For each $r\in\mathbb N$ there is a constant $C_r>0$ such that every $f\in C(\mathbb T)$ and every $n\ge 1$ satisfy
\begin{align*}
E_{n,\operatorname{trig}}(f)_\infty\le C_r\,\omega_r\left(f,\frac{1}{n}\right).
\end{align*}
Knowledge Status
Analysis
Discussion
A theorem in [approximation theory](/page/Approximation%20Theory) concerning Jackson Theorem For Periodic Functions, used to organise the analytic structure of approximation methods and convergence results.