markov and mod-erate growth. These properties are both satisfied for the Lebesguemeasure in a convex set. The case of interpolating families in convexsets was not considered, since there were several technical hurdles toapply the same technique.Our aim in this paper is to fill this gap and obtain necessary geomet-ric conditions for interpolating families in the two settings mentionedabove. The geometric conditions that usually appear in this type ofproblem come into three flavours: • A separation condition. This is implied by the Riesz-Fischercondition i.e. the left hand side of (1.1). The fact that oneshould be able to interpolate the values one and zero impliesthat different points λ, λ (cid:48) ∈ Λ k with λ (cid:54) = λ (cid:48) cannot be too close.The separation conditions in our settings are studied in Section3.1. • A Carleson type condition. This is a condition that ensuresthe continuity of the embedding as in (1.2). A geometric char-acterization of the Carleson is given in Theorem 6 for convexdomains and the Lebesgue measure, and in Theorem 7 for theball and the measures µ a . • A density condition. This is a global condition that usually fol-lows from both the Bessel and the Riesz-Fischer condition. Adensity necessary condition for interpolating sequences is pro-vided in Theorem 9 for convex sets endowed with the Lebesguemeasure, and in Theorem 10 for the ball and the measures µ a .Moreover, in this last setting we get an extension of the densityresults proved in [2] for sampling sequences.Finally, a natural question is whether or not there exists a family { Λ k } that is both sampling and interpolating. To answer this questionis very difficult in general [13]. A particular case is when { κ k,λ } λ ∈ Λ k form an orthonormal basis. In the last section we study the existence JORGE ANTEZANA, JORDI MARZO, AND JOAQUIM ORTEGA-CERD `A of orthonormal basis of reproducing kernels in the case of the ballwith the measures µ a . More precisely, if the spaces P k endowed withthe inner product of L ( µ a ), then in Theorem 14 we prove that for k big enough the space P k does not admit an orthonormal basis ofreproducing kernels. To determine whether or not there exists a family { Λ k } that is both sampling and interpolating for ( P k , µ a ) remains anopen problem. 2. Technical results
Before stating and proving our results we will recall the behaviourof the kernel in the diagonal, or equivalently the Christoffel function,we will define an appropriate metric and introduce some needed tools.2.1.
Christoffel functions and equilibrium measures.
To writeexplicitly the sampling and interpolating conditions we need an esti-mate of the Christoffel function. In [2] it was observed that in the caseof the measure dµ ( x ) = χ Ω ( x ) dV ( x ) it is possible to obtain preciseestimates for the size of the reproducing kernel on the diagonal: Theorem 1.
Let Ω be a smoothly bounded convex domain in R n . Thenthe reproducing kernel for ( P k , χ Ω dV ) satisfies (2.1) β k ( x ) = K k ( x, x ) (cid:39) min (cid:16) k n (cid:112) d ( x, ∂ Ω) , k n +1 (cid:17) ∀ x ∈ Ω . where d ( x, ∂ Ω) denotes the Euclidean distance of x ∈ Ω to the boundaryof Ω . For the weight (1 − | x | ) a − / in the ball B the asymptotic behaviourof the Christoffel is well known. Proposition 2.
For any a ≥ and d ≥ let dµ a ( x ) = (1 − | x | ) a − / χ B ( x ) dV ( x ) . Then the reproducing kernel for ( P k , dµ a ) satisfies (2.2) β k ( µ a , x ) = K k ( µ a , x, x ) (cid:39) min (cid:16) k n d ( x, ∂ B ) a , k n +2 a (cid:17) ∀ x ∈ Ω . The proof follows from [14, Prop 4.5 and 5.6], Cauchy–Schwarz in-equality and the extremal characterization of the kernel K k ( µ a ; x, x ) = (cid:26) | P ( x ) | : P ∈ P k , ˆ | P | dµ a ≤ (cid:27) . To define the equilibrium measure we have to introduce a few con-cepts from pluripotential theory, see [9]. Given a non pluripolar com-pact set K ⊂ R n ⊂ C n the pluricomplex Green function is the semi-continuous regularization G ∗ K ( z ) = lim sup ξ → z G K ( ξ ) , NTERPOLATION BY POLYNOMIALS IN CONVEX DOMAINS 5 where G K ( ξ ) = sup (cid:26) log + | p ( ξ ) | deg( p ) : p ∈ P ( C n ) , sup K | p ( ξ ) | ≤ (cid:27) . The pluripotential equilibrium measure for of K is the (probability)Monge-Amp`ere Borel measure dµ eq = ( dd c G ∗ K ) n . In the general case, when Ω is a smooth bounded convex domain theequilibrium measure is very well understood, see [3] and [5]. It behavesroughly as dµ eq (cid:39) / (cid:112) d ( x, ∂ Ω) dV . In particular, the pluripotentialequilibrium measure for the ball B is given (up to normalization) by dµ ( x ) = √ −| x | dV ( x ) . An anisotropic distance.
