The professor has completely abandoned mathematical accuracy for the sake of a pun! In reality, a Fourier transform is a sophisticated mathematical technique that converts signals between time and frequency domains—not whatever this numerical "four"-play is.
The joke shows a tenured professor who's clearly given up on teaching actual math and instead created an elaborate, nonsensical definition based on the number 4 appearing in "Fourier." The chalkboard even "proves" his point with 4444 in base 5 being the "fouriest" number.
This is what happens when mathematicians stop caring about academic rigor and start caring about dad jokes. Tenure really is mathematical freedom!