I can't figure out how to derive equation A.4 from this paper,
Simon, Barry. "Some Jacobi matrices with decaying potential and dense point spectrum." Communications in Mathematical Physics 87, no. 2 (1982): 253-258.
Let $r(x)$ be a function in $L^1(\mathbb R)\cap L^\infty(\mathbb R)$. We define the operator $K(r)$ on $L^2(\mathbb R)$ as
$$(K(r)f)(x)=\int r(x-y)f(y)dy,$$ where $f$ is a function in $L^2(\mathbb R)$.
Let $\mathcal F$ be the conventional Fourier transform (not defined explicitly in the paper).
How do we show that for any $g\in L^2(\mathbb R)$,
$$(\mathcal F K(r)\mathcal F^{-1}g)(k)=g(k)\int r(x)e^{-ikx}dx $$ ?
Let us assume \begin{align} \mathcal{F}(f) = \frac{1}{2\pi}\int^\infty_{-\infty} e^{-ikx}f(x)\ dx. \end{align}
Then by direct calculation, we see that \begin{align} \mathcal{F}K(r)\mathcal{F}^{-1}[g](k) =&\ \frac{1}{2\pi}\int^\infty_{-\infty} \int^\infty_{-\infty} \int^\infty_{-\infty} r(x-y)e^{-i(kx-\xi y)}g(\xi)\ d\xi dy dx\\ =&\ \frac{1}{2\pi}\int^\infty_{-\infty} \int^\infty_{-\infty} \int^\infty_{-\infty} r(x') e^{-i(kx'+(k-\xi) y)} g(\xi)\ d\xi dy dx' \\ =&\ \left(\frac{1}{2\pi} \int^\infty_{-\infty}\int^\infty_{-\infty} e^{-i(k-\xi)y}g(\xi) d\xi dy\right) \left(\int^\infty_{-\infty} r(x') e^{-ikx'}\ dx'\right)\\ =&\ \left(\int^\infty_{-\infty} \delta(k-\xi) g(\xi) d\xi\right)\left(\int^\infty_{-\infty} r(x') e^{-ikx'}\ dx'\right)\\ =&\ g(k)\left(\int^\infty_{-\infty} r(x') e^{-ikx'}\ dx'\right). \end{align}