The natural distance to formulate theseparation condition and the Carleson condition is not the Euclideandistance. Consider in the unit ball B ⊂ R n the following distance: ρ ( x, y ) = arccos (cid:110) (cid:104) x, y (cid:105) + (cid:112) − | x | (cid:112) − | y | (cid:111) . This is the geodesic distance of the points x (cid:48) , y (cid:48) in the sphere S n de-fined as x (cid:48) = ( x, (cid:112) − | x | ) and y (cid:48) = ( y, (cid:112) − | y | ). If we consideranisotropic balls B ( x, ε ) = { y ∈ B : ρ ( x, y ) < ε } , they are comparableto a box centered at x (a product of intervals) which are of size ε in thetangent directions and size ε + ε (cid:112) − | x | in the normal direction. Ifwe want to refer to a Euclidean ball of center x and radius ε we woulduse the notation B ( x, ε ).The Euclidean volume of a ball B ( x, ε ) is comparable to ε n (cid:112) − | x | if (1 − | x | ) > ε and ε n +1 otherwise.This distance ρ can be extended to an arbitrary smooth convex do-main Ω by using Euclidean balls contained in Ω and tangent to theboundary of Ω. This can be done in the following way. Since Ω issmooth, there is a tubular neighbourhood U ⊂ R n of the boundary ofΩ where each point x ∈ U has a unique closest point ˜ x in ∂ Ω and thenormal line to ∂ Ω at ˜ x passes by x . There is a fixed small radius r > x ∈ U ∩ Ω it is contained in a ball of radius r , B ( p, r ) ⊂ Ω and such that it is tangent to ∂ Ω at ˜ x . We define on x a Riemannian metric which comes from the pullback of the standardmetric on ∂ ˜ B ( p, r ) where ˜ B ( p, r ) is a ball in R n +1 centered at ( p,
0) andof radius r > R n +1 onto the first n -variables. Inthis way we have defined a Riemannian metric in the domain Ω ∩ U . Inthe core of Ω, i.e. far from the boundary we use the standard Euclideanmetric. We glue the two metrics with a partition of unity.The resulting metric ρ on Ω has the relevant property that the ballsof radius (cid:15) behave as in the unit ball, that is a ball B ( x, ε ) of center x JORGE ANTEZANA, JORDI MARZO, AND JOAQUIM ORTEGA-CERD `A and of radius ε in this metric is comparable to a box of size ε in thetangent directions and size ε + ε (cid:112) d ( x, ∂ Ω) in the normal direction.2.3.
Well localized polynomials.
