Interpolation by multivariate polynomials in convex domains

Let $\Omega$ be a convex open set in $\mathbb R^n$ and let $\Lambda_k$ be a finite subset of $\Omega$. We find necessary geometric conditions for $\Lambda_k$ to be interpolating for the space of multivariate polynomials of degree at most $k$. Our results are asymptotic in $k$. The density conditions obtained match precisely the necessary geometric conditions that sampling sets are known to satisfy, and they are expressed in terms of the equilibrium potential of the convex set. Moreover, we prove that in the particular case of the unit ball, for $k$ large enough, there is no family of orthogonal reproducing kernels in the space of polynomials of degree at most $k$.


Introduction
Given a measure µ in R n we consider the space P k of polynomials of total degree at most k in n-variables endowed with the natural scalar product in L 2 (µ). We assume that L 2 (µ) is a norm for P k , i.e. the support of µ is not contained in the zero set of any p ∈ P k , p = 0. In this case the point evaluation at any given point x ∈ R n is a bounded linear functional and (P k , L 2 (µ)) becomes a reproducing kernel Hilbert space, i.e for any x ∈ R n , there is a unique function K k (µ, x, ·) ∈ P k such that p(x) = p, K k (µ, x, ·) =ˆp(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. Following Shapiro and Shields in [15] we define sampling and interpolating 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 2 (µ)) if the associated family of normalized reproducing kernels at the points λ ∈ Λ k , i.e. κ k,λ , is a Riesz sequence in the Hilbert space P k , uniformly in k, i.e there is a constant C > 0 independent of k such that for any linear combination of the normalized reproducing kernels we have: 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 that the following moment problem is solvable: for arbitrary values {v λ } λ∈Λ k there exists a polynomial p ∈ P k such that p(λ)/ β k (λ) = p, κ k,λ = v λ for all λ ∈ Λ k and This is the reason Λ is called an interpolating family. The right hand side inequality in (1.1) is called the Bessel property for the normalized reproducing kernels {κ k,λ } λ∈Λ k . The Bessel property is equivalent to have (1.2) λ∈Λ k |p(λ)| 2 β k (λ) ≤ C p 2 for all p ∈ P k . That is, if we denote µ k := λ∈Λ k δ λ β k (λ) , we are requiring that the identity is a continuous embedding of (P k , L 2 (µ)) into (P k , L 2 (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 2 (µ)) if the 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.e there is a constant C > 0 independent of k such that for any polynomial p ∈ P k : Observe that the left hand side inequality in (1.3) is the Bessel condition mentioned above. If we were considering a single space of polynomials P k 0 then the notion of interpolating family amounts to say that the corresponding reproducing kernels are independent. On the other hand, the notion of sampling family corresponds to the reproducing kernels span the whole space P k 0 .
In this work we will restrict our attention to two classes of measures: where Ω is a smooth bounded convex domain and dV is the Lebesgue measure.
where a ≥ 0 and B is the unit ball B = {x ∈ R n : |x| ≤ 1}.
In these two cases there are good explicit estimates for the size of the reproducing kernel on the diagonal K k (µ, x, x), and therefore both notions, interpolation and sampling families, become more tangible. In [2] the authors obtained necessary geometric conditions for sampling families in bounded smooth convex sets with weights when the weights satisfy two technical conditions: Bernstein-Markov and moderate growth. These properties are both satisfied for the Lebesgue measure in a convex set. The case of interpolating families in convex sets was not considered, since there were several technical hurdles to apply the same technique.
