Along with the Laplace transform, it is often used to analyze differential equations.
Over the space of Schwartz functions, this operation is an endomorphism. This operation taken on the extension of a function is the subject of the Poisson summation formula, and Hankel generalized this operation to radial functions. This operation is unitary by Parseval's theorem, and the Cooley- Tukey algorithm is used to implement a "fast" version of this operation. Along with the Laplace transform, it is often used to analyze differential equations. Also useful in analog signal processing, FTP, name this transform that takes a function to its frequency domain, which is useful in decomposing waves and is named for a French mathematician.
Over the space of Schwartz functions, this operation is an endomorphism.
{Fourier Transform}