This question is motivated by Fourier and Laplace transforms. Recall that both transforms are intuitively representation of a function as a combination of $g_k(t) = e^{kt}$, i.e., a function $f(t)$ can be written as $\int F(k) g_k(t) dk$, where the domain of $k$ is anything you want it to be.
Now I want to pick different functions for expansion, say $h_k(t) = \cos(ke^{-t})$ for example, and I try to express $f$ on some domain by $f(t) = \int \hat F(k)h_k(t) dk$. What can I say about $\hat F(k)$? What might be conditions to impose on $f$ and/or $h_k$ that make calculation of $\hat F(k)$ possible? For example, when is it true that $\hat F(k) = \int \frac{f(t)}{h_k(t)}w(t) dt$ for some $w(t)$? (This is "heuristically" true for both Fourier and Laplace transforms, so I definitely accept heuristic suggestions.)
The reason I want to do this is because I expect, for this particular example, that representing $f$ by $\hat F$ would simplify the calculation involving this linear operator: $M(f)(t) = e^{2t}(f''(t) + f'(t))$. This is because $h_k$ is an eigenvector of $M$. I believe the motivation for Fourier and Laplace transforms is similar as $e^{kt}$ is an eigenvector of the first derivative operator.