I've got a non-negative function $f(x)=a+b\cos(\omega_1 x)$ for $a,b,\omega_1 \in \mathbb R$ and I need to represent $b^2$ as a square of the Fourier-transform of a whatever function.
As a physicist I do my best first without (very much) checking if the transformations are equivalent:
\begin{align}
b&={\mathscr{F}\{f(x)\}(\omega)}\vert_{\omega=\omega_1}\tag{1}\\\\
&=\int_{-\infty}^{+\infty}\mathscr{F}\{f(x)\}\delta(\omega-\omega_1)d\omega\tag{2}\\\\
&=\frac 1{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\mathscr{F}\{f(x)\}\mathscr{F}\{e^{-i\omega_1 x}\}d\omega\tag{3}\\\\
&=\frac 1{2\pi}\int_{-\infty}^{+\infty}\mathscr{F}\{f(x)*e^{-i\omega_1 x}\}d\omega\tag{4}\\\\
&=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(-i \omega_{1}\left(x-x^{\prime}\right)\right) d x^{\prime}\right) \exp (i \omega x) d x\right) d \omega\tag{5}\\\\
&=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) \exp \left(-i \omega_{1} x\right) d x^{\prime}\right) \exp (i \omega x) d x\right) d \omega\tag{6}\\\\
&=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}\right) \exp \left(-i \omega_{1} x\right) \exp (i \omega x) d x\right) d \omega\tag{7}\\\\
&=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}\right) \exp \left(i (\omega-\omega_{1}) x\right) d x\right) d \omega\tag{8}\\\\
&=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left(\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}\right)\left(\int_{-\infty}^{+\infty} \exp \left(i\left(\omega-\omega_{1}\right) x\right) d x\right)\right) d \omega\tag{9}\\\\
&=\frac{1}{2 \pi}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}\right) \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} \exp \left(i\left(\omega-\omega_{1}\right) x\right) d x\right) d \omega\tag{10}\\\\
&=\frac{1}{2 \pi}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}\right) \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} \exp \left(i\left(\omega-\omega_{1}\right) x\right) d \omega\right) d x\tag{11}\\\\
&=\frac{1}{2 \pi}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}\right) \iint_{-\infty}^{+\infty} \exp (i \omega x) \exp \left(-i \omega_{1} x\right) d \omega d x\tag{12}\\\\
&=\frac{1}{2 \pi}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}\right) \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} e x p\left(-i \omega_{1} x\right) d \omega\right) \exp (i \omega x) d x\tag{13}\\\\
&=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}\right) \exp \left(-i \omega_{1} x\right) d \omega\right) \exp (i \omega x) d x\tag{14}\\\\
&=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left(\int_{-\infty}^{+\infty} {\left[f(x) * \exp \left(-i \omega_{1} x\right)\right](x)} d \omega\right) \exp (i \omega x) d x\tag{15}\\\\
&=\frac{1}{2 \pi} \int_{-\infty}^{+\infty}\left[f(x) * \exp \left(-i \omega_{1} x\right)\right](x)\left(\int_{-\infty}^{+\infty} d \omega\right) \exp (i \omega x) d x\tag{16}\\\\
&=\frac{1}{2 \pi} \mathcal{F}\left\{\left[f(x) * \exp \left(-i \omega_{1} x\right)\right](x)\right\}(\omega) \cdot \int_{-\infty}^{+\infty} d \omega\tag{17}\\\\
\end{align}
Where the last factor is the overall volume of the reciprocal to $x$ space. This quantity has a prefactor of $2\pi$ and units of those inverse to $\int_{-\infty}^{+\infty}dx$.
From (6) to (7) was used, that $\exp \left(-i \omega_{1} x\right)$ is not a function of $x'$; from (8) to (9) - that $\int_{-\infty}^{+\infty} f\left(x^{\prime}\right) \exp \left(i \omega_{1} x^{\prime}\right) d x^{\prime}$ doesn't depend on $x, x'$ and $\omega$; throughout (10)-(12) - the Fubini's (or Fubini-Tonelli's?) theorem is applied; from (12) to (13) the action is performed as, for example, here in the proof of the multiplication theorem.
My questions are
- Can either of Fubini's or Fubini-Tonelli's theorems be applied where they are applied in this derivation?
- Are all the transformations legit (and in which sense)?
My fear is, that either the integrability of $\exp \left(i\left(\omega-\omega_{1}\right) x\right)$ is not met here, or, possibly, other requirements. I'd appreciate any help.