= Fourier-cutoff recovery of expectations from a characteristic function
Let $f$ be compactly supported and continuous, let $g,\widehat g\in L^1(\mathbb R)$, and use $\widehat g(t)=\int g(x)e^{itx}\,dx$. If $g(0)\ne0$, then
$$
\mathbb E[f(Z)]
=\lim_{\varepsilon\downarrow0}
\frac1{2\pi g(0)}
\int_{\mathbb R}g(\varepsilon s)\widehat f(-s)\varphi_Z(s)\,ds.
$$
This follows by <Fubini's theorem>, a change of variables, <Fourier inversion theorem>, and <dominated convergence theorem>. It gives a direct proof of the <uniqueness theorem for characteristic functions> because the right side depends only on $\varphi_Z$.
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