Fourier transform

To simplify the math, it is desirable to write the Fourier series as a sum of complex exponentials (see Fourier series § Exponential form). Each complex exponential or complex sinusoid of frequency ξ can be expressed using Euler's formula as the sum of a cosine wave of frequency ξ for the real component plus a sine wave also of frequency ξ for the imaginary component:

${\displaystyle {\begin{aligned}e^{2\pi i\xi x}&=\cos(2\pi \xi x)+i\sin(2\pi \xi x)\end{aligned}}}$

Expressing real sinusoids as complex sinusoids makes it necessary for the Fourier coefficients ${\displaystyle c_{n}}$to be complex valued, but has the benefit of compactly representing all necessary information about each frequency. The usual interpretation of this complex number is that ${\displaystyle \left\vert c_{n}\right\vert }$ (its magnitude) gives the amplitude and ${\displaystyle \arg(c_{n})}$ (its argument) gives the phase of the complex sinusoid for that coefficient.

Illustrating a complex exponential in three-dimensions. The real component is a cosine wave. The imaginary component is a sine wave. Together they form a helix. Negating the frequency can be understood as changing the handedness of the helix; rotating with the opposite orientation but with the same number of rotations per second.

These complex exponentials may have negative frequency. For example, both complex sinusoids e^2πiξx and e^−2πiξx complete one cycle per unit of x, but the first represents a positive frequency while the second represents a negative frequency. Positive frequency can be understood as rotating counter-clockwise about the complex plane while negative frequency can be understood as rotating clockwise about the complex plane. When complex sinusoids are interpreted as a helix in three-dimensions (with the third dimension being the imaginary component), negating the frequency simply changes the handedness of the helix.[12]

Real sine and cosine waves can be recovered from the complex exponential representation of sinusoids. For example, a corollary to Euler's formula allows expressing cosine and sine waves as either the real or imaginary part of a complex sinusoid or as a weighted sum of two complex sinusoids of opposite frequency:

${\displaystyle {\begin{aligned}\cos(2\pi \xi x)&=\operatorname {Re} \left(e^{2\pi i\xi x}\right)={\tfrac {1}{2}}e^{2\pi i\xi x}+{\tfrac {1}{2}}e^{-2\pi i\xi x},\\\sin(2\pi \xi x)&=\operatorname {Im} \left(e^{2\pi i\xi x}\right)={\tfrac {1}{2i}}e^{2\pi i\xi x}-{\tfrac {1}{2i}}e^{-2\pi i\xi x}.\end{aligned}}}$

Consequently, a general form of any real sinusoid (with frequency ξ, phase shift θ, and amplitude A) can be expressed as the sum of two complex sinusoids of opposite frequency (ξ and -ξ) but equal magnitude (A/2) and with the phase shift θ embedded in both their complex coefficients:

${\displaystyle {\begin{aligned}A\cos(2\pi \xi x+\theta )&={\tfrac {A}{2}}e^{2\pi i\xi x+i\theta }+{\tfrac {A}{2}}e^{-2\pi i\xi x-i\theta }={\tfrac {Ae^{i\theta }}{2}}e^{2\pi i\xi x}+{\tfrac {Ae^{-i\theta }}{2}}e^{-2\pi i\xi x}.\end{aligned}}}$

Hence, every real sinusoid (and real signal) can be considered to consist of a positive and negative frequency, whose imaginary components cancel but whose real components contribute equally to form the real signal.

To avoid the use of complex numbers and negative frequencies, the sine and cosine transforms together can be used as an equivalent alternative form of the Fourier transform.

There is a close connection between the definition of Fourier series and the Fourier transform for functions f that are zero outside an interval. For such a function, we can calculate its Fourier series on any interval that includes the points where f is not zero. The Fourier transform is also defined for such a function. As we increase the length of the interval in which we calculate the Fourier series, then the Fourier series coefficients begin to resemble the Fourier transform and the sum of the Fourier series of f begins to resemble the inverse Fourier transform. More precisely, suppose T is large enough that the interval [−T/2, T/2] contains the interval in which f is not zero. Then, the nth series coefficient c_n is given by:

${\displaystyle c_{n}={\frac {1}{T}}\int _{-{\frac {T}{2}}}^{\frac {T}{2}}f(x)\,e^{-2\pi i\left({\frac {n}{T}}\right)x}\,dx.}$

Comparing this to the definition of the Fourier transform, it follows that:

${\displaystyle c_{n}={\frac {1}{T}}{\hat {f}}\left({\frac {n}{T}}\right)}$

since f(x) is zero outside [−T/2, T/2]. Thus, the Fourier coefficients are equal to the values of the Fourier transform sampled on a grid of width 1/T, multiplied by the grid width 1/T.

