Equidistribution of the Fekete points on the sphere

The Fekete points are the points that maximize a Vandermonde-type determinant that appears in the polynomial Lagrange interpolation formula. They are well suited points for interpolation formulas and numerical integration. We prove the asymptotic equidistribution of the Fekete points in the sphere. The way we proceed is by showing their connection with other array of points, the Marcinkiewicz-Zygmund arrays and the interpolating arrays, that have been studied recently.


INTRODUCTION
For any integer ℓ ≥ 0, let H ℓ be the space of spherical harmonics of degree ℓ in S d . For any integer L ≥ 0 we denote the space of spherical harmonics of degree not exceeding L by Π L These vector spaces have dimensions Let {Q L 1 , . . . , Q L π L } be any basis in Π L . The points Z L = {z L1 , . . . , z L,π L } maximizing the determinant |∆(x 1 , . . . , x π L )| = | det(Q L i (x j )) i,j | are called the Fekete points of degree L for S d (this points are sometimes called extremal fundamental systems of points as in [SW04]). They are not to be confused with the elliptic Fekete points which are a system of points that minimize the potential energy. The extremal fundamental system of points are better suited nodes for cubature formulas and for polynomial interpolation, see [SW04] and the references therein.
The geometric properties of the distribution of the Fekete points on the sphere has been the subject of research, see for instance [Rei90], [LBW08] or [SW04]. One natural problem is the limiting distribution of the points as L → ∞. If we denote by µ L = 1 π L j δ x j it has been known long ago that for the elliptic Fekete points µ L converges vaguely to the uniform distribution of the sphere even for a wide class of potentials, see [HS04] for a very nice survey. For the Fekete points in compacts K ⊂ C this is a classical result. Much less is known in higher dimensions. On the paper [BB08] the authors have found the limiting distribution in the context of line bundles over complex manifolds. The techniques there are very different from ours, they rely on a careful study of the weighted transfinite diameter and its differentiability. We will rather emphasize the connection of the Fekete points with the Marcinkiewicz-Zygmund arrays and the interpolating arrays (see below for the definitions). As long as the density of these arrys is understood we can obtain the equidistribution of the Fekete points. This is the case of the sphere where we can build on the work [Mar07], where the M-Z arrays and interpolating arrays are studied. The same approach is being pursued by R. Berman in line bundles over complex manifolds.
1.1. Marcinkiewicz-Zygmund inequalities and interpolation. We consider arrays of points on the sphere S d that determine the norm of the polynomials, and also arrays of points where we are free to interpolate arbitrary values by polynomials,i.e. interpolating arrays. More precisely, For any degree L we take m L points in S d (1) Then the L p −norm in S d of a polynomial of degree L is comparable to the discrete version given by the weighted ℓ p −norm of its restriction to Z(L). Definition 1.2. Let Z = {Z(L)} L≥0 be a triangular array with m L ≤ π L for all L. We say that Z is L p −interpolating, if for all arrays {c Lj } L≥0,1≤j≤m L of values such that there exists a sequence of polynomials Q L ∈ Π L uniformly bounded in L p such that Roughly speaking in order to recover the L p −norm of a polynomial of degree L from the evaluation at the points in Z(L) we need a sufficiently big number of points in Z(L). Thus, intuitively, the M-Z arrays must have high density. On the other hand, in an interpolating array it is possible to have a spherical harmonic of degree at most L attaining some prescribed values on Z(L). Intuitively this is possible only when Z(L) is sparse.
In dimension one the roots of unity are simultaneously an interpolating and an M-Z array when 1 < p < ∞. On higher dimension the situation is more delicate. It has been proved, [Mar07, Theorem 1.7] that there are not arrays which are simultaneously L p -MZ and interpolating when d > 2 and p = 2 and most likely even when p = 2. We will prove that the Fekete points are a very reasonable substitute. If we perturb them slightly they are interpolating sequences and a different perturbation makes them MZ-arrays. Thus, in a sense, they behave like the roots of unity in higher dimensions. Since the densities of the MZ-arrays and the interpolating arrays are well understood, see [Mar07, Theorem 1.6], then we will get some geometric information on the Fekete points.
In the next section we will provide the connection of Fekete points and interpolating and MZarrays. On the last section we will draw some geometric/metric consequences.
Acknowledgment This paper has its origins in a conversation of the second author with Robert Berman at the Mittag-Leffler Institute over the possibility of connecting Fekete points and sampling sequences. It is a pleasure to thank him for sharing his thoughts and to the Institute for the warm hospitality and great atmosphere.

FEKETE POINTS, MZ-ARRAYS, INTERPOLATING ARRAYS
Proof. Assume that Z is a collection of Fekete points. It satisfies a nice separation property that is convenient to prove the first inequality of (1.1).