Our aim in this paper is to fill this gap and obtain necessary geometric conditions for interpolating families in the two settings mentioned above. The geometric conditions that usually appear in this type of problem come into three flavours: • A separation condition. This is implied by the Riesz-Fischer condition i.e. the left hand side of (1.1). The fact that one should be able to interpolate the values one and zero implies that different points λ, λ ∈ Λ k with λ = λ cannot be too close. The separation conditions in our settings are studied in Section 3.1. • A Carleson type condition. This is a condition that ensures the continuity of the embedding as in (1.2). A geometric characterization of the Carleson is given in Theorem 6 for convex domains and the Lebesgue measure, and in Theorem 7 for the ball and the measures µ a . • A density condition. This is a global condition that usually follows from both the Bessel and the Riesz-Fischer condition. A density necessary condition for interpolating sequences is provided in Theorem 9 for convex sets endowed with the Lebesgue measure, and in Theorem 10 for the ball and the measures µ a . Moreover, in this last setting we get an extension of the density results 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 question is 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 of orthonormal basis of reproducing kernels in the case of the ball with the measures µ a . More precisely, if the spaces P k endowed with the inner product of L 2 (µ a ), then in Theorem 14 we prove that for k big enough the space P k does not admit an orthonormal basis of reproducing kernels. To determine whether or not there exists a family {Λ k } that is both sampling and interpolating for (P k , µ a ) remains an open problem.

Technical results
Before stating and proving our results we will recall the behaviour of the kernel in the diagonal, or equivalently the Christoffel function, we will define an appropriate metric and introduce some needed tools.

Christoffel functions and equilibrium measures.
To write explicitly the sampling and interpolating conditions we need an estimate of the Christoffel function. In [2] it was observed that in the case of the measure dµ(x) = χ Ω (x)dV (x) it is possible to obtain precise estimates for the size of the reproducing kernel on the diagonal: Let Ω be a smoothly bounded convex domain in R n . Then the reproducing kernel for (P k , χ Ω dV ) satisfies where d(x, ∂Ω) denotes the Euclidean distance of x ∈ Ω to the boundary of Ω.
For the weight (1 − |x| 2 ) a−1/2 in the ball B the asymptotic behaviour of the Christoffel is well known. Then the reproducing kernel for (P k , dµ a ) satisfies The proof follows from [14, Prop 4.5 and 5.6], Cauchy-Schwarz inequality and the extremal characterization of the kernel To define the equilibrium measure we have to introduce a few concepts from pluripotential theory, see [9]. Given a non pluripolar compact set K ⊂ R n ⊂ C n the pluricomplex Green function is the semicontinuous regularization The pluripotential equilibrium measure for of K is the (probability) Monge-Ampère Borel measure dµ eq = (dd c G * K ) n . In the general case, when Ω is a smooth bounded convex domain the equilibrium measure is very well understood, see [3] and [5]. It behaves roughly as dµ eq 1/ d(x, ∂Ω)dV . In particular, the pluripotential equilibrium measure for the ball B is given (up to normalization) by

2.2.
An anisotropic distance. The natural distance to formulate the separation condition and the Carleson condition is not the Euclidean distance. Consider in the unit ball B ⊂ R n the following distance: This is the geodesic distance of the points x , y in the sphere S n defined as x = (x, 1 − |x| 2 ) and y = (y, 1 − |y| 2 ). If we consider anisotropic balls B(x, ε) = {y ∈ B : ρ(x, y) < ε}, they are comparable to a box centered at x (a product of intervals) which are of size ε in the tangent directions and size ε 2 + ε 1 − |x| 2 in the normal direction. If we want to refer to a Euclidean ball of center x and radius ε we would use the notation B(x, ε). The Euclidean volume of a ball B(x, ε) is comparable to ε n 1 − |x| 2 if (1 − |x| 2 ) > ε 2 and ε n+1 otherwise.
This distance ρ can be extended to an arbitrary smooth convex domain Ω by using Euclidean balls contained in Ω and tangent to the boundary of Ω. This can be done in the following way. Since Ω is smooth, there is a tubular neighbourhood U ⊂ R n of the boundary of Ω where each point x ∈ U has a unique closest pointx in ∂Ω and the normal line to ∂Ω atx passes by x. There is a fixed small radius r > 0 such that for any point x ∈ U ∩ Ω it is contained in a ball of radius r, B(p, r) ⊂ Ω and such that it is tangent to ∂Ω atx. We define on x a Riemannian metric which comes from the pullback of the standard metric on ∂B(p, r) whereB(p, r) is a ball in R n+1 centered at (p, 0) and of radius r > 0 by the projection of R n+1 onto the first n-variables. In this way we have defined a Riemannian metric in the domain Ω ∩ U . In the core of Ω, i.e. far from the boundary we use the standard Euclidean metric. We glue the two metrics with a partition of unity.
