# Fourier transform

From AMS Glossary

## Fourier transform

An analytical transformation of a function where

*f*(*x*) obtained (if it exists) by multiplying the function by*e*^{-iux}and integrating over all*x*,*u*is the new variable of the transform*F*(*u*) and*i*^{2}= -1.If the Fourier transform of a function is known, the function itself may be recovered by use of the inversion formula: The Fourier transform has the same uses as the Fourier series: For example, the integrand F(

*u*) exp (*iux*) is a solution of a given linear equation, so that the integral sum of these solutions is the most general solution of the equation. When the variable*u*is complex, the Fourier transform is equivalent to the Laplace transform.*See also*Fourier integral, spectral function.