Under appropriate conditions, the Fourier series of f will equal the function f. In other words, f can be written:

${\displaystyle f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,e^{2\pi i\left({\frac {n}{T}}\right)x}=\sum _{n=-\infty }^{\infty }{\hat {f}}(\xi _{n})\ e^{2\pi i\xi _{n}x}\Delta \xi ,}$

where the last sum is simply the first sum rewritten using the definitions ξ_n = n/T, and Δξ = n + 1/T − n/T = 1/T.

This second sum is a Riemann sum. By letting T → ∞ it will converge to the integral for the inverse Fourier transform as expressed above. Under suitable conditions, this argument may be made precise.[13]

The Fourier transform has the following basic properties:[14]

For any complex numbers a and b, if h(x) = af (x) + bg(x), then ĥ(ξ) = a · f̂(ξ) + b · ĝ(ξ).

Animation showing the Fourier Transform of a time shifted signal. [Top] the original signal (yellow), is continuously time shifted (blue). [Bottom] The resultant Fourier Transform of the time shifted signal. Note how the higher frequency components revolve in complex plane faster than the lower frequency components.

For any real number x₀, if h(x) = f(x − x₀), then ĥ(ξ) = e^{−2πix₀ξ} f̂(ξ).

For any real number ξ₀, if h(x) = e^2πixξ₀ f(x), then ĥ(ξ) = f̂(ξ − ξ₀).

For a non-zero real number a, if h(x) = f(ax), then

$${\displaystyle {\hat {h}}(\xi )={\frac {1}{|a|}}{\hat {f}}\left({\frac {\xi }{a}}\right).}$$

The case a = −1 leads to the time-reversal property, which states: if h(x) = f(−x), then ĥ(ξ) = f̂(−ξ).

If h(x) = f(x), then

$${\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(-\xi )}}.}$$

In particular, if f is real, then one has the reality condition

$${\displaystyle {\hat {f}}(-\xi )={\overline {{\hat {f}}(\xi )}},}$$

that is, f̂ is a Hermitian function. And if f is purely imaginary, then

$${\displaystyle {\hat {f}}(-\xi )=-{\hat {f}}(\xi ).}$$

• If ${\displaystyle h(x)=\Re {(f(x))}}$, then ${\displaystyle {\hat {h}}(\xi )={\frac {1}{2}}\left({\hat {f}}(\xi )+{\overline {{\hat {f}}(-\xi )}}\right)}$.
• If ${\displaystyle h(x)=\Im {(f(x))}}$, then ${\displaystyle {\hat {h}}(\xi )={\frac {1}{2i}}\left({\hat {f}}(\xi )-{\overline {{\hat {f}}(-\xi )}}\right)}$.

Substituting ξ = 0 in the definition, we obtain

$${\displaystyle {\hat {f}}(0)=\int _{-\infty }^{\infty }f(x)\,dx.}$$

That is the same as the integral of f over all its domain and is also known as the average value or DC bias of the function.

Under suitable conditions on the function ${\displaystyle f}$, it can be recovered from its Fourier transform ${\displaystyle {\hat {f}}}$. Indeed, denoting the Fourier transform operator by ${\displaystyle {\mathcal {F}}}$, so ${\displaystyle {\mathcal {F}}f:={\hat {f}}}$, then for suitable functions, applying the Fourier transform twice simply flips the function: ${\displaystyle ({\mathcal {F}}^{2}f)(x)=f(-x)}$, which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yields ${\displaystyle {\mathcal {F}}^{4}(f)=f}$, so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times: ${\displaystyle {\mathcal {F}}^{3}({\hat {f}})=f}$. In particular the Fourier transform is invertible (under suitable conditions).

More precisely, defining the parity operator ${\displaystyle {\mathcal {P}}}$ such that ${\displaystyle ({\mathcal {P}}f)(x)=f(-x)}$, we have:

${\displaystyle {\begin{aligned}{\mathcal {F}}^{0}&=\mathrm {id} ,\\{\mathcal {F}}^{1}&={\mathcal {F}},\\{\mathcal {F}}^{2}&={\mathcal {P}},\\{\mathcal {F}}^{3}&={\mathcal {F}}^{-1}={\mathcal {P}}\circ {\mathcal {F}}={\mathcal {F}}\circ {\mathcal {P}},\\{\mathcal {F}}^{4}&=\mathrm {id} \end{aligned}}}$

These equalities of operators require careful definition of the space of functions in question, defining equality of functions (equality at every point? equality almost everywhere?) and defining equality of operators – that is, defining the topology on the function space and operator space in question. These are not true for all functions, but are true under various conditions, which are the content of the various forms of the Fourier inversion theorem.