The basic tool that we will useto prove the Carleson condition and the separation are well localizedpolynomials. These were studied by Petrushev and Xu in the unit ballwith the measure dµ a = (1 − | x | ) a − dV, for a ≥ . We recall theirbasic properties:
Theorem 3 (Petrushev and Xu) . Let dµ a = (1 − | x | ) a − dV for a ≥ . For any k ≥ entire and any y ∈ B ⊂ R n there are polynomials L ak ( · , y ) ∈ P k that satisfy: (1) L ak as a variable of x is a polynomial of degree k . (2) L ak ( x, y ) = L ak ( y, x ) . (3) L ak reproduces all the polynomials of degree k , i.e. (2.3) p ( y ) = b an ˆ B L ak ( x, y ) p ( x ) dµ a ( x ) . ∀ p ∈ P k . (4) For any γ > there is a c γ such that (2.4) | L ak ( x, y ) | ≤ c γ (cid:112) β k ( µ a , x ) β k ( µ a , y )(1 + kρ ( x, y )) γ . (5) The kernels L ak are Lispchitz with respect to the metric ρ , moreconcretely, for all x ∈ B ( y, /k ) : (2.5) | L ak ( w, x ) − L ak ( w, y ) | ≤ c γ kρ ( x, y ) (cid:112) β k ( µ a , w ) β k ( µ a , y )(1 + kρ ( w, y )) γ (6) There is ε > such that L ak ( x, y ) (cid:39) K k ( µ a ; y, y ) for all x ∈ B ( y, ε/k ) .Proof. All the properties are proved in [14, Thm 4.2, Prop 4.7 and4.8] except the behaviour near the diagonal number 6. Let us start byobserving that by the Lipschitz condition (2.5) it is enough to provethat L ak ( x, x ) (cid:39) K k ( µ a ; x, x ).This follows from the definition of L ak which is done as follows. Thesubspace V k ⊂ L ( B ) are the polynomials of degree k that are orthog-onal to lower degree polynomials in L ( B ) with respect to the measure dµ a . Consider the kernels P k ( x, y ) which are the kernels that give theorthogonal projection on V k . If f , . . . , f r is an orthonormal basis for V k then P k ( x, y ) = (cid:80) rj =1 f j ( x ) f j ( y ). The kernel L ak is defined as L ak ( x, y ) = ∞ (cid:88) j =0 ˆ a (cid:18) jk (cid:19) P j ( x, y ) . We assume that ˆ a is compactly supported, ˆ a ≥
0, ˆ a ∈ C ∞ ( R ), supp ˆ a ⊂ [0 , a ( t ) = 1 on [0 ,
1] and ˆ a ( t ) ≤ ,
2] as in the picture:
NTERPOLATION BY POLYNOMIALS IN CONVEX DOMAINS 7 k k a ( x/k )Then, all the terms are positive in the diagonal. Hence, we get β k ( µ a , x ) = K k ( µ a ; x, x ) ≤ L ak ( x, x ) ≤ K k ( µ a ; x, x ) = β k ( µ a , x ) . Since β k ( µ a , x ) (cid:39) β k ( µ a , x ) we obtain the desired estimate. (cid:3) They also proved the following integral estimate [14, Lemma 4.6]
Lemma 4.
Let α > and a ≥ . If γ > is big enough we have ˆ B K k ( µ a , y, y ) α (1 + kρ ( x, y )) γ dµ a ( y ) (cid:46) K k ( µ a , x, x ) − α . main results Separation.
In our first result we prove that for Λ = { Λ k } inter-polating there exist (cid:15) > λ,λ (cid:48) ∈ Λ k ,λ (cid:54) = λ (cid:48) ρ ( λ, λ (cid:48) ) ≥ (cid:15)k . Theorem 5. If Ω is a smooth convex set and Λ = { Λ k } is an interpolat-ing sequence then there is an ε > such that the balls { B ( λ, ε/k ) } λ ∈ Λ k are pairwise disjoint.Proof. Consider the metric in Ω defined in section 2.2. We can re-strict the argument to a ball, of a fixed radius r (Ω) , in one of the twocases: tangent to the boundary or at a positive distance to the com-plement R n \ Ω . Let us assume that there is another point from Λ k , λ (cid:48) ∈ B ( λ, ε/k ). Since it is interpolating we can build a polynomial p ∈ P k such that p ( λ (cid:48) ) = 0, p ( λ ) = 1 and (cid:107) p (cid:107) (cid:46) /K k ( µ , λ, λ ). Takea ball Ω such that it contains λ and λ (cid:48) and that it is tangent to ∂ Ωat a closest point to λ . To simplify the notation assume that radius ofthis ball is one, and it is denoted by B . In this ball the kernel L k fromTheorem 3, for the Lebesgue measure a = , is reproducing so(3.1) 1 = ˆ B ( L k ( λ, w ) − L k ( λ (cid:48) , w )) p ( w ) dV ( w ) . We can use the estimate | p ( w ) | ≤ (cid:113) β k ( µ , w ) (cid:107) p (cid:107) ≤ (cid:113) β k ( µ , w ) /β k ( µ , λ )) JORGE ANTEZANA, JORDI MARZO, AND JOAQUIM ORTEGA-CERD `A
Figure 3.1. and the inequality (2.5) to obtain1 (cid:46) kρ ( λ, λ (cid:48) ) ˆ B β k ( µ , w ) dV ( w )(1 + kρ ( y, λ )) γ , Taking α = 1 and a = in Lemma 4 we obtain 1 (cid:46) kρ ( λ, λ (cid:48) ) asstated. (cid:3) Observe that considering the general case L ak in (3.1), one can provethe corresponding result for interpolating sequences for P k with weight dµ a ( x ) = (1 − | x | ) a − dV ( x ) in the ball B . Carleson condition.
