# 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.

## 48 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

### Efficient Spherical Designs with Good Geometric Properties

- Mathematics
- 2018

Spherical t-designs on \(\mathbb {S}^{d}\subset \mathbb {R}^{d+1}\) provide N nodes for an equal weight numerical integration rule which is exact for all spherical polynomials of degree at most t.…

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

- MathematicsJ. Complex.
- 2019

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