**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.

Read more about this topic: Fourier Transform

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