I'm not a mathematician, so I may abuse some notation here. Please comment for any clarification.
Let's define a general linear transform as
$$\int_XK(\mathbf{\omega},x)f(x)dx$$
where $X$ is some region, and both $x$ and $\omega$ can be vectors. In some usual cases, such as those of Laplace and Fourier transforms, $x \in \mathbb{R}$, $\omega \in \mathbb{R} \text{ or } \mathbb{C}$, and $X = (-\infty, \infty)$. In case of a wavelet transform, $\omega$ is represented by two numbers, namely scale and shift.
My question is, given the transfomration kernel $K(\omega, x)$, and given that the transformation is invertible, is there a standard way of finding its inverse kernel $\bar{K}(\omega,x)$ such that
$$\int_\Omega \bar{K}(\omega, x)\left(\int_XK(\mathbf{\omega},y)f(y)dy\right)d\omega = f(x)$$ with some reasonable constraints, e.g. $f(x)$ behaves well in some sense, or perhaps the equivalence holds almost everywhere?
I understand that this question may be a bit generic, but perhaps there are some special cases for which this is easy? For example, if $K(\omega, x)$ is a unitary transformation, then finding the inverse transform is nearly trivial. Are there other such cases? At the very least, where should I look?