The resulting metric ρ on Ω has the relevant property that the balls of radius behave as in the unit ball, that is a ball B(x, ε) of center x and of radius ε in this metric is comparable to a box of size ε in the tangent directions and size ε 2 + ε d(x, ∂Ω) in the normal direction.
2.3. Well localized polynomials. The basic tool that we will use to prove the Carleson condition and the separation are well localized polynomials. These were studied by Petrushev and Xu in the unit ball with the measure dµ a = (1 − |x| 2 ) a− 1 2 dV, for a ≥ 0. We recall their basic properties: For any k ≥ 1 entire and any y ∈ B ⊂ R n there are polynomials L a k (·, y) ∈ P k that satisfy: (5) The kernels L a k are Lispchitz with respect to the metric ρ, more concretely, for all x ∈ B(y, 1/k): Proof. All the properties are proved in [14, Thm 4.2, Prop 4.7 and 4.8] except the behaviour near the diagonal number 6. Let us start by observing that by the Lipschitz condition (2.5) it is enough to prove that L a k (x, x) K k (µ a ; x, x). This follows from the definition of L a k which is done as follows. The subspace V k ⊂ L 2 (B) are the polynomials of degree k that are orthogonal to lower degree polynomials in L 2 (B) with respect to the measure dµ a . Consider the kernels P k (x, y) which are the kernels that give the orthogonal projection on V k . If f 1 , . . . , f r is an orthonormal basis for We assume thatâ is compactly supported,â ≥ 0,â ∈ C ∞ (R), suppâ ⊂ [0, 2],â(t) = 1 on [0, 1] andâ(t) ≤ 1 on [1,2] as in the picture: Then, all the terms are positive in the diagonal. Hence, we get x, x) = β 2k (µ a , x). Since β k (µ a , x) β 2k (µ a , x) we obtain the desired estimate.

Theorem 5.
If Ω is a smooth convex set and Λ = {Λ k } is an interpolating sequence then there is an ε > 0 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 restrict the argument to a ball, of a fixed radius r(Ω), in one of the two cases: tangent to the boundary or at a positive distance to the complement R n \ Ω. Let us assume that there is another point from Λ k , λ ∈ B(λ, ε/k). Since it is interpolating we can build a polynomial p ∈ P k such that p(λ ) = 0, p(λ) = 1 and p 2 1/K k (µ1 2 , λ, λ). Take a ball Ω such that it contains λ and λ and that it is tangent to ∂Ω at a closest point to λ. To simplify the notation assume that radius of this ball is one, and it is denoted by B. In this ball the kernel L 1 2 k from Theorem 3, for the Lebesgue measure a = 1 2 , is reproducing so We can use the estimate (1 + kρ(y, λ)) γ , Taking α = 1 and a = 1 2 in Lemma 4 we obtain 1 kρ(λ, λ ) as stated.
Observe that considering the general case L a k in (3.1), one can prove the corresponding result for interpolating sequences for P k with weight dµ a (x) = (1 − |x| 2 ) a− 1 2 dV (x) in the ball B.

Carleson condition.
Let us deal with condition (1.2). For a convex smooth set Ω ⊂ R n is a particular instance of the following definition.
Definition 3. A sequence of measures µ k ∈ M(Ω) are called Carleson measures for (P k , dµ) if there is a constant C > 0 such that for all p ∈ P k .
