I want to calculate the expected value of the deconvolution kernel density estimator. Delaigle and Hall did this in their paper "Kernel methods and minimum contrast estimators for empirical deconvolution", but there is one step which I don't unterstand:
$ \dfrac{1}{2 \pi} \int e^{-itx} \phi_K(ht) \phi_X(t) \, dt = \dfrac{1}{h} \int K(u/h) f_X(x - u) \, du $
with $\phi_K (z)$ characteristic function of a Kernel $K$ and $\phi_X(z)$ characteristic function of a random variable $X$. How do I get from the left statement to the right statement? It seems like there was used substitution.
Let us consider the Fourier transform under characteristic function convention $E[e^{iaX}]$. We have that $$\begin{aligned}\mathcal{F}(f_X(a-x))(t)&=\int_\mathbb{R}f_X(a-x)e^{ixt}dx=\\&=\int_\mathbb{R}f_X(y)e^{i(a-y)t}dy=\\ &=e^{iat}\int_\mathbb{R}f_X(y)e^{-iyt}dy=\\ &=e^{iat}\overline{\varphi_X}(t) \end{aligned}$$ and for $h>0$ $$\begin{aligned}\mathcal{F}(f_K(x/h))(t)&=\int_\mathbb{R}f_K(x/h)e^{ixt}dx=\\&=\int_\mathbb{R}f_K(y)e^{iy(ht)}(hdy)=\\ &=h\varphi_K(ht) \end{aligned}$$ and finally by Plancherel/Parseval $$\begin{aligned}\frac{1}{2\pi}\int_\mathbb{R}\overline{\mathcal{F}(f_X(a-x))(t)}\mathcal{F}(f_K(x/h))(t)dt&=\frac{h}{2\pi}\int_\mathbb{R}e^{-iat}\varphi_X(t)\varphi_K(ht)dt=\\ &=\int_\mathbb{R}f_X(a-x)f_K(x/h)dx\end{aligned}$$ where $\overline{z}$ is the complex conjugate of $z \in \mathbb{C}$.