This fourfold periodicity of the Fourier transform is similar to a rotation of the plane by 90°, particularly as the two-fold iteration yields a reversal, and in fact this analogy can be made precise. While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in the time–frequency domain (considering time as the x-axis and frequency as the y-axis), and the Fourier transform can be generalized to the fractional Fourier transform, which involves rotations by other angles. This can be further generalized to linear canonical transformations, which can be visualized as the action of the special linear group SL₂(R) on the time–frequency plane, with the preserved symplectic form corresponding to the uncertainty principle, below. This approach is particularly studied in signal processing, under time–frequency analysis.

The frequency variable must have inverse units to the units of the original function's domain (typically named t or x). For example, if t is measured in seconds, ξ should be in cycles per second. If the scale of time is in units of 2π seconds, then another greek letter ω typically is used instead to represent angular frequency (where ω = 2πξ) in units of radians per second. If using x for units of length, then ξ must be in inverse length, e.g., wavenumbers. That is to say, there are two versions of the real line: one which is the range of t and measured in units of t, and the other which is the range of ξ and measured in inverse units to the units of t. These two distinct versions of the real line cannot be equated with each other. Therefore, the Fourier transform goes from one space of functions to a different space of functions: functions which have a different domain of definition.

In general, ξ must always be taken to be a linear form on the space of its domain, which is to say that the second real line is the dual space of the first real line. See the article on linear algebra for a more formal explanation and for more details. This point of view becomes essential in generalisations of the Fourier transform to general symmetry groups, including the case of Fourier series.

That there is no one preferred way (often, one says "no canonical way") to compare the two versions of the real line which are involved in the Fourier transform—fixing the units on one line does not force the scale of the units on the other line—is the reason for the plethora of rival conventions on the definition of the Fourier transform. The various definitions resulting from different choices of units differ by various constants.

Let ${\displaystyle {\hat {f}}_{1}(\xi )}$ be the form of the Fourier transform in terms of ordinary frequency ξ.

Because ${\displaystyle \xi ={\tfrac {\omega }{2\pi }}}$, the alternative form ${\displaystyle {\hat {f}}_{3}(\omega )}$ (which Fourier transform § Other conventions calls the non-unitary form in angular frequency) has no factor in its definition

${\displaystyle {\begin{aligned}{\hat {f}}_{3}(\omega )\ &{\stackrel {\mathrm {def} }{=}}\ \int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega \cdot x}\,dx={\hat {f}}_{1}\left({\frac {\omega }{2\pi }}\right)\end{aligned}}}$

but has a factor of ${\displaystyle {\tfrac {1}{2\pi }}}$ in its corresponding inversion formula

${\displaystyle {\begin{aligned}f(x)&={\frac {1}{2\pi }}\int _{-\infty }^{\infty }{\hat {f}}_{3}(\omega )\cdot e^{i\omega \cdot x}\,d\omega .\end{aligned}}}$

An alternative form ${\displaystyle {\hat {f}}_{2}(\omega )}$ (which Fourier transform § Other conventions calls the unitary form in angular frequency) has a factor of ${\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}$ in its definition

${\displaystyle {\begin{aligned}{\hat {f}}_{2}(\omega )\ &{\stackrel {\mathrm {def} }{=}}\ {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(x)\cdot e^{-i\omega \cdot x}\,dx\end{aligned}}}$

and also has that same factor of ${\displaystyle {\tfrac {1}{\sqrt {2\pi }}}}$ in its corresponding inversion formula, producing a symmetrical relationship

${\displaystyle {\begin{aligned}f(x)&={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\hat {f}}_{2}(\omega )\cdot e^{i\omega \cdot x}\,d\omega .\end{aligned}}}$

In other conventions, the Fourier transform has i in the exponent instead of −i, and vice versa for the inversion formula. Effectively that defines  ${\displaystyle e^{i2\pi \xi x},\xi >0}$  as a negative frequency, because the corresponding spectral component is ${\displaystyle {\hat {f}}(-\xi ).}$ But many of the identities involving the Fourier transform remain valid in those conventions, provided all terms that explicitly involve i have it replaced by −i. To add further confusion, be aware that since electrical engineers use the letter i to represent current, their form of the transform typically uses the letter j for the imaginary unit instead of i.

When using dimensionless units, the constant factors might not even be written in the transform definition. For instance, in probability theory, the characteristic function Φ of the probability density function f of a random variable X of continuous type is defined without a negative sign in the exponential, and since the units of x are ignored, there is no 2π either:

${\displaystyle \phi (\lambda )=\int _{-\infty }^{\infty }f(x)e^{i\lambda x}\,dx.}$

(In probability theory, and in mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because so many random variables are not of continuous type, and do not possess a density function, and one must treat not functions but distributions, i.e., measures which possess "atoms".)

