Geodesic distance Riesz energy on the sphere

Abstract

We study energy integrals and discrete energies on the sphere, in particular, analogs of the Riesz energy with the geodesic distance in place of Euclidean, and observe that the range of exponents for which the uniform distribution optimizes such energies is different from the classical case. We also obtain a general form of the Stolarsky principle, which relates discrete energies to certain L 2 discrepancies. This leads to new proofs of discrepancy estimates, as well as the sharp asymptotics of the difference between optimal discrete and continuous energies in the geodesic case.

Quadrature Points via Heat Kernel Repulsion

“…A now classical result of Montgomery [37,38] implies that a set of N points cannot be orthogonal to more than the first ∼ c d N trigonometric functions. Bilyk & Dai [4] have established the analogous result on S d−1 with trigonometric functions replaced by spherical harmonics (also in the context of irregularities of distribution). We refer to [50] for a recent refinement of Montgomery's result and to recent work of Bilyk, Dai and the third author [5] for general refinements.…”

“…The d−dimensional torus T d vastly exceeds any other manifold in simplicity: in terms of numerical integration, many sets of points have been proposed and studied. We will compare our approach to both deterministic (1)(2)(3)(4) and stochastic (5)(6)…”

On the Fejes Tóth problem about the sum of angles between lines

“…F (x · y) = x − y ). Some generalizations of this principle were obtained in [2,3]. Moreover, in [3] a version of this principle has been used to characterize the extremizers of the energy integral and discrete energy with the potential F (t) = arccos t (i.e.…”

Generalized designs on graphs: Sampling, spectra, symmetries

“…It is impossible to summarize the field, we mention the seminal papers by Bondarenko, Radchenko, and Viazovska and Yudin and refer to a recent survey of Brauchart and Grabner . A lemma of Montgomery (see also ) may be understood as the study of the analogue of spherical designs on , where polynomials are replaced by trigonometric polynomials—this result does not seem to be very well known in this community since the relevant statement appears as a lemma and is used for a very different purpose. To the best of our knowledge, the first upper bound for weighted spherical designs on Riemannian manifolds is due to the author ; the eigenfunctions of the Laplacian on are polynomials, the classical setting is therefore included as a special case.…”