 Everywhere-defined Fourier multipliers on continuous periodic functions are bounded

Suppose every continuous periodic function $f$ has a unique continuous periodic $Tf$ with Fourier coefficients $\widehat{Tf}(n)={\alpha }_n\hat{f}(n)$. Uniqueness first makes $T$ linear. If $f_j\to f$ and $T{f}_j\to g$ uniformly, continuity of each coefficient functional gives $\hat{g}(n)={\alpha }_n\hat{f}(n)$ for every fixed $n$, hence $g=Tf$. Thus $T$ has a closed graph and the closed graph theorem makes it bounded. Applying it to normalized exponentials gives $|{\alpha }_n|\le \Vert T\Vert$.

For $e_n(t)=(2\pi {)}^{-1/2}{e}^{int}$, the bounded functional $\varphi (f)=\sqrt{2\pi }(Tf)(0)$ satisfies $\varphi (e_n)={\alpha }_n$. The normalization factor follows from $e_n(0)=(2\pi {)}^{-1/2}$.