Extremal Systems of Points and Numerical Integration on the Sphere

Abstract

This paper considers extremal systems of points on the unit sphere S r ⊆ R r+1 , related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d n = dim P n points, where P n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S 2 of degrees up to 191 (36, 864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.

Equidistribution of Fekete Points on the Sphere

“…. , 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.…”

“…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 → ∞.…”

Support of the logarithmic equilibrium measure on sets of revolution in R3

“…Then T (t) = γ ′ (t) denotes the unit tangent vector, κ(t) = |T ′ (t)| denotes the curvature, and N(t) = T ′ (t)/|T ′ (t)| = γ ′′ (t)/κ(t) denotes the unit normal vector to the curve γ for t ∈ [a, b]. Substituting these expressions into (13) and (14) we obtain…”

“…For further extensions of these results, see [3]. Related results and applications appear in [5] (coding theory), [13] (cubature on the sphere), and [1] (finite normalized tight frames).…”