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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}) = \ …
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