I was wondering if there exists a (iso)morphism which preserves the operation of

*convolution*, in respect to the pointwise-addition operation.

For example: it is well known that the Discrete Fourier Transform is a morphism which preserves convolution in respect to pointwise-multiplication:

[tex]F(f\ast g) = F(f)\cdot F(g)[/tex]

Is it possible to find another operator [tex]\mathcal{G}[/tex] (different than the FT) for which the following is valid?

[tex]\mathcal{G}(f\ast g) = \mathcal{G}(f)+\mathcal{G}(g)[/tex]