Prove that if $f \in C^r(T)$, then $\hat{f}(n) = o(\frac{1}{|n|^r} )$ as $n \rightarrow ±∞$

Question

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Answer 1

$$\begin{align} \widehat f(n)&=\int_{\mathbb{T}}f(x)\,e^{-inx}\,dx\\ &=\int_{\mathbb{T}}f\Bigl(x+\frac{\pi}{n}\Bigr)\,e^{-in\bigl(x+\tfrac{\pi}{n}\bigr)}\,dx\\ &=-\int_{\mathbb{T}}f\Bigl(x+\frac{\pi}{n}\Bigr)\,e^{-inx}\,dx. \end{align}$$ Averaging the two expressions for $\widehat f(n)$ we get $$ |\widehat f(n)|\le\frac12\int_{\mathbb{T}}\Bigl|f(x)-f\Bigl(x+\frac{\pi}{n}\Bigr)\Bigr|\,dx\le\frac{C\,\pi^r}{2\,n^r}\,2\,\pi. $$