Let $r \in \mathbb{N}$, and let $\mathbb{T} := \mathbb{R}/2\pi\mathbb{Z}$ be equipped with its normalized Haar measure $m_{\mathbb{T}}$. For $t \in \mathbb{T}$, let $|t|$ denote the geodesic distance from $t$ to $0$. For $f \in C(\mathbb{T})$ and $h \in \mathbb{T}$, define the $r$-th forward difference
Assume that there exist a constant $A_r>0$ and, for each $n\in\mathbb{N}$, a finite signed complex Borel measure $\mu_n$ on $\mathbb{T}$ such that, for every $f\in C(\mathbb{T})$ and every $x\in\mathbb{T}$,