Convolution of fraction function

Question

I know that convolution is defined: $$f*g=\int f(x-y)\cdot g(y) \, dy $$

How to develop below functions to convolution equation

$$\int {f(x-y) \over g(y)} \, dy =\text{ ???}$$

and

$$\int {f(x-y) \over g^2(y)} \, dy=\text{ ???} $$

Thank you so much

Answer 1

Assume $h(x)=\frac{1}{g(x)}$ then: $$f*\frac{1}{g}=f*h=\int {f(x-y)h(y)} \, dy \int {f(x-y)\frac{1}{g(y)}} \, dy$$ Second assume $h(x)=\frac{1}{g^2(x)}$ then: $$f*\frac{1}{g^2}=f*h=\int {f(x-y)h(y)} \, dy \int {f(x-y)\frac{1}{g^2(y)}} \, dy$$