Let us deal with condition (1.2). For aconvex smooth set Ω ⊂ R n is a particular instance of the followingdefinition. Definition 3.
A sequence of measures µ k ∈ M (Ω) are called Carlesonmeasures for ( P k , dµ ) if there is a constant C > ˆ Ω | p ( x ) | dµ k ( x ) ≤ C (cid:107) p (cid:107) L ( µ ) , for all p ∈ P k .In particular if Λ k is a sequence of interpolating sets then the se-quence of measures µ k = (cid:80) λ ∈ Λ k δ λ β k ( λ ) is Carleson.The geometric characterization of the Carleson measures when Ω isa smooth convex domain is in terms of anisotropic balls. Theorem 6.
A sequence of measures µ k is Carleson for the polynomi-als P k in a smooth bounded convex domain Ω if and only if there is aconstant C such that for all points x ∈ Ω(3.2) µ k ( B ( x, /k )) ≤ CV ( B ( x, /k )) . Proof.
We prove the necessity. For any x ∈ Ω there is a cube Q thatcontains Ω which is tangent to ∂ Ω at a closest point to x as in thepicture: This cube has fixed dimensions independent of the point x ∈ Ω. We can construct a polynomial Q xk of degree at most kn taking NTERPOLATION BY POLYNOMIALS IN CONVEX DOMAINS 9 the product of one dimensional polynomials L k . We test against thesepolynomials that peak at B ( x, /k ) ˆ B ( x, /k ) | Q xk | dµ k ≤ ˆ Ω | Q xk | dµ k ≤ C (cid:107) Q xk (cid:107) L ( Q ) , by property (6) in Theorem 3 and the estimate (2.2) the necessarycondition follows.For the sufficiency we use the reproducing property of L k ( z, y ). Thatis for any point x ∈ Ω there is a Euclidean ball B x contained in Ω suchthat x ∈ B x and it is tangent to ∂ Ω in the closest point to x as in thepicture. Moreover since Ω is a smoothly bounded convex domain wecan assume that the radius B has a lower bound independent of x . Inthis ball we can reconstruct any polynomial p ∈ P k using L k . That is ˆ Ω | p ( x ) | dµ k ( x ) ≤ ˆ Ω (cid:12)(cid:12)(cid:12)(cid:12) ˆ B x L k ( x, y ) p ( y ) dV ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dµ k ( x ) . We use the estimate (2.4) and we get ˆ Ω | p ( x ) | dµ k ( x ) (cid:46) ˆ Ω ˆ B x (cid:112) β k ( x ) β k ( y )(1 + kρ ( x, y )) γ | p ( y ) | dV ( y ) dµ k ( x ) . We break the integral in two regions, when ρ ( x, y ) < k is big enough we obtain: ˆ Ω | p ( x ) | dµ k ( x ) ≤ ˆ Ω ˆ B x ∩ ρ ( x,y ) > | p ( y ) | dV ( y ) dµ k ( x )+ C ˆ Ω ˆ B x ∩ ρ ( x,y ) < (cid:112) β k ( x ) β k ( y )(1 + kρ ( x, y )) γ | p ( y ) | dV ( y ) dµ k ( x )The first integral in the right hand side is bounded by ´ Ω | p ( y ) | dV ( y )since µ k (Ω) is bounded by hypothesis (it is possible to cover Ω by balls { B ( x n , /k ) } with controlled overlap).In the second integral, observe that if w ∈ B ( x, /k ) then ρ ( w, x ) ≤ /k and therefore (cid:112) β k ( x ) β k ( y )(1 + kρ ( x, y )) γ (cid:46) V ( B ( x, /k )) ˆ B ( x, /k ) (cid:112) β k ( w ) β k ( y )(1 + kρ ( w, y )) γ dV ( w ) . We plug this inequality in the second integral and we can bound it by C ˆ Ω | p ( y ) | ˆ ρ ( w,y ) < (cid:112) β k ( w ) β k ( y )(1 + kρ ( w, y )) γ µ k ( B ( w, /k )) V ( B ( w, /k )) dV ( w ) dV ( y ) . We use the hypothesis (3.2) and Lemma 4 with α = 1 / C ´ Ω | p ( y ) | dV ( y ). (cid:3) The weighted case in the unit ball is simpler.