Definition 2.2. A triangular array
Reimer, [Rei90] observed that the Fekete points are uniformly separated. More precisely, just by using Marcel Riesz result for trigonometric polynomials on great circles.Thus we know that and therefore the array Z ε is uniformly separated. This implies the following Plancherel-Polya type inequality for any 1 ≤ p < ∞ The right hand side inequality in (1.1) is more delicate, we need an appropriate representation formula for the polynomials in terms of the values at the points. The most naive approach is to start by the Lagrange interpolation formula. Let ℓ Li (z) = ∆(z L1 , . . . , z L,i−1 , z, z L,i+1 , . . . , z L,π L ) ∆(z L1 , . . . , z L,π L ) then sup z∈S d |ℓ Li (z)| ≤ 1 and the Lagrange interpolation operator defined in C(S d ) as We want better control of the norms. So we need a slightly bigger set of points and a weighted representation formula. Let p be a polynomial in one variable of degree [Lε] and such that p(1) = 1. Then given Q ∈ Π L one has for a fixed z ∈ S d R(w) = Q(w)p( z, w ) ∈ Π Lε and therefore we obtain our weighted representation formula: We define the operator Q L from C π Lε → Π L 2ε as We want to prove that with constants uniform in L which is the righthand sided of (1.1). We need to choose the weight p with care. We need a polynomial p that peaks at one point, has degree [εL] and decays fast far away from the picking point. For this purpose we will use powers of the Jacobi polynomials which are natural in this context because they are the reproducing kernels in Π L , see [Mar07]. The Jacobi polynomials P (α,β) L of degree L and index (α, β) are the orthogonal polynomials on [−1, 1] with respect to the weight function (1 − x) α (1 + x) β with α, β > −1. We take the normalization We can use the estimates in [Sze75, Section 7.34] to obtain, for any v ∈ S d We will use as auxiliary polynomial then p(1) ≃ 1 and by the estimate (3) |v j | and also for any fixed z ∈ S d Then the result follows by the Riesz-Thorin interpolation theorem.
The corresponding result for interpolation reads as follows:

Proof. Given an array of values {v
and R L (z L −ε j ) = v L −ε j . This time the map R L is from C π L −ε → Π L and the L p -estimates on the norm of R L follow exactly as the estimates of Q L in the previous Theorem.

GEOMETRIC PROPERTIES OF THE FEKETE POINTS
We will draw some geometric information on the Fekete points. For a given z ∈ S d and 0 < R < 1 we denote by B(z, R) the spherical cap B(z, R) = {w ∈ S d ; d(z, w) < R}. We will prove that as L → ∞ the number of Fekete points in B(z, R) gets closer to π Lσ (B(z, R)) wherẽ σ is the normalized Lebesgue measure on S d , i.e.σ = σ/σ(S d ). We need first information on the density of M-Z and interpolation arrays. Let Z(L) be the set of Fekete points of degree L. We know that for any ε > 0 the array Z ε = {Z ε (L)} L≥0 is L 2 -MZ, so if we unwind the definitions corresponding to the densities, we get that for any ε > 0, there is a big α = α(ε) such that for all L and z ∈ S d (4) Similarly since Z −ε is interpolating, whenever Z is a Fekete array, from the density condition we get that there is a big α = α(ε) such that for all L and z ∈ S d (5) 1 π L #(Z(L) ∩ B(z, α L )) σ(B(z, α L )) ≤ (1 + ε).

Vague convergence.
Let us see how the inequalities (4) and (5) imply that the normalized counting measure converges vaguely to the Lebesgue measure. Indeed, defining and (σ * χ B(N,α/L) )(z) = σ(B(z, α/L)) and finally for any ε > 0 there is a big α such that for any z and L ≥ 1.
We take an arbitrary spherical cap B(z, r). We want to check that µ L (B(z, r)) →σ(B(z, r)) as L → ∞. We fix an ε > 0 and we take the convolution of (6) with the function χ B(N,r) σ(B(N,α/L)) with a very big L and this proves for any z and L big. We take limits as L → ∞ and since Z is uniformly separated that means that µ L (B(z, r + α/L) \ B(z, r − α/L)) → 0 uniformly in z ∈ S d as L → ∞. Thus This is for an arbitrary spherical cap. This already implies the vague convergence of the measures, see [Blü90], i.e.
for any f ∈ C(S d ).
From the uniform densities condition on the Fekete points we may obtain other geometric consequences on the distribution of the Fekete points Z(L). We give one example. It is well known that using the bound given by by L. Fejes Tóth [FT49] for the maximum of the minimal spherical distance between any set of π L = (L + 1) 2 points on S 2 , there exist at least two points in Z(L) at distance d L with d L ≤ arccos cot 2 ω L − 1 2 , ω L = (L + 1) 2 (L + 1) 2 − 2 π 6 , but as L arccos cot 2 ω L − 1 2 ր κ = 3.80925 . . . , when L → ∞ one has min However the numerical results in [SW04] suggest that the right bound should be To bound the maximal number, N, of disjoints spherical caps in S 2 of radius η/L to be found in a larger spherical cap of radius α/L we use the following result due to J. Molnár , see [Mol52]. Then substituting in the density condition one gets η ≤ 4 π √ 12 = κ.
So, we get not only that there exists a couple of points in each generation Z(L) at distance smaller than κ/L but a more uniform estimate: for any ε > 0 there is an α such that: inf z Li ,z Lj ∈B(z,α/L) d(z Li , z Lj ) ≤ κ + ε L for any spherical cap B(z, α/L).
Up to now we have drawn information from the MZ-arrays and interpolating arrays to get new information on the Fekete points, but the reverse trend can also be useful. For instance, since any Fekete array has density one and a small perturbation makes it a MZ-array (or interpolation), then we obtain the following corollary Corollary 3.3. Given any ǫ > 0, there are arrays Z ε and Z −ε with densities D + (Z ε ) = D − (Z ε ) = 1 + ε, D + (Z −ε ) = D − (Z −ε ) = 1 − ε, such that Z ε is an L p -MZ array for any p ∈ [1, ∞] and Z −ε is an L p -interpolating array for any p ∈ [1, ∞].
Thus the necessary density conditions that MZ-arrays and interpolating arrays satisfy are sharp.