On the existence and estimates of nested spherical designs

Ruigang Zheng; Xiaosheng Zhuang

doi:10.1016/j.acha.2024.101672

On the existence and estimates of nested spherical designs

Ruigang Zheng
, Xiaosheng Zhuang^*

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › Letter › peer-review

2 Citations (Scopus)

Abstract

In this paper, we prove the existence of a spherical t-design formed by adding extra points to an arbitrarily given point set on the sphere and, subsequently, deduce the existence of nested spherical designs. Estimates on the number of required points are also given. For the case that the given point set is a spherical t₁-design such that t₁< t and the number of points is of optimal order t₁^d, we show that the upper bound of the total number of extra points and given points for forming nested spherical t-design is of order t^2d+1. A brief discussion concerning the optimal order in nested spherical designs is also given. © 2024 Elsevier Inc.

Original language	English
Article number	101672
Journal	Applied and Computational Harmonic Analysis
Volume	73
Online published	4 Jun 2024
DOIs	https://doi.org/10.1016/j.acha.2024.101672
Publication status	Published - Nov 2024

Funding

We would like to thank anonymous reviewers for their valuable comments and suggestions that greatly helped us to improve the presentation of this paper. This work was supported in part by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Project CityU 11309122 and Project CityU 11302023.

Research Keywords

Nested structure
Optimal order
Spherical t-design

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RGC-funded

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AU - Zheng, Ruigang

AU - Zhuang, Xiaosheng

PY - 2024/11

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N2 - In this paper, we prove the existence of a spherical t-design formed by adding extra points to an arbitrarily given point set on the sphere and, subsequently, deduce the existence of nested spherical designs. Estimates on the number of required points are also given. For the case that the given point set is a spherical t1-design such that t1 < t and the number of points is of optimal order t1d, we show that the upper bound of the total number of extra points and given points for forming nested spherical t-design is of order t2d+1. A brief discussion concerning the optimal order in nested spherical designs is also given. © 2024 Elsevier Inc.

AB - In this paper, we prove the existence of a spherical t-design formed by adding extra points to an arbitrarily given point set on the sphere and, subsequently, deduce the existence of nested spherical designs. Estimates on the number of required points are also given. For the case that the given point set is a spherical t1-design such that t1 < t and the number of points is of optimal order t1d, we show that the upper bound of the total number of extra points and given points for forming nested spherical t-design is of order t2d+1. A brief discussion concerning the optimal order in nested spherical designs is also given. © 2024 Elsevier Inc.

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KW - Optimal order

KW - Spherical t-design

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DO - 10.1016/j.acha.2024.101672

M3 - Letter

AN - SCOPUS:85195588778

SN - 1063-5203

VL - 73

JO - Applied and Computational Harmonic Analysis

JF - Applied and Computational Harmonic Analysis

M1 - 101672

ER -

On the existence and estimates of nested spherical designs

Abstract

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GRF: Framelets for Geometric Deep Learning: Spheres, Graphs, and Neural Networks

GRF: Directional Framelets on Compact Sets: Theory, Construction, Realization, and Applications

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On the existence and estimates of nested spherical designs

Abstract

Funding

Research Keywords

RGC Funding Information

Access Output

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Projects

GRF: Framelets for Geometric Deep Learning: Spheres, Graphs, and Neural Networks

GRF: Directional Framelets on Compact Sets: Theory, Construction, Realization, and Applications

Cite this