Theorem 7.
Let dµ a ( x ) = (1 − | x | ) a − dV ( x ) for a ≥ the weight inthe unit ball B ⊂ R n . A sequence of measures { µ k } are Carleson for ( P k , µ a ) if there is a constant C such that for all points x ∈ B (3.3) µ k ( B ( x, /k )) ≤ C µ a ( B ( x, /k )) . Proof.
Supose { µ k } are Carleson. Then for any x ∈ B (3.4) ˆ B ( x, /k ) | L ak ( x, w ) | dµ k ( w ) ≤ C (cid:107) L ak ( x, · ) (cid:107) µ . By property (6) in Theorem 3 and the estimate K k ( µ, x, x ) ≤ (cid:107) L ak ( x, · ) (cid:107) µ ≤ K k ( µ, x, x ) , the result follows. The necessity follows exactly like in the unweightedcase with the obvious changes. (cid:3) Density condition.
In [2, Theorem 4] a necessary density con-dition for sampling sequences for polynomials in convex domains wasobtained. It states the following:
Theorem 8.
Let Ω be a smooth convex domain in R n , and let Λ be asampling sequence. Then for any B ( x, r ) ⊂ Ω the following holds: lim sup k →∞ k ∩ B ( x, r )dim P k ≥ µ eq ( B ( x, r )) . Here µ eq is the equilibrium measure associated to Ω . Let us see how, with a similar technique, a corresponding densitycondition can be obtained as well in the case of interpolating sequences.
Theorem 9.
Let Ω be a smooth convex domain in R n , and let Λ be aninterpolating sequence. Then for any B ( x, r ) ⊂ Ω the following holds: lim sup k →∞ k ∩ B ( x, r )dim P k ≤ µ eq ( B ( x, r )) . Here µ eq is the equilibrium measure associated to Ω .Remark. In the statements of Theorems 8 and 9 we could have replaced B ( x, r ) by any open set, in particular they could have been formulatedwith balls B ( x, r ) in the anisotropic metric. Proof.
