### 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}.

Language | English |
---|---|

Pages | 128-143 |

Number of pages | 16 |

Journal | Journal of approximation theory |

Volume | 239 |

DOIs | |

Status | Published - 2019 |

### Fingerprint

### 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

### Cite this

*Journal of approximation theory*,

*239*, 128-143. https://doi.org/10.1016/j.jat.2018.12.001

**Comparison of probabilistic and deterministic point sets on the sphere.** / Grabner, Peter; Stepaniuk, Tetiana.

Research output: Contribution to journal › Article › Research › peer-review

*Journal of approximation theory*, vol. 239, pp. 128-143. https://doi.org/10.1016/j.jat.2018.12.001

}

TY - JOUR

T1 - Comparison of probabilistic and deterministic point sets on the sphere

AU - Grabner, Peter

AU - Stepaniuk, Tetiana

PY - 2019

Y1 - 2019

N2 - 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 dt 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.

AB - 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 dt 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.

KW - Discrete energy

KW - Energy integral

KW - Equal-area partition

KW - Equal-weight numerical integration

KW - Sphere

KW - t-design

KW - The s-energy

KW - Well-separated point sets

UR - http://www.scopus.com/inward/record.url?scp=85059021559&partnerID=8YFLogxK

U2 - 10.1016/j.jat.2018.12.001

DO - 10.1016/j.jat.2018.12.001

M3 - Article

VL - 239

SP - 128

EP - 143

JO - Journal of approximation theory

T2 - Journal of approximation theory

JF - Journal of approximation theory

SN - 0021-9045

ER -