I wanted to understand why this text talks about applying the Fourier transform on H(f) to obtain h(t). I view Fourier transform as moving from the time or spatial domain to the frequency / spatial frequency domain. Is it normal for Fourier transform and inverse Fourier transform to be used interchangeably?
the first image pertains to my question.
The second image is for additional context which appear before the the paragraph of interest.
The Fourier transform doesn't move only from time to frequency (or space etc.), it moves between these domains. The only difference between the Fourier transform and its inverse is a sign in the exponent and possibly the scaling. But these are conventions, and you may have noticed that the Fourier transform can be defined differently in different fields.
With the definition of the Fourier transform and its inverse as it is common in signal processing
\begin{align*} X(\omega) &= \int_{-\infty}^{\infty}x(t)e^{-j\omega t}dt \\ x(t) &= \frac{1}{2\pi} \int_{-\infty}^{\infty}X(\omega)e^{j\omega t}d\omega \\ \end{align*}
we have
\begin{align*} \mathscr{F}\big\{X(\omega)\big\} &= \int_{-\infty}^{\infty}X(\omega)e^{-j\omega t}d\omega \\ &= 2\pi x(-t) \end{align*}
which shows that applying the Fourier transform twice gets us back to the time domain.
Also take a look at this related question.