From the higher point of view of group characters, which is much more abstract, all these arbitrary choices disappear, as will be explained in the later section of this article, which treats the notion of the Fourier transform of a function on a locally compact Abelian group.

The rectangular function is Lebesgue integrable.

The sinc function, which is the Fourier transform of the rectangular function, is bounded and continuous, but not Lebesgue integrable.

The Fourier transform may be defined in some cases for non-integrable functions, but the Fourier transforms of integrable functions have several strong properties.

The Fourier transform f̂ of any integrable function f is uniformly continuous and[15]

${\displaystyle \left\|{\hat {f}}\right\|_{\infty }\leq \left\|f\right\|_{1}}$

By the Riemann–Lebesgue lemma,[16]

${\displaystyle {\hat {f}}(\xi )\to 0{\text{ as }}|\xi |\to \infty .}$

However, ${\displaystyle {\hat {f}}}$ need not be integrable. For example, the Fourier transform of the rectangular function, which is integrable, is the sinc function, which is not Lebesgue integrable, because its improper integrals behave analogously to the alternating harmonic series, in converging to a sum without being absolutely convergent.

It is not generally possible to write the inverse transform as a Lebesgue integral. However, when both f and ${\displaystyle {\hat {f}}}$ are integrable, the inverse equality

${\displaystyle f(x)=\int _{-\infty }^{\infty }{\hat {f}}(\xi )e^{2i\pi x\xi }\,d\xi }$

holds almost everywhere. That is, the Fourier transform is injective on L¹(R). (But if f is continuous, then equality holds for every x.)

Let f(x) and g(x) be integrable, and let f̂(ξ) and ĝ(ξ) be their Fourier transforms. If f(x) and g(x) are also square-integrable, then the Parseval formula follows:[17]

${\displaystyle \langle f,g\rangle _{L^{2}}=\int _{-\infty }^{\infty }f(x){\overline {g(x)}}\,dx=\int _{-\infty }^{\infty }{\hat {f}}(\xi ){\overline {{\hat {g}}(\xi )}}\,d\xi ,}$

where the bar denotes complex conjugation.

The Plancherel theorem, which follows from the above, states that[18]

${\displaystyle \|f\|_{L^{2}}=\int _{-\infty }^{\infty }\left|f(x)\right|^{2}\,dx=\int _{-\infty }^{\infty }\left|{\hat {f}}(\xi )\right|^{2}\,d\xi .}$

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on L²(R). On L¹(R) ∩ L²(R), this extension agrees with original Fourier transform defined on L¹(R), thus enlarging the domain of the Fourier transform to L¹(R) + L²(R) (and consequently to L^p(R) for 1 ≤ p ≤ 2). Plancherel's theorem has the interpretation in the sciences that the Fourier transform preserves the energy of the original quantity. The terminology of these formulas is not quite standardised. Parseval's theorem was proved only for Fourier series, and was first proved by Lyapunov. But Parseval's formula makes sense for the Fourier transform as well, and so even though in the context of the Fourier transform it was proved by Plancherel, it is still often referred to as Parseval's formula, or Parseval's relation, or even Parseval's theorem.

See Pontryagin duality for a general formulation of this concept in the context of locally compact abelian groups.

The Poisson summation formula (PSF) is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. The Poisson summation formula says that for sufficiently regular functions f,

${\displaystyle \sum _{n}{\hat {f}}(n)=\sum _{n}f(n).}$

It has a variety of useful forms that are derived from the basic one by application of the Fourier transform's scaling and time-shifting properties. The formula has applications in engineering, physics, and number theory. The frequency-domain dual of the standard Poisson summation formula is also called the discrete-time Fourier transform.

Poisson summation is generally associated with the physics of periodic media, such as heat conduction on a circle. The fundamental solution of the heat equation on a circle is called a theta function. It is used in number theory to prove the transformation properties of theta functions, which turn out to be a type of modular form, and it is connected more generally to the theory of automorphic forms where it appears on one side of the Selberg trace formula.

Suppose f(x) is an absolutely continuous differentiable function, and both f and its derivative f′ are integrable. Then the Fourier transform of the derivative is given by

${\displaystyle {\widehat {f'\,}}(\xi )={\mathcal {F}}\left\{{\frac {d}{dx}}f(x)\right\}=2\pi i\xi {\hat {f}}(\xi ).}$

More generally, the Fourier transformation of the nth derivative f⁽ⁿ⁾ is given by

${\displaystyle {\widehat {f^{(n)}}}(\xi )={\mathcal {F}}\left\{{\frac {d^{n}}{dx^{n}}}f(x)\right\}=(2\pi i\xi )^{n}{\hat {f}}(\xi ).}$

By applying the Fourier transform and using these formulas, some ordinary differential equations can be transformed into algebraic equations, which are much easier to solve. These formulas also give rise to the rule of thumb "f(x) is smooth if and only if f̂(ξ) quickly falls to 0 for |ξ| → ∞." By using the analogous rules for the inverse Fourier transform, one can also say "f(x) quickly falls to 0 for |x| → ∞ if and only if f̂(ξ) is smooth."

