# Well-Separated Spherical Designs

@article{Bondarenko2013WellSeparatedSD, title={Well-Separated Spherical Designs}, author={Andriy V. Bondarenko and Danylo V. Radchenko and Maryna S. Viazovska}, journal={Constructive Approximation}, year={2013}, volume={41}, pages={93-112} }

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.

## 51 Citations

### Estimates for Logarithmic and Riesz Energies of Spherical t-Designs

- MathematicsSpringer 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

- Mathematics
- 2018

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

- Mathematics
- 2015

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

- MathematicsInternational Mathematics Research Notices
- 2019

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 $\mathbb{S}^2$

- Mathematics
- 2016

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…

### Numerical computation of triangular complex spherical designs with small mesh ratio

- MathematicsJ. Comput. Appl. Math.
- 2023

### Upper and lower estimates for numerical integration errors on spheres of arbitrary dimension

- MathematicsJ. Complex.
- 2019

## References

SHOWING 1-10 OF 22 REFERENCES

### Spherical Designs via Brouwer Fixed Point Theorem

- MathematicsSIAM J. Discret. Math.
- 2010

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

- Mathematics
- 2010

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

- MathematicsSIAM J. Numer. Anal.
- 2010

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

- MathematicsSIAM J. Numer. Anal.
- 2006

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

- Mathematics
- 1992

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

- MathematicsAdv. Comput. Math.
- 2009

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

- MathematicsAdv. Comput. Math.
- 2008

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

- Mathematics
- 1995

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…

### On averaging sets

- Mathematics
- 1991

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…