Suppose that we have the following integral equation $$ \int_0^\infty F(q) \cos q z \, \mathrm{d} q = \int_0^\infty G(q) \cos q z \, \mathrm{d} q \, , $$ wherein $F(q)$ and $G(q)$ are arbitrary functions of the wavenumber $q$ which they are not necessary even functions. I was wondering whether $F(q) = G(q)$ holds in this case. I think that such equality would be guaranteed if $F$ and $G$ are even functions in virtue of Fourier transforms. Any help would be highly appreciated. Thank you.
a