Fourier Transform

The Fourier transform is a generalization of the complex Fourier series in the limit as . Replace the discrete with the continuous while letting . Then change the sum to an integral, and the equations become

Here,

is called the forward () Fourier transform, and

is called the inverse () Fourier transform. The notation is introduced in Trott (2004, p. xxxiv), and and are sometimes also used to denote the Fourier transform and inverse Fourier transform, respectively (Krantz 1999, p. 202).

Note that some authors (especially physicists) prefer to write the transform in terms of angular frequency instead of the oscillation frequency . However, this destroys the symmetry, resulting in the transform pair

To restore the symmetry of the transforms, the convention

is sometimes used (Mathews and Walker 1970, p. 102).

In general, the Fourier transform pair may be defined using arbitrary real constants and , with , as

The Fourier transform of a function is implemented in the Wolfram Language as FourierTransform[f, x, k], and different choices of and can be used by passing the optional FourierParameters -> a, b option. By default, the Wolfram Language takes FourierParameters -> 0, 1. Unfortunately, a number of other conventions are in widespread use. For example, is used in modern physics, is used in pure mathematics and systems engineering, is used in probability theory for the computation of the characteristic function, is used in classical physics, and is used in signal processing. In this work, following Bracewell (1999, pp. 6-7), it is always assumed that and unless otherwise stated. This choice often results in greatly simplified transforms of common functions such as 1, , etc.

Since any function can be split up into even and odd portions and ,

for an absolutely integrable function , its Fourier transform can be expressed in terms of the Fourier cosine transform and Fourier sine transform as

A function that is absolutely integrable on and of bounded variation in a neighborhood of satisfies the Fourier inversion formula

where and are the one-sided limits (NIST DLMF, §1.14(i)). At a point of continuity, the right-hand side equals . At a discontinuity, it need not equal the assigned value . Decay estimates for the Fourier transform require appropriate integrability of the derivatives, as in the calculation below.

The Fourier transform is linear, since if and have Fourier transforms and , then

Therefore,

The Fourier transform is also symmetric since implies .

For Schwartz functions and , let denote their convolution. Then

The first of these is derived as follows:

where .

There is also a somewhat surprising and extremely important relationship between the autocorrelation and the Fourier transform known as the Wiener-Khinchin theorem. Let , and denote the complex conjugate of , then the inverse Fourier transform of the absolute square of is given by

The Fourier transform of a derivative of a function is simply related to the transform of the function itself. Assume is a continuously differentiable function and both and are absolutely integrable on . Consider

Now use integration by parts

with

and

then

Absolute integrability of implies that has finite limits at both ends of the real line. Absolute integrability of forces these limits to be zero,

so the boundary term vanishes, leaving

If is times continuously differentiable and are absolutely integrable, this process can be iterated to yield

The important modulation theorem of Fourier transforms allows to be expressed in terms of as follows,

If both and are absolutely integrable, the derivative of the Fourier transform is given by

it follows that

For a nonnegative integer , if and are absolutely integrable, iteration gives the general formula

When is a probability density function with a finite second raw moment, its variance is

For two such probability density functions and , variances add under convolution,

If has the Fourier transform , then the Fourier transform has the shift property

so has the Fourier transform

If has a Fourier transform , then for a real constant , the Fourier transform obeys a similarity theorem.

so has the Fourier transform

The "equivalent width" of a Fourier transform is

The "autocorrelation width" is

where denotes the cross-correlation of and and is the complex conjugate.

Any operation on which leaves its area unchanged leaves unchanged, since

The following table summarized some common Fourier transform pairs.

In two dimensions, the Fourier transform becomes

Similarly, the -dimensional Fourier transform can be defined for , by