On Spherical $T$-Designs in $\mathbb{R}^2$

TL;DR

This paper studies spherical $T$-designs on the unit circle $S^1$ through the harmonic strength $\mathrm{Hst}(X)$, linking design conditions to Fourier-type moments $P_k(X)=\sum_{\xi\in X}\xi^{k}$. It shows that for every $t\ge1$, there exist uncountably many $5$-point designs with $\mathrm{Hst}(X)=\{t\}$, and that for any finite $T\subset\mathbb{N}$ there exists a design with $\mathrm{Hst}(X)=T$; in particular, $N(\{t\})=5$. The construction relies on a 5-point parametrization $X(x)$ on $S^1$ and the observation that $P_k(X(x))=f_k(x)$ for explicit analytic functions, enabling $P_k(X^{1/t}(x))=0$ exactly for $k=t$. A multiplicative product construction $X_1\cdot X_2$ preserves harmonic strengths, yielding an upper bound $N(T)\le 5^{|T|}$, and the paper ends with open questions about exact minimal sizes and potential optimizations.

Abstract

In this paper, we study spherical $T$-designs and their harmonic strength $\text{Hst}(X)$ on the unit circle $S^1$. For any finite set $T\subset\mathbb{N}$, we constructively demonstrate the existence of a finite design $X$ such that $\text{Hst}(X)=T$.

Theorems & Definitions (11)

• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• proof : Proof of Theorem A
• Remark 2.4
• Lemma 2.5
• proof