## Abstract

In this paper we make a comparison between certain probabilistic and deterministic point sets and show that some deterministic constructions (especially spherical t-designs) are better or as good as probabilistic ones like the jittered sampling model. We find asymptotic equalities for the discrete Riesz s-energy of sequences of well separated t-designs on the unit sphere S
^{d}⊂R
^{d+1}, d≥2. The case d=2 was studied in Hesse (2009) and Hesse and Leopardi (2008). In Bondarenko et al., (2015) it was established that for d≥2, there exists a constant c
_{d}, such that for every N>c
_{d}t
^{d} there exists a well-separated spherical t-design on S
^{d} with N points. This paper gives results, based on recent developments that there exists a sequence of well separated spherical t-designs such that t and N are related by N≍t
^{d}.

Original language | English |
---|---|

Pages (from-to) | 128-143 |

Number of pages | 16 |

Journal | Journal of approximation theory |

Volume | 239 |

DOIs | |

Publication status | Published - 2019 |

## Keywords

- Discrete energy
- Energy integral
- Equal-area partition
- Equal-weight numerical integration
- Sphere
- t-design
- The s-energy
- Well-separated point sets

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Mathematics(all)
- Numerical Analysis