In particular if Λ k is a sequence of interpolating sets then the sequence of measures µ k = λ∈Λ k Proof. We prove the necessity. For any x ∈ Ω there is a cube Q that contains Ω which is tangent to ∂Ω at a closest point to x as in the picture: This cube has fixed dimensions independent of the point x ∈ Ω. We can construct a polynomial Q x k of degree at most kn taking the product of one dimensional polynomials L 1 2 k . We test against these polynomials that peak at B(x, 1/k) by property (6) in Theorem 3 and the estimate (2.2) the necessary condition follows. For the sufficiency we use the reproducing property of L 1 2 k (z, y). That is for any point x ∈ Ω there is a Euclidean ball B x contained in Ω such that x ∈ B x and it is tangent to ∂Ω in the closest point to x as in the picture. Moreover since Ω is a smoothly bounded convex domain we can assume that the radius B has a lower bound independent of x. In this ball we can reconstruct any polynomial p ∈ P k using L We use the estimate (2.4) and we get We break the integral in two regions, when ρ(x, y) < 1 and otherwise. When k is big enough we obtain: The first integral in the right hand side is bounded by´Ω |p(y)| 2 dV (y) since µ k (Ω) is bounded by hypothesis (it is possible to cover Ω by balls {B(x n , 1/k)} with controlled overlap).
In the second integral, observe that if w ∈ B(x, 1/k) then ρ(w, x) ≤ 1/k and therefore We plug this inequality in the second integral and we can bound it by We use the hypothesis (3.2) and Lemma 4 with α = 1/2 to bound it finally by C´Ω |p(y)| 2 dV (y).
The weighted case in the unit ball is simpler.
Proof. Supose {µ k } are Carleson. Then for any x ∈ B By property (6) in Theorem 3 and the estimate , x), the result follows. The necessity follows exactly like in the unweighted case with the obvious changes.
3.3. Density condition. In [2, Theorem 4] a necessary density condition for sampling sequences for polynomials in convex domains was obtained. It states the following: Let Ω be a smooth convex domain in R n , and let Λ be a sampling sequence. Then for any B(x, r) ⊂ Ω the following holds: , r)).
Here µ eq is the equilibrium measure associated to Ω.
Let us see how, with a similar technique, a corresponding density condition 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 an interpolating sequence. Then for any B(x, r) ⊂ Ω the following holds: 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 formulated with balls B(x, r) in the anisotropic metric.
Proof. Let F k ⊂ P k be the subspace spanned by Denote by g λ the dual (biorthogonal) basis to κ λ in F k . We have clearly that • We can span any function in F k in terms of κ λ , thus: where K k (x, y) is the reproducing kernel of the subspace F k .
• The norm of g λ is uniformly bounded since κ λ was a uniform Riesz sequence. • g λ (λ) = β k (λ). This is due to the biorthogonality and the reproducing property.
We are going to prove that the measure σ k = 1 dim P k λ∈Λ k δ λ , and the measure ν k = 1 dim P k K k (x, x)dµ(x) are very close to each other. This are two positive measures that are not probability measures but they have the same mass (equal to #Λ k dim P k ≤ 1). Therefore, there is a way to quantify the closeness through the Vaserstein 1-distance. For an introduction to Vaserstein distance see for instance [16]. We want to prove that W (σ k , ν k ) → 0 because the Vaserstein distance metrizes the weak-* topology.
In this case, it is known that K k (x, x) ≤ K k (x, x) and 1 dim P k β k (x) → µ eq in the weak-* topology, where µ eq is the normalized equilibrium measure 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 ) → 0 we use a non positive transport plan as in [11]: It has the right marginals, σ k and ν k and we can estimate the integral The only point that merits a clarification is that we need an inequality: This is problematic. We know that Λ k is an interpolating sequence for the polynomials of degree k. Thus the normalized reproducing kernels at λ ∈ Λ k form a Bessel sequence for P k but the inequality that we need is applied to K k (x, y)(y i − x i ) for all i = 1, . . . , n. That is to a polynomial of degree k + 1. We are going to show that if Λ k is an interpolating sequence for the polynomials of degree k it is also a Carleson sequence for the polynomials of degree k + 1.
Thus µ k is a Carleson measure for P k+1 .
Finally in [2,Theorem 17] it was proved that From the behaviour on the diagonal of the kernel (2.2) its easy to check that the kernel is both Bernstein-Markov (sub-exponential) and has moderate growth, see definitions in [2]. From the characterization for sampling sequences proved in [2, Theorem 1] and with the obvious changes in the proof of the previous theorem we deduce the following: Theorem 10. Consider the space of polynomials P k restricted to the ball B ⊂ R n with the measure dµ a (x) = (1 − |x| 2 ) a− 1 2 dV. Let Λ = {Λ k } be a sequence sets of points in B.
