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

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