Fourier Transform - Definition

Definition

There are several common conventions for defining the Fourier transform ƒ̂ of an integrable function ƒ: R → C (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.