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2
votes
How to get Fourier–Stieltjes transform on $\mathbb R$ from the nice function on $\mathbb T$ ...
I suspect that you meant
\begin{equation*}
\widehat{f}(n)
=
\frac{1}{2\pi}
\int_{-\pi}^{\pi}
f(e^{i\theta})
e^{-in\theta}
d\theta
\end{equation*}
in which case
\begin{equation*}
F(x)
=
f(e^{ix})
=
\ …