Both the Laplace transform and the Fourier transform in some sense decode the "spectrum" of a function. The Laplace transform gives a power-series decomposition whereas the Fourier transform gives a harmonic (or loop-based) decomposition.
Are there deep connections between these two transforms? The formulaic connection is clear, but is there something deeper?
(Maybe the answer will involve spectral theory?)
Best Answer
I don't know what answer you are looking for but for example both Laplace and Fourier transform are a so called Gelfand Transform.
You can find good introduction to Gelfand Transform in nice book Functional analysis for probability and stochastic processes: an introduction, A. Bobrowski. Look into Chapter 6.