Does there exist a Laplace series for the Laplace transform?

Question

I am trying to get a better understanding of what the Laplace transform is. When I learnt about the Fourier transform I found it useful to learn about the Fourier series first. Does there exist some kind of Laplace series which is to the Laplace transform what the Fourier series is to the Fourier transform?

Answer 1

By introducing the dirac comb,

$\sum\limits_{t=0}^\infty f(t)e^{-st}=\int_0^\infty\operatorname{\text{Ш}}_1(t)f(t)e^{-st}~dt$