Let F k ⊂ P k be the subspace spanned by κ λ ( x ) = K k ( λ, x ) / (cid:112) β k ( λ ) ∀ λ ∈ Λ k . Denote by g λ the dual (biorthogonal) basis to κ λ in F k . We have clearlythat • We can span any function in F k in terms of κ λ , thus: (cid:88) λ ∈ Λ k κ λ ( x ) g λ ( x ) = K k ( x, x ) , where K k ( x, y ) is the reproducing kernel of the subspace F k . NTERPOLATION BY POLYNOMIALS IN CONVEX DOMAINS 11 • The norm of g λ is uniformly bounded since κ λ was a uniformRiesz sequence. • g λ ( λ ) = (cid:112) β k ( λ ). This is due to the biorthogonality and thereproducing property.We are going to prove that the measure σ k = P k (cid:80) λ ∈ Λ k δ λ , and themeasure ν k = P k K k ( x, x ) dµ ( x ) are very close to each other. Thisare two positive measures that are not probability measures but theyhave the same mass (equal to k dim P k ≤ W ( σ k , ν k ) → K k ( x, x ) ≤ K k ( x, x ) and P k β k ( x ) → µ eq in the weak-* topology, where µ eq is the normalized equilibriummeasure associated to Ω (see for instance [1]). Therefore, lim sup k σ k ≤ µ eq .In order to prove that W ( σ k , ν k ) → ρ k ( x, y ) = 1dim P k (cid:88) λ ∈ Λ k δ λ ( y ) × g λ ( x ) κ λ ( x ) dµ ( x )It has the right marginals, σ k and ν k and we can estimate the integral W ( σ k , ν k ) ≤ ¨ Ω × Ω | x − y | d | ρ k | = O (1 / √ k ) . The only point that merits a clarification is that we need an inequality:1dim P k (cid:88) λ ∈ Λ k ˆ Ω | λ − x | | K k ( λ, x ) | K k ( x, x ) dµ ( x ) ≤ P k ¨ Ω × Ω | y − x | | K k ( y, x ) | dµ ( x ) dµ ( y ) . This is problematic. We know that Λ k is an interpolating sequence forthe polynomials of degree k . Thus the normalized reproducing kernelsat λ ∈ Λ k form a Bessel sequence for P k but the inequality that weneed is applied to K k ( x, y )( y i − x i ) for all i = 1 , . . . , n . That is toa polynomial of degree k + 1. We are going to show that if Λ k isan interpolating sequence for the polynomials of degree k it is also aCarleson sequence for the polynomials of degree k + 1.Observe that since it is interpolating then it is uniformly separated,i.e. B ( λ, ε/k ) are disjoint. That means that in particular µ k ( B ( z, / ( k + 1)) (cid:46) V ( B ( z, / ( k + 1)) . Thus µ k is a Carleson measure for P k +1 . Finally in [2, Theorem 17] it was proved that1dim P k ¨ Ω × Ω | y − x | | K k ( y, x ) | dµ ( x ) dµ ( y ) = O (1 /k ) . (cid:3) From the behaviour on the diagonal of the kernel (2.2) its easy tocheck that the kernel is both Bernstein-Markov (sub-exponential) andhas moderate growth, see definitions in [2]. From the characterizationfor sampling sequences proved in [2, Theorem 1] and with the obviouschanges in the proof of the previous theorem we deduce the following:
Theorem 10.
Consider the space of polynomials P k restricted to theball B ⊂ R n with the measure dµ a ( x ) = (1 − | x | ) a − dV. Let
Λ = { Λ k } be a sequence sets of points in B . • If Λ is a sampling sequence lim inf k →∞ k ∩ B ( x, r ))dim P k ≥ µ eq ( B ( x, r )) . • If Λ is interpolating lim sup k →∞ k ∩ B ( x, r ))dim P k ≤ µ eq ( B ( x, r )) . Remark.
One can construct interpolation or sampling sequences withdensity arbitrary close to the critical density with sequences of points { Λ k } such that the corresponding Lagrange interpolating polynomialsare uniformly bounded. In particular de above inequalities are sharp,for a similar construction on the sphere see [12].3.4. Orthonormal basis of reproducing kernels.
Sampling andinterpolation are somehow dual concepts. Sequences which are bothsampling and interpolating (i.e. complete interpolating sequences) areoptimal in some sense because they are at the same time minimalsampling sequences and maximal interpolating sequences. They willsatisfy the equality in Theorem 10. In general domains, to prove ordisprove the existence of such sequences is a difficult problem [13].If Λ = { Λ k } is a complete interpolating sequence the correspondingreproducing kernels { κ k,λ } is a Riesz basis in the space of polynomials(uniformly in the degree). An obvious example of complete interpo-lating sequences would be sequences providing an orthonormal basisof reproducing kernels. In dimension 1, with the weight (1 − x ) a − / , a basis of Gegenbauer polynomials { G ( a ) j } j =0 ,...,k is orthogonal and thereproducing kernel in P k evaluated at the zeros of the polynomial G ( a ) k +1 gives an orthogonal sequence. In our last result we prove that forgreater dimensions there are no orthogonal basis of P k of reproducingkernels with the measure dµ a ( x ) = (1 − | x | ) a − / dV ( x ) . NTERPOLATION BY POLYNOMIALS IN CONVEX DOMAINS 13
Our first goal is to show that sampling sequences are dense enough,Theorem 12. Recall that in the bulk (i.e. at a fixed positive distancefrom the boundary) the Euclidean metric and the metric ρ are equiva-lent. In our first result we prove that the right hand side of (1.3) andthe separation imply that there are points of the sequence in any ball(of the bulk) of big enough radius. Proposition 11.