The Fourier transform translates between convolution and multiplication of functions. If f(x) and g(x) are integrable functions with Fourier transforms f̂(ξ) and ĝ(ξ) respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms f̂(ξ) and ĝ(ξ) (under other conventions for the definition of the Fourier transform a constant factor may appear).

This means that if:

${\displaystyle h(x)=(f*g)(x)=\int _{-\infty }^{\infty }f(y)g(x-y)\,dy,}$

where ∗ denotes the convolution operation, then:

${\displaystyle {\hat {h}}(\xi )={\hat {f}}(\xi )\cdot {\hat {g}}(\xi ).}$

In linear time invariant (LTI) system theory, it is common to interpret g(x) as the impulse response of an LTI system with input f(x) and output h(x), since substituting the unit impulse for f(x) yields h(x) = g(x). In this case, ĝ(ξ) represents the frequency response of the system.

Conversely, if f(x) can be decomposed as the product of two square integrable functions p(x) and q(x), then the Fourier transform of f(x) is given by the convolution of the respective Fourier transforms p̂(ξ) and q̂(ξ).

In an analogous manner, it can be shown that if h(x) is the cross-correlation of f(x) and g(x):

${\displaystyle h(x)=(f\star g)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}g(x+y)\,dy}$

then the Fourier transform of h(x) is:

${\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}\cdot {\hat {g}}(\xi ).}$

As a special case, the autocorrelation of function f(x) is:

${\displaystyle h(x)=(f\star f)(x)=\int _{-\infty }^{\infty }{\overline {f(y)}}f(x+y)\,dy}$

for which

${\displaystyle {\hat {h}}(\xi )={\overline {{\hat {f}}(\xi )}}{\hat {f}}(\xi )=\left|{\hat {f}}(\xi )\right|^{2}.}$

One important choice of an orthonormal basis for L²(R) is given by the Hermite functions

${\displaystyle \psi _{n}(x)={\frac {\sqrt[{4}]{2}}{\sqrt {n!}}}e^{-\pi x^{2}}\mathrm {He} _{n}\left(2x{\sqrt {\pi }}\right),}$

where He_n(x) are the "probabilist's" Hermite polynomials, defined as

${\displaystyle \mathrm {He} _{n}(x)=(-1)^{n}e^{\frac {x^{2}}{2}}\left({\frac {d}{dx}}\right)^{n}e^{-{\frac {x^{2}}{2}}}}$

Under this convention for the Fourier transform, we have that

${\displaystyle {\hat {\psi }}_{n}(\xi )=(-i)^{n}\psi _{n}(\xi )}$.

In other words, the Hermite functions form a complete orthonormal system of eigenfunctions for the Fourier transform on L²(R).[14] However, this choice of eigenfunctions is not unique. There are only four different eigenvalues of the Fourier transform (±1 and ±i) and any linear combination of eigenfunctions with the same eigenvalue gives another eigenfunction. As a consequence of this, it is possible to decompose L²(R) as a direct sum of four spaces H₀, H₁, H₂, and H₃ where the Fourier transform acts on He_k simply by multiplication by i^k.

Since the complete set of Hermite functions provides a resolution of the identity, the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed. This approach to define the Fourier transform was first done by Norbert Wiener.[19] Among other properties, Hermite functions decrease exponentially fast in both frequency and time domains, and they are thus used to define a generalization of the Fourier transform, namely the fractional Fourier transform used in time–frequency analysis.[20] In physics, this transform was introduced by Edward Condon.[21]

The Heisenberg group is a certain group of unitary operators on the Hilbert space L²(R) of square integrable complex valued functions f on the real line, generated by the translations (T_y f)(x) = f (x + y) and multiplication by e^2πixξ, (M_ξ f)(x) = e^2πixξ f (x). These operators do not commute, as their (group) commutator is

${\displaystyle \left(M_{\xi }^{-1}T_{y}^{-1}M_{\xi }T_{y}f\right)(x)=e^{2\pi iy\xi }f(x)}$

which is multiplication by the constant (independent of x) e^2πiyξ ∈ U(1) (the circle group of unit modulus complex numbers). As an abstract group, the Heisenberg group is the three-dimensional Lie group of triples (x, ξ, z) ∈ R² × U(1), with the group law

