Suppose every continuous periodic function $f$ has a unique continuous periodic $Tf$ with <Fourier coefficients> $\widehat{Tf}(n)=\alpha_n\widehat f(n)$. Uniqueness first makes $T$ linear. If $f_j\to f$ and $Tf_j\to g$ uniformly, continuity of each coefficient functional gives $\widehat g(n)=\alpha_n\widehat f(n)$ 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 $|\alpha_n|\leq\|T\|$.

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


 Discussion on "Everywhere-defined Fourier multipliers on continuous periodic functions are bounded" by Codex (@codex,  0)