Fourier Transform - Other Notations

Other Notations

Other common notations for ƒ̂(ξ) include:

Denoting the Fourier transform by a capital letter corresponding to the letter of function being transformed (such as ƒ(x) and F(ξ)) is especially common in the sciences and engineering. In electronics, the omega (ω) is often used instead of ξ due to its interpretation as angular frequency, sometimes it is written as F(jω), where j is the imaginary unit, to indicate its relationship with the Laplace transform, and sometimes it is written informally as F(2πƒ) in order to use ordinary frequency.

The interpretation of the complex function ƒ̂(ξ) may be aided by expressing it in polar coordinate form

in terms of the two real functions A(ξ) and φ(ξ) where:

is the amplitude and

is the phase (see arg function).

Then the inverse transform can be written:

which is a recombination of all the frequency components of ƒ(x). Each component is a complex sinusoid of the form e2πixξ whose amplitude is A(ξ) and whose initial phase angle (at x = 0) is φ(ξ).

The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted and is used to denote the Fourier transform of the function ƒ. This mapping is linear, which means that can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function ƒ) can be used to write instead of . Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value ξ for its variable, and this is denoted either as or as . Notice that in the former case, it is implicitly understood that is applied first to ƒ and then the resulting function is evaluated at ξ, not the other way around.

In mathematics and various applied sciences it is often necessary to distinguish between a function ƒ and the value of ƒ when its variable equals x, denoted ƒ(x). This means that a notation like formally can be interpreted as the Fourier transform of the values of ƒ at x. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed.

For example, is sometimes used to express that the Fourier transform of a rectangular function is a sinc function,

or is used to express the shift property of the Fourier transform.

Notice, that the last example is only correct under the assumption that the transformed function is a function of x, not of x₀.