Is there an integral transform that simplifies functional composition?

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Laplace, Fourier and Mellin integral transforms simplify operations like multiplication by a variable, taking derivatives, shifts, dilatations. Are there integral transforms that, in some way, simplify functional composition (by substituting it with a "simpler" operation on images, like multiplication, or summation, or even convolution)? I am aware of existence this question, and don't find neither its formulation nor the answer satisfactory.