A transform is chosen according to the problem. For every problem that a transform can solve, there is one best transform for that problem.

The Laplace transform is related to the Fourier transform, but whereas the Fourier transform expresses a function or signal as a series of modes ofvibration (frequencies), the Laplace transform resolves a function into its moments. Like the Fourier transform, the Laplace transform is used for solving differential and integral equations.

Transform boundary

A synonym for transform is change. An antonym for transform is preserve.

They are similar. In many problems, both methods can be used. You can view Fourier transform is the Laplace transform on the circle, that is |z|=1. When you do Fourier transform, you don't need to worry about the convergence region. However, you need to find the convergence region for each Laplace transform.

The discrete version of Fourier transform is discrete Fourier transform, and the discrete version of Laplace transform is Z-transform.