${\displaystyle \left(x_{1},\xi _{1},t_{1}\right)\cdot \left(x_{2},\xi _{2},t_{2}\right)=\left(x_{1}+x_{2},\xi _{1}+\xi _{2},t_{1}t_{2}e^{2\pi i\left(x_{1}\xi _{1}+x_{2}\xi _{2}+x_{1}\xi _{2}\right)}\right).}$

Denote the Heisenberg group by H₁. The above procedure describes not only the group structure, but also a standard unitary representation of H₁ on a Hilbert space, which we denote by ρ : H₁ → B(L²(R)). Define the linear automorphism of R² by

${\displaystyle J{\begin{pmatrix}x\\\xi \end{pmatrix}}={\begin{pmatrix}-\xi \\x\end{pmatrix}}}$

so that J² = −I. This J can be extended to a unique automorphism of H₁:

${\displaystyle j\left(x,\xi ,t\right)=\left(-\xi ,x,te^{-2\pi ix\xi }\right).}$

According to the Stone–von Neumann theorem, the unitary representations ρ and ρ ∘ j are unitarily equivalent, so there is a unique intertwiner W ∈ U(L²(R)) such that

${\displaystyle \rho \circ j=W\rho W^{*}.}$

This operator W is the Fourier transform.

Many of the standard properties of the Fourier transform are immediate consequences of this more general framework.[22] For example, the square of the Fourier transform, W², is an intertwiner associated with J² = −I, and so we have (W²f)(x) = f (−x) is the reflection of the original function f.

The Fourier transform f̂(ξ) is related to the Laplace transform F(s), which is also used for the solution of differential equations and the analysis of filters.

It may happen that a function f for which the Fourier integral does not converge on the real axis at all, nevertheless has a complex Fourier transform defined in some region of the complex plane.

For example, if f(t) is of exponential growth, i.e.,

${\displaystyle \vert f(t)\vert <Ce^{a\vert t\vert }}$

for some constants C, a ≥ 0, then[29]

${\displaystyle {\hat {f}}(i\tau )=\int _{-\infty }^{\infty }e^{2\pi \tau t}f(t)\,dt,}$

convergent for all 2πτ < −a, is the two-sided Laplace transform of f.

The more usual version ("one-sided") of the Laplace transform is

${\displaystyle F(s)=\int _{0}^{\infty }f(t)e^{-st}\,dt.}$

If f is also causal, and analytical, then: ${\displaystyle {\hat {f}}(i\tau )=F(-2\pi \tau ).}$ Thus, extending the Fourier transform to the complex domain means it includes the Laplace transform as a special case in the case of causal functions—but with the change of variable s = 2πiξ.

From another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term which lets it converge outside of the imaginary line where the Fourier transform is defined. As such it can converge for at most exponentially divergent series and integrals, whereas the original Fourier decomposition cannot, enabling analysis of systems with divergent or critical elements. Two particular examples from linear signal processing are the construction of allpass filter networks from critical comb and mitigating filters via exact pole-zero cancellation on the unit circle. Such designs are common in audio processing, where highly nonlinear phase response is sought for, as in reverb.

Furthermore, when extended pulselike impulse responses are sought for signal processing work, the easiest way to produce them is to have one circuit which produces a divergent time response, and then to cancel its divergence through a delayed opposite and compensatory response. There, only the delay circuit in-between admits a classical Fourier description, which is critical. Both the circuits to the side are unstable, and do not admit a convergent Fourier decomposition. However, they do admit a Laplace domain description, with identical half-planes of convergence in the complex plane (or in the discrete case, the Z-plane), wherein their effects cancel.

In modern mathematics the Laplace transform is conventionally subsumed under the aegis Fourier methods. Both of them are subsumed by the far more general, and more abstract, idea of harmonic analysis.

If f̂ is complex analytic for a ≤ τ ≤ b, then

${\displaystyle \int _{-\infty }^{\infty }{\hat {f}}(\sigma +ia)e^{2\pi i\xi t}\,d\sigma =\int _{-\infty }^{\infty }{\hat {f}}(\sigma +ib)e^{2\pi i\xi t}\,d\sigma }$

by Cauchy's integral theorem. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.[30]

Theorem: If f(t) = 0 for t < 0, and |f(t)| < Ce^a|t| for some constants C, a > 0, then

${\displaystyle f(t)=\int _{-\infty }^{\infty }{\hat {f}}(\sigma +i\tau )e^{2\pi i\xi t}\,d\sigma ,}$

for any τ < −a/2π.

This theorem implies the Mellin inversion formula for the Laplace transformation,[29]

${\displaystyle f(t)={\frac {1}{2\pi i}}\int _{b-i\infty }^{b+i\infty }F(s)e^{st}\,ds}$

for any b > a, where F(s) is the Laplace transform of f(t).

