For Fourier transform, it has been ingrained in my head that all we are doing is projecting a function onto its Fourier basis, namely $(1, cos(t), sin(t),...cos(nt), sin(nt) ...)$
Can anyone comment whether we can also think of the laplace transform and the z-transform as projections on their respective basis?
On some level this is more difficult because the kernel of the laplace transform is $e^{st}$ which is not intuitive as to what the basis would be.
More difficult is the z-transform, which has the kernel $z^{-k}$. In this case can we say that we are projecting our function onto a set of discrete basis $(1, z^{-1}, z^{-2}...z^{-n})$, where each $z$ is a complex number. It is not clear what these basis are.
Can someone clarify this issue? Thanks.