I understand the fact that the Laplace Transform is a 'generalization' of the Fourier Transform and I feel like I've wrapped my head around the Fourier Transform decently enough. I'm trying to translate my understanding to the Laplace Transform, though I'd just like to make sure I'm not walking into a rabbit hole (perhaps I'm already in one).
I understand that via Fourier decomposition, one can turn a function into a summation of complex sinusoids of various frequencies and amplitudes, where the Fourier coefficient of a particular sinusoid can be determined by evaluating the Fourier Transform at the particular sinusoid's frequency. Is there an equivalent 'Laplace' decomposition, where one could convert a function into a sum of complex decaying sinusoids where a particular complex decaying sinusoid's 'Laplace coefficient' can be determined via evaluating the Laplace Transform at that particular value of 's'?
(Apologies for my possible incorrect use of language, I'm new to this)