The hypotheses can be weakened, as in the results of Carleman and Hunt, to f(t) e^−at being L¹, provided that f is of bounded variation in a closed neighborhood of t (cf. Dirichlet-Dini theorem), the value of f at t is taken to be the arithmetic mean of the left and right limits, and provided that the integrals are taken in the sense of Cauchy principal values.[31]

L² versions of these inversion formulas are also available.[32]

Generally speaking, the more concentrated f(x) is, the more spread out its Fourier transform f̂(ξ) must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in x, its Fourier transform stretches out in ξ. It is not possible to arbitrarily concentrate both a function and its Fourier transform.

The trade-off between the compaction of a function and its Fourier transform can be formalized in the form of an uncertainty principle by viewing a function and its Fourier transform as conjugate variables with respect to the symplectic form on the time–frequency domain: from the point of view of the linear canonical transformation, the Fourier transform is rotation by 90° in the time–frequency domain, and preserves the symplectic form.

Suppose f(x) is an integrable and square-integrable function. Without loss of generality, assume that f(x) is normalized:

${\displaystyle \int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1.}$

It follows from the Plancherel theorem that f̂(ξ) is also normalized.

The spread around x = 0 may be measured by the dispersion about zero[33] defined by

${\displaystyle D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx.}$

In probability terms, this is the second moment of |f(x)|² about zero.

The uncertainty principle states that, if f(x) is absolutely continuous and the functions x·f(x) and f′(x) are square integrable, then[14]

${\displaystyle D_{0}(f)D_{0}\left({\hat {f}}\right)\geq {\frac {1}{16\pi ^{2}}}}$.

The equality is attained only in the case

${\displaystyle {\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}}$

where σ > 0 is arbitrary and C₁ = ⁴√2/√σ so that f is L²-normalized.[14] In other words, where f is a (normalized) Gaussian function with variance σ², centered at zero, and its Fourier transform is a Gaussian function with variance σ⁻².

In fact, this inequality implies that:

${\displaystyle \left(\int _{-\infty }^{\infty }(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }(\xi -\xi _{0})^{2}\left|{\hat {f}}(\xi )\right|^{2}\,d\xi \right)\geq {\frac {1}{16\pi ^{2}}}}$

for any x₀, ξ₀ ∈ R.[13]

In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, to within a factor of Planck's constant. With this constant properly taken into account, the inequality above becomes the statement of the Heisenberg uncertainty principle.[34]

A stronger uncertainty principle is the Hirschman uncertainty principle, which is expressed as:

${\displaystyle H\left(\left|f\right|^{2}\right)+H\left(\left|{\hat {f}}\right|^{2}\right)\geq \log \left({\frac {e}{2}}\right)}$

where H(p) is the differential entropy of the probability density function p(x):

${\displaystyle H(p)=-\int _{-\infty }^{\infty }p(x)\log {\bigl (}p(x){\bigr )}\,dx}$

where the logarithms may be in any base that is consistent. The equality is attained for a Gaussian, as in the previous case.

Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable function f for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically[35]) λ by

${\displaystyle f(t)=\int _{0}^{\infty }{\bigl (}a(\lambda )\cos(2\pi \lambda t)+b(\lambda )\sin(2\pi \lambda t){\bigr )}\,d\lambda .}$

This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions a and b can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised):

${\displaystyle a(\lambda )=2\int _{-\infty }^{\infty }f(t)\cos(2\pi \lambda t)\,dt}$

and

${\displaystyle b(\lambda )=2\int _{-\infty }^{\infty }f(t)\sin(2\pi \lambda t)\,dt.}$

Older literature refers to the two transform functions, the Fourier cosine transform, a, and the Fourier sine transform, b.

The function f can be recovered from the sine and cosine transform using

${\displaystyle f(t)=2\int _{0}^{\infty }\int _{-\infty }^{\infty }f(\tau )\cos {\bigl (}2\pi \lambda (\tau -t){\bigr )}\,d\tau \,d\lambda .}$

together with trigonometric identities. This is referred to as Fourier's integral formula.[29][36][37][38]

Let the set of homogeneous harmonic polynomials of degree k on Rⁿ be denoted by A_k. The set A_k consists of the solid spherical harmonics of degree k. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if f(x) = e^−π|x|2P(x) for some P(x) in A_k, then f̂(ξ) = i^−k f(ξ). Let the set H_k be the closure in L²(Rⁿ) of linear combinations of functions of the form f(|x|)P(x) where P(x) is in A_k. The space L²(Rⁿ) is then a direct sum of the spaces H_k and the Fourier transform maps each space H_k to itself and is possible to characterize the action of the Fourier transform on each space H_k.[16]

Let f(x) = f₀(|x|)P(x) (with P(x) in A_k), then

${\displaystyle {\hat {f}}(\xi )=F_{0}(|\xi |)P(\xi )}$

where

${\displaystyle F_{0}(r)=2\pi i^{-k}r^{-{\frac {n+2k-2}{2}}}\int _{0}^{\infty }f_{0}(s)J_{\frac {n+2k-2}{2}}(2\pi rs)s^{\frac {n+2k}{2}}\,ds.}$

Here J_{(n + 2k − 2)/2} denotes the Bessel function of the first kind with order n + 2k − 2/2. When k = 0 this gives a useful formula for the Fourier transform of a radial function.[39] This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases n + 2 and n[40] allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.