Remark. One can construct interpolation or sampling sequences with density arbitrary close to the critical density with sequences of points {Λ k } such that the corresponding Lagrange interpolating polynomials are 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 and interpolation are somehow dual concepts. Sequences which are both sampling and interpolating (i.e. complete interpolating sequences) are optimal in some sense because they are at the same time minimal sampling sequences and maximal interpolating sequences. They will satisfy the equality in Theorem 10. In general domains, to prove or disprove the existence of such sequences is a difficult problem [13]. If Λ = {Λ k } is a complete interpolating sequence the corresponding reproducing kernels {κ k,λ } is a Riesz basis in the space of polynomials (uniformly in the degree). An obvious example of complete interpolating sequences would be sequences providing an orthonormal basis of reproducing kernels. In dimension 1, with the weight (1 − x 2 ) a−1/2 , a basis of Gegenbauer polynomials {G (a) j } j=0,...,k is orthogonal and the reproducing kernel in P k evaluated at the zeros of the polynomial G gives an orthogonal sequence. In our last result we prove that for greater dimensions there are no orthogonal basis of P k of reproducing kernels with the measure dµ a (x) = (1 − |x| 2 ) a−1/2 dV (x).
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 distance from the boundary) the Euclidean metric and the metric ρ are equivalent. In our first result we prove that the right hand side of (1.3) and the separation imply that there are points of the sequence in any ball (of the bulk) of big enough radius.
for a ≥ 0 the weight in the unit ball B ⊂ R n . Let Λ k ⊂ B be a finite subset and C, > 0 be constants such that , Let |x 0 | = C 0 < 1 4 , < M and k ≥ 1 be such that Λ k ∩ B(x 0 , M/k) = ∅. Then M < A for a certain constant A depending only on C, , n and a.
Proof. By the construction of function L a (x, y), it is clear that for any ≥ 0 K (µ a ; x, x) ≤ˆB L a (x, y) 2 dµ a (y) ≤ K 2 (µ a ; x, x).
Let P (x) = L a [k/2] (x, x 0 ) ∈ P k . From the property above, the hypothesis and Proposition 2 we get (3.6) .
Assume that M/k ≤ 1 2 . For λ ∈ Λ k ∩ B(0, M/k) we have V (B(λ, k )) ∼ ( k ) n and therefore For the other inequality, take the constant A (assume A > ) given in Proposition 11 depending on the sampling and the separation constants of Λ and n. For M > A and k > 0 such that B(0, M k ) ⊂ B(0, 1 4 ) one can find N disjoint balls B(x j , A k ) for j = 1, . . . N included in B(0, M/k) and such that Observe that each ball B(x j , A k ) contains by Proposition 11 at least one point from Λ k and therefore # (Λ k ∩ B(0, M/k)) ≥ N M A n .
We will use the following result from [8].
Theorem 13. Let B ⊂ R n , n > 1, be the unit ball. There do not exist infinite subsets Λ ⊂ R n such that the exponentials e i x,λ , λ ∈ Λ, are pairwise orthogonal in L 2 (B). Or, equivalently, there do not exist infinite subsets Λ ⊂ R n such that |λ − λ | is a zero of J n/2 , the Bessel function of order n/2, for all distinct λ, λ ∈ Λ.
Following ideas from [7] we can prove now our main result about orthogonal basis. A similar argument can be used on the sphere to study tight spherical designs.
Observe that this last result would be a direct consequence of the necessary density condition for complete interpolating sets if we could take balls of radius r/n for a fixed r > 0 in the condition. Finally, we obtain an infinite set X such that for λ = λ ∈ X J * n/2 (|λ − λ |) = 0, in contradiction with Theorem 13.
Remark. Note that the fact that the interpolating sequence {Λ k } is complete was used only to guarantee that # (Λ 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.