Can we see the Fourier transform as a filtering operation?
Hint:

The DFT matrix only contains powers of a complex root of unit. Every row or column holds increasing powers of a common power ($\omega^{mn}=(\omega^m)^n$); hence the real and imaginary parts follow (co)sinusoids of increasing frequencies.

https://www.dsprelated.com/freebooks/mdft/DFT_Sinusoids.html

Every component of the DFT of a signal is a convolution of the signal with those sinusoids.