linguistic:paraphrasing
Definition

The Fourier transform is an analysis process, decomposing a complex-valued function f (x) {\displaystyle \textstyle f(x)} into its constituent frequencies and their amplitudes. The inverse process is synthesis, which recreates f (x) {\displaystyle \textstyle f(x)} from its transform.

We can start with an analogy, the Fourier series, which analyzes f (x) {\displaystyle \textstyle f(x)} on a bounded interval x ∈, {\displaystyle \textstyle x\in,} for some positive real number P. {\displaystyle P.} The constituent frequencies are a discrete set of harmonics at frequencies n P, n ∈ Z, {\displaystyle {\tfrac {n}{P}},n\in \mathbb {Z},} whose amplitude and phase are given (up to multiplicative constants) by the analysis formula: c n = ∫ − P / 2 P / 2 f (x) e − i 2 π n P x d x. {\displaystyle c_{n}=\int _{-P/2}^{P/2}f(x)\,e^{-i2\pi {\frac {n}{P}}x}\,dx.} The actual Fourier series is the synthesis formula: f (x) = 1 P ∑ n = − ∞ ∞ c n e i 2 π n P x, x ∈. {\displaystyle f(x)={\tfrac {1}{P}}\sum _{n=-\infty }^{\infty }c_{n}\,e^{i2\pi {\tfrac {n}{P}}x},\quad \textstyle x\in.}

The analogy for a function f (x) {\displaystyle \textstyle f(x)} can be obtained formally from the analysis formula by taking the limit as P → ∞ {\displaystyle P\to \infty }, while at the same time taking n {\displaystyle n} so that n P → ξ ∈ R. {\displaystyle {\tfrac {n}{P}}\to \xi \in \mathbb {R}.} Formally carrying this out, we obtain:

This integral converges when f (x) {\displaystyle f(x)} decays with all derivatives as x → ± ∞ {\displaystyle x\to \pm \infty }: lim x → ∞ f (n) (x) = 0, n = 0, 1, 2, … {\displaystyle \lim _{x\to \infty }f^{(n)}(x)=0,n=0,1,2,\dots }. (See Schwartz function). By the Riemann–Lebesgue lemma, the transformed function f ^ {\displaystyle {\widehat {f}}} also decays with all derivatives. The validity of this definition for classes of functions f {\displaystyle f} that do not necessarily decay like this is discussed later in this section.

Evaluating Eq.1 for all values of ξ {\displaystyle \xi } produces the frequency-domain function. The complex number f ^ (ξ) {\displaystyle {\widehat {f}}(\xi)}, in polar coordinates, conveys both amplitude and phase of frequency ξ. {\displaystyle \xi.} The intuitive interpretation of Eq.1 is that the effect of multiplying f (x) {\displaystyle f(x)} by e − i 2 π ξ x {\displaystyle e^{-i2\pi \xi x}} is to subtract ξ {\displaystyle \xi } from every frequency component of function f (x). {\displaystyle f(x).} Only the component that was at frequency ξ {\displaystyle \xi } can produce a non-zero value of the infinite integral, because (at least formally) all the other shifted components are oscillatory and integrate to zero. (see § Example)

The corresponding synthesis formula for such a function is:

Eq.2 is a representation of f (x) {\displaystyle f(x)} as a weighted summation of complex exponential functions.

This is also known as the Fourier inversion theorem, and was first introduced in Fourier's Analytical Theory of Heat.

The functions f {\displaystyle f} and f ^ {\displaystyle {\widehat {f}}} are referred to as a Fourier transform pair. A common notation for designating transform pairs is: f (x) ⟷ F f ^ (ξ), {\displaystyle f(x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ {\widehat {f}}(\xi),} for example rect (x) ⟷ F sinc (ξ). {\displaystyle \operatorname {rect} (x)\ {\stackrel {\mathcal {F}}{\longleftrightarrow }}\ \operatorname {sinc} (\xi).}