The algebra of vectors convolution has nothing to do with Fourier,
to the contrary of convolution of two kernels, Fourier stuff.

Fourier Convol Kernel Algebra.sm (48.54 KiB) downloaded 751 time(s).

... last example for convenience.

Fourier Convol Kernel Algebra.sm (86.59 KiB) downloaded 769 time(s).