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

TL;DR

This paper investigates spherical T-designs on the unit circle, establishing the existence of finite designs with specified harmonic strength for any finite subset of natural numbers, advancing the understanding of design construction.

Contribution

It provides a constructive proof for the existence of finite spherical T-designs with arbitrary harmonic strength sets on the unit circle.

Findings

Existence of finite T-designs with prescribed harmonic strength.

Construction method for spherical T-designs on the circle.

Extension of design theory to harmonic strength specifications.

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 $\text{Hst}(X)=T$.