Fourier Transform - Definition


There are several common conventions for defining the Fourier transform ƒ̂ of an integrable function ƒ: RC (Kaiser 1994, p. 29), (Rahman 2011, p. 11). This article will use the definition:

, for every real number ξ.

When the independent variable x represents time (with SI unit of seconds), the transform variable ξ represents frequency (in hertz). Under suitable conditions, ƒ is determined by ƒ̂ via the inverse transform:

for every real number x.

The statement that ƒ can be reconstructed from ƒ̂ is known as the Fourier integral theorem, and was first introduced in Fourier's Analytical Theory of Heat (Fourier 1822, p. 525), (Fourier & Freeman 1878, p. 408), although what would be considered a proof by modern standards was not given until much later (Titchmarsh 1948, p. 1). The functions ƒ and ƒ̂ often are referred to as a Fourier integral pair or Fourier transform pair (Rahman 2011, p. 10).

For other common conventions and notations, including using the angular frequency ω instead of the frequency ξ, see Other conventions and Other notations below. The Fourier transform on Euclidean space is treated separately, in which the variable x often represents position and ξ momentum.

Read more about this topic:  Fourier Transform

Famous quotes containing the word definition:

    Scientific method is the way to truth, but it affords, even in
    principle, no unique definition of truth. Any so-called pragmatic
    definition of truth is doomed to failure equally.
    Willard Van Orman Quine (b. 1908)

    The man who knows governments most completely is he who troubles himself least about a definition which shall give their essence. Enjoying an intimate acquaintance with all their particularities in turn, he would naturally regard an abstract conception in which these were unified as a thing more misleading than enlightening.
    William James (1842–1910)

    I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.”
    Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)