I have this function $f(at)$, and I want to Fourier-tranform it. I proceed in the following way, for $\quad\alpha<0 \Longrightarrow a=-|a|$:

\begin{align} \ \mathcal{F}_{t \rightarrow \xi}[f(at)]= \hat{f}(at)&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+ \infty} f(at) e^{-i t \xi} dt \\ &= \text{...for at=u}\\ &= \frac{1}{|a|}\frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+ \infty} f(u) e^{-i u \xi/|a|} du \qquad \star\\ \hat{f}(at)&= \frac{1}{|a|}\hat{f}(-\frac{\xi}{|a|}) \end{align}

Now, the steps above seem clear and correct and can be found in any book (where $\hat{f}(-\xi/|a| = \hat{f}(\xi/a))$). Nonetheless, I can't quite convince myself of the last passage. I would like to apply an Inverse Fourier transform to the last line, in order to retrieve the second last. In other words I would like to get to $\star$ by:

\begin{align} \mathcal{F}_{\xi \rightarrow t}[\hat{f}(at)]&= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+ \infty} \hat{f}(at) e^{i t \xi} dt \\ &= \frac{1}{\sqrt{2 \pi}} \int_{-\infty}^{+ \infty} \frac{1}{|a|}\hat{f}(-\frac{\xi}{|a|}) e^{i t \xi} dt \\ \end{align}

I feel like there is something I can't get, anyone can help?


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