Knowing that the convolution operation (*), when defined, is commutative, associative and distributive with respect to addition, what happens with multiplication (.) in both time and frequency (Fourier transform of time domain) domains? Example:

y(t) = [a(t).b(t)]*h(t)

or

Y(jw) = [A(jw)*B(jw)].H(jw)

Is it possible to perform any operation to a(t), b(t) and/or h(t), so that:

y(t) = [a'(t)*h'(t)].[b'(t)*h'(t)]

or

Y(jw) = [A'(jw).H'(jw)]*[B'(jw).H'(jw)]

in summary:

I want that,
[a(t).b(t)]*h(t) = [a'(t)*h'(t)].[b'(t)*h'(t)]

and/or

[A(jw)*B(jw)].H(jw) = [A'(jw).H'(jw)]*[B'(jw).H'(jw)]

if possible, of course.

The x' should represent some sort of modification of the original function and not a differentiation.

Thanks, Pires