Can the following functional equation be solved for H(x) in terms of F(x)? $$ x \cdot H(x) = [F * H](x) $$ where $*$ denotes convolution.
Of ancillary interest, can anything be said about the solution to function equations which mix multiplication and convolution of the form: (for given F,G) $$ G(x) \cdot H(x) = [F * H](x) $$
The main equation arises from attempting to describe the fourier transform of the following $h(x)$ in terms of the transform of $f(x)$. $$h(x) = \exp \left ( i \int_{0}^{x} f(t) dt \right )$$