Fourier transform

The continuous Fourier transform is a linear operator which maps functions to other functions. Loosely, the Fourier transform decomposes a function into a continous spectrum of the frequencies that comprise that function. This is similar to the basic idea of the various other Fourier transforms inclucing the Fourier series of a periodic function.

The precise definition varies, with a number of equivalent definitions in use which usually differ by a constant factor. Suppose f : R -> C is a Lebesgue integrable function. We then define its continuous Fourier transform F : R -> C as

F(s) = ∫ e^ist f(t) dt

for every real number s. We think of s as a frequency and F(s) as the complex number which encodes amplitude and phase of the signal f(t) at that frequency.

The Fourier transform is close to self-inverse: if F(s) is defined as above, then

f(t) = (2π)^-1 ∫ e^-ist F(s) ds

for every real number t (provided that F(s) is integrable, which need not be the case).

Fourier transforms, and the closely related Laplace transforms are widely used in solving differential equations. In mathematical physics, the Fourier transform of a signal f(t) can be thought of as that signal in the "frequency domain".

mention compatibility with differentiation and convolution

The most general and natural context for studying the continuous Fourier transform is given by the tempered distributions; these include all the integrable functions mentioned above but have the added advantage that the Fourier transform of any tempered distribution is again a tempered distribution. Furthermore, the useful Dirac delta is a tempered distribution but not a function. The Fourier transform of the Dirac delta is the constant function 1, and vice versa.

See also: Fourier transform, Fourier series, Laplace transform, Discrete Fourier transform