The Inverse Fourier Transform

Abstract

The Fourier transform allows us to pass from the time domain to the frequency domain. It is remarkable that the inverse operation is obtained very simply from ℱ itself. In fact, it is just . However, one must be cautious, for as we have seen in the last lesson, f being integrable does not imply that $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}$ is integrable (Section 17.1.2). We will need additional hypotheses on f to invert f ↦ $\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{f}$