Suppose I have some function with \begin{align} \tilde{F}[t] = \int dx F(x) G(x,t) \end{align} I want to know what is the condition for the inverse transformation of $F$, exists, i.e., \begin{align} F[x]= \int dt \tilde{F}[t] K[x,t] \end{align} and if exists how to find $K[x,t]$ with $G(x,t)$.
There are lots of transformations such as Fourier, Laplace, Mellin, and in each case, the kernel $G(x,t)$ and $K[x,t]$ is somewhat well-defined and explicitly known.
What are the conditions for $F, G, \tilde{F}, K$ for which the above relation(?) [I mean transformation and its inverse transformation works] holds?