Well-Separated Spherical Designs

  title={Well-Separated Spherical Designs},
  author={Andriy V. Bondarenko and Danylo V. Radchenko and Maryna S. Viazovska},
  journal={Constructive Approximation},
For each $$N\ge C_dt^d$$N≥Cdtd, we prove the existence of a well-separated spherical $$t$$t-design in the sphere $$S^d$$Sd consisting of $$N$$N points, where $$C_d$$Cd is a constant depending only on $$d$$d. 

Estimates for Logarithmic and Riesz Energies of Spherical t-Designs

  • T. Stepanyuk
  • Mathematics
    Springer Proceedings in Mathematics & Statistics
  • 2020
In this paper we find asymptotic equalities for the discrete logarithmic energy of sequences of well separated spherical $t$-designs on the unit sphere ${\mathbb{S}^{d}\subset\mathbb{R}^{d+1}}$,

Comparison of probabilistic and deterministic point sets

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (spherical $t$-designs) are better or as good as

Relation between spherical designs through a Hopf map

Cohn--Conway--Elkies--Kumar [Experiment. Math. (2007)] described that one can construct a family of designs on $S^{2n-1}$ from a design on $\mathbb{CP}^{n-1}$. In this paper, we prove their claim for

Spectral Limitations of Quadrature Rules and Generalized Spherical Designs

We study manifolds $M$ equipped with a quadrature rule $$\begin{equation} \int_{M}{\phi(x)\,\mathrm{d}x} \simeq \sum_{i=1}^{n}{a_i \phi(x_i)}.\end{equation*}$$We show that $n$-point quadrature

A Comparison of Popular Point Configurations on $\mathbb{S}^2$

There are many ways to generate a set of nodes on the sphere for use in a variety of problems in numerical analysis. We present a survey of quickly generated point sets on $\mathbb{S}^2$, examine

Distributing many points on spheres: Minimal energy and designs




Spherical Designs via Brouwer Fixed Point Theorem

It is shown that c_{d}t is a constant depending only on d, and the existence of a spherical design on S^{d} consisting of N points is proved.

Optimal asymptotic bounds for spherical designs

In this paper we prove the conjecture of Korevaar and Meyers: for each $N\ge c_dt^d$ there exists a spherical $t$-design in the sphere $S^d$ consisting of $N$ points, where $c_d$ is a constant

Well Conditioned Spherical Designs for Integration and Interpolation on the Two-Sphere

This paper shows how to construct well conditioned spherical designs with $N\geq(t+1)^2$ points by maximizing the determinant of a matrix while satisfying a system of nonlinear constraints.

Existence of Solutions to Systems of Underdetermined Equations and Spherical Designs

It is shown that the construction of spherical designs is equivalent to solution of underdetermined equations and a new verification method for underd determined equations is derived using Brouwer’s fixed point theorem.

Construction of spherical t-designs

Spherical t-designs are Chebyshev-type averaging sets on the d-dimensional unit sphere Sd−1, that are exact for polynomials of degree at most t. The concept of such designs was introduced by

The s-energy of spherical designs on S2

This paper investigates the s-energy of (finite and infinite) well separated sequences of spherical designs on the unit sphere S2. A spherical n-design is a point set on S2 that gives rise to an

The Coulomb energy of spherical designs on S2

If the sequence of well separated spherical designs is such that m and n are related by m = O(n2), then the Coulomb energy of each m-point spherical n-design has an upper bound with the same first term and a second term of the same order as the bounds for the minimum energy of point sets on S2.

Asymptotics for minimal discrete energy on the sphere

We investigate the energy of arrangements of N points on the surface of the unit sphere Sd in Rd+1 that interact through a power law potential V = 1/rs, where s > 0 and r is Euclidean distance. With

Spherical codes and designs

On averaging sets

AbstractFor any set Φ={f1,f2,...,fs} ofC3-functions on the interval [−1, 1], and for any weight functionw(x) satisfyingL1≥w(x)≥L2(1−|x|)β(L1,L2>0, β≥0) and $$\int_{ - 1}^1 {w(x)dx = 1} $$ , we give a