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I got the following paragraph from the book "A wavelet tour of signal processing" chapter one, page 2.

The Fourier transform is everywhere in physics and mathematics because it diagonalizes time-invariant convolution operators. It rules over linear time-invariant signal processing, the building blocks of which are frequency filtering operators.

How is it (illustrated) formulated mathematically?

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    $\begingroup$ this is just the statement that the Fourier transform $\hat{C}(\omega,\omega')$ of the integral operator $C(t,t')$ -- if it is only a function of $t-t'$ (time translation invariant), is of the form $\hat{C}(\omega,\omega')=f(\omega)\delta(\omega-\omega')$, so only nonvanishing on the "diagonal" where $\omega=\omega'$. $\endgroup$
    – Carlo Beenakker
    Commented Nov 6, 2020 at 9:38

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