Let dµ a ( x ) = (1 −| x | ) a − dV ( x ) for a ≥ the weightin the unit ball B ⊂ R n . Let Λ k ⊂ B be a finite subset and C, (cid:15) > beconstants such that (3.5) ˆ B | P ( x ) | dµ a ( x ) ≤ C (cid:88) λ ∈ Λ k | P ( λ ) | K k ( µ a ; λ, λ ) , for all P ∈ P k and inf λ,λ (cid:48) ∈ Λ k λ (cid:54) = λ (cid:48) ρ ( λ, λ (cid:48) ) ≥ (cid:15)k . Let | x | = C < , (cid:15) < M and k ≥ be such that Λ k ∩ B ( x , M/k ) = ∅ . Then
M < A for a certain constant A depending only on C, (cid:15), n and a. Proof.
By the construction of function L a(cid:96) ( x, y ) , it is clear that for any (cid:96) ≥ K (cid:96) ( µ a ; x, x ) ≤ ˆ B L a(cid:96) ( x, y ) dµ a ( y ) ≤ K (cid:96) ( µ a ; x, x ) . Let P ( x ) = L a [ k/ ( x, x ) ∈ P k . From the property above, the hypothesisand Proposition 2 we get(3.6) k n ∼ K [ k/ ( µ a ; x , x ) ≤ ˆ B P ( y ) dµ a ( y ) (cid:46) (cid:88) | λ − x | >M/k | P ( λ ) | K k ( µ a ; λ, λ ) . From [6, Lemma 11.3.6.], given x ∈ B and 0 < r < π (3.7) µ a ( B ( x, r )) ∼ r n ( (cid:112) − | x | + r ) a , and therefore(3.8) µ a ( B ( x, r )) ∼ (cid:40) r n +2 a if 1 − | x | < r ,r n (1 − | x | ) a otherwise , and(3.9) µ a ( B ( x, r )) (cid:38) (cid:40) r n +2 a if | x | > ,r n otherwise . From (4) in Theorem 3, the separation of the sequence, and theestimate (3.9) we get0 < c ≤ (cid:88) | λ − x | >M/k k/ ρ ( x , λ )) γ = (cid:88) | λ − x | >M/k µ a ( B ( λ, (cid:15)/ k )) ˆ B ( λ,(cid:15)/ k ) dµ a ( x )(1 + [ k/ ρ ( x , λ )) γ (cid:46) (cid:88) Mk < | λ − x | < + (cid:88) < | λ − x | µ a ( B ( λ, (cid:15)/ k )) ˆ B ( λ,(cid:15)/ k ) dµ a ( x )(1 + 2 kρ ( x , x )) γ (cid:46) (cid:18) k(cid:15) (cid:19) n ˆ Mk r n − ( kr ) γ dr + k a + n − γ (cid:15) a + n µ a ( B (0 , / c ) . (3.10)Now, for γ = n + a we get0 < c ≤ k n +2 a (cid:20) − r n +2 a (cid:21) r = Mk + 1 k n , and then a uniform (i.e. independent of k ) upper bound for M < A = A ( C, (cid:15), n, a ) . (cid:3) Proposition 12.
Let
Λ = { Λ k } be a separated sampling sequence for B ⊂ R n . Then there exist M , k > such that for any M > M andall k ≥ k k ∩ B (0 , M/k )) ∼ M n . Proof.
