Equidistribution of Fekete Points on the Sphere

Abstract

ABSTRACT. 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.

Low cardinality admissible meshes on quadrangles, triangles and disks

“…We recall that in the case of the sphere, existence of admissible meshes (norming sets) with O(n 2 ) cardinality was proved in [12] and also [14], using other methods.…”

“…[12,14]). Such meshes are nearly optimal for least-squares approximation, and contain interpolation sets (discrete extremal sets) that distribute asymptotically as Fekete points of the domain and can be computed by basic numerical linear algebra [3,4,6,16].…”

Beurling–Landauʼs density on compact manifolds

“…A natural problem is to find the limiting distribution of points as L → ∞. In [13], J. Marzo and J. Ortega-Cerdà proved that as L → ∞, the number of Fekete points of degree L for S m in a spherical cap B(z, R) gets closer to k Lσ (B(z, R)), whereσ is the normalized Lebesgue measure on S m . They emphasize the connection of the Fekete points with the M-Z and interpolating arrays.…”

Marcinkiewicz–Zygmund inequalities and interpolation by spherical polynomials with respect to doubling weights

“…In order to use the density results proved in [11] and [12], we prove in Section 4 that by enlarging or diminishing slightly the number of points we can get an L q,w -MZ or L q,w -interpolating families starting from any other an L p -MZ or L p -interpolating family and vice versa. This turns out to be crucial because by it we can use the proved density results.…”

“…The precise formulation of the sparsity requirement is expressed in terms of the following Beurling type densities (see [18,11] with a slight different notation, and [12]): Definition 1.6. For a triangular family Z in S d we define the upper and lower densities, respectively, as…”