Let (cid:15) > λ,λ (cid:48) ∈ Λ k λ (cid:54) = λ (cid:48) ρ ( λ, λ (cid:48) ) ≥ (cid:15)k . Assume that
M/k ≤ . For λ ∈ Λ k ∩ B (0 , M/k ) we have V ( B ( λ, (cid:15)k )) ∼ ( (cid:15)k ) n and therefore(3.11) k ∩ B (0 , M/k )) (cid:16) (cid:15)k (cid:17) n (cid:46) (cid:18) Mk (cid:19) n . For the other inequality, take the constant A (assume A > (cid:15) ) given inProposition 11 depending on the sampling and the separation constantsof Λ and n. For
M > A and k > B (0 , Mk ) ⊂ B (0 , ) one canfind N disjoint balls B ( x j , Ak ) for j = 1 , . . . N included in B (0 , M/k )and such that N V ( B (0 , Ak )) > V ( B (0 , Mk )) . NTERPOLATION BY POLYNOMIALS IN CONVEX DOMAINS 15
Observe that each ball B ( x j , Ak ) contains by Proposition 11 at least onepoint from Λ k and therefore k ∩ B (0 , M/k )) ≥ N (cid:38) (cid:18) MA (cid:19) n . (cid:3) We will use the following result from [8].
Theorem 13.
Let B ⊂ R n , n > , be the unit ball. There do notexist infinite subsets Λ ⊂ R n such that the exponentials e i (cid:104) x,λ (cid:105) , λ ∈ Λ , are pairwise orthogonal in L ( B ) . Or, equivalently, there do not existinfinite subsets Λ ⊂ R n such that | λ − λ (cid:48) | is a zero of J n/ , the Besselfunction of order n/ , for all distinct λ, λ (cid:48) ∈ Λ . Following ideas from [7] we can prove now our main result aboutorthogonal basis. A similar argument can be used on the sphere tostudy tight spherical designs.
Theorem 14.
Let B ⊂ R n be the unit ball and n > . There is nosequence of finite sets
Λ = { Λ k } ⊂ B such that the reproducing kernels { K k ( µ : x, λ ) } λ ∈ Λ k form an orthogonal basis of P k with respect to themeasure dµ a = (1 − | x | ) a − dV .Theorem 14. The following result can be easily deduced from [10, The-orem 1.7]:Given { u k } k , { v k } k convergent sequences in R n and u k → u, v k → v, when k → ∞ . Thenlim k →∞ K k ( µ ; u k k , v k k ) K k ( µ ; 0 ,
0) = J ∗ n/ ( | u − v | ) J ∗ n/ (0) . Let Λ k be such that { κ λ } λ ∈ Λ k is an orthonormal basis of P k withrespect to the measure dµ a = (1 − | x | ) a − dV . Then K k ( µ ; λ ( k ) , λ (cid:48) ( k ) ) = 0 , for λ ( k ) (cid:54) = λ (cid:48) ( k ) ∈ Λ k . We know that Λ k is uniformly separated for some (cid:15) > ρ ( λ ( k ) , λ (cid:48) ( k ) ) ≥ (cid:15)k . Then the sets X k = k (Λ k ∩ B (0 , / ⊂ R n are uniformly separated | λ − λ (cid:48) | (cid:38) (cid:15), λ (cid:54) = λ (cid:48) ∈ X, and X k converges weakly to some uniformly separated set X ⊂ R n . The limit is not empty because by Proposition 12 for any
M > , k ∩ B (0 , M/k )) ∼ M d . Observe that this last result would be a direct consequence of the nec-essary density condition for complete interpolating sets if we could take balls of radius r/n for a fixed r > X such that for λ (cid:54) = λ (cid:48) ∈ XJ ∗ n/ ( | λ − λ (cid:48) | ) = 0 , in contradiction with Theorem 13. (cid:3) Remark.
Note that the fact that the interpolating sequence { Λ k } iscomplete was used only to guarantee that k ∩ B (0 , M/k )) ∼ M d .So, the above result could be extended to sequences { Λ k } such that { κ k,λ } λ ∈ Λ k is orthonormal (but not necessarily a basis for P k ) if Λ k ∩ B (0 , M/k ) contains enough points. References [1] R. Berman, S. Boucksom, D.W. Nystr¨om
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Departamento de Matem´atica, Universidad Nacional de La Plata,and Instituto Argentino de Matem´atica “Alberto P. Calder´on” (IAM-CONICET), Buenos Aires, Argentina
Email address : [email protected] Dept. Matem`atica i Inform`atica, Universitat de Barcelona and BGS-Math, Gran Via 585, 08007 Barcelona, Spain
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