In higher dimensions it becomes interesting to study restriction problems for the Fourier transform. The Fourier transform of an integrable function is continuous and the restriction of this function to any set is defined. But for a square-integrable function the Fourier transform could be a general class of square integrable functions. As such, the restriction of the Fourier transform of an L²(Rⁿ) function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in L^p for 1 < p < 2. Surprisingly, it is possible in some cases to define the restriction of a Fourier transform to a set S, provided S has non-zero curvature. The case when S is the unit sphere in Rⁿ is of particular interest. In this case the Tomas–Stein restriction theorem states that the restriction of the Fourier transform to the unit sphere in Rⁿ is a bounded operator on L^p provided 1 ≤ p ≤ 2n + 2/n + 3.

One notable difference between the Fourier transform in 1 dimension versus higher dimensions concerns the partial sum operator. Consider an increasing collection of measurable sets E_R indexed by R ∈ (0,∞): such as balls of radius R centered at the origin, or cubes of side 2R. For a given integrable function f, consider the function f_R defined by:

${\displaystyle f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi )e^{2\pi ix\cdot \xi }\,d\xi ,\quad x\in \mathbb {R} ^{n}.}$

Suppose in addition that f ∈ L^p(Rⁿ). For n = 1 and 1 < p < ∞, if one takes E_R = (−R, R), then f_R converges to f in L^p as R tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for n > 1. In the case that E_R is taken to be a cube with side length R, then convergence still holds. Another natural candidate is the Euclidean ball E_R = {ξ : |ξ| < R}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in L^p(Rⁿ). For n ≥ 2 it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless p = 2.[19] In fact, when p ≠ 2, this shows that not only may f_R fail to converge to f in L^p, but for some functions f ∈ L^p(Rⁿ), f_R is not even an element of L^p.

The definition of the Fourier transform by the integral formula

${\displaystyle {\hat {f}}(\xi )=\int _{\mathbb {R} ^{n}}f(x)e^{-2\pi i\xi \cdot x}\,dx}$

is valid for Lebesgue integrable functions f; that is, f ∈ L¹(Rⁿ).

The Fourier transform F : L¹(Rⁿ) → L^∞(Rⁿ) is a bounded operator. This follows from the observation that

${\displaystyle \left\vert {\hat {f}}(\xi )\right\vert \leq \int _{\mathbb {R} ^{n}}\vert f(x)\vert \,dx,}$

which shows that its operator norm is bounded by 1. Indeed, it equals 1, which can be seen, for example, from the transform of the rect function. The image of L¹ is a subset of the space C₀(Rⁿ) of continuous functions that tend to zero at infinity (the Riemann–Lebesgue lemma), although it is not the entire space. Indeed, there is no simple characterization of the image.

Since compactly supported smooth functions are integrable and dense in L²(Rⁿ), the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in L²(Rⁿ) by continuity arguments. The Fourier transform in L²(Rⁿ) is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, here meaning that for an L² function f,

${\displaystyle {\hat {f}}(\xi )=\lim _{R\to \infty }\int _{|x|\leq R}f(x)e^{-2\pi ix\cdot \xi }\,dx}$

where the limit is taken in the L² sense. (More generally, you can take a sequence of functions that are in the intersection of L¹ and L² and that converges to f in the L²-norm, and define the Fourier transform of f as the L² -limit of the Fourier transforms of these functions.[41])

Many of the properties of the Fourier transform in L¹ carry over to L², by a suitable limiting argument.

Furthermore, F : L²(Rⁿ) → L²(Rⁿ) is a unitary operator.[42] For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product, so in this case these follow from the Fourier inversion theorem combined with the fact that for any f, g ∈ L²(Rⁿ) we have

${\displaystyle \int _{\mathbb {R} ^{n}}f(x){\mathcal {F}}g(x)\,dx=\int _{\mathbb {R} ^{n}}{\mathcal {F}}f(x)g(x)\,dx.}$

In particular, the image of L²(Rⁿ) is itself under the Fourier transform.

The definition of the Fourier transform can be extended to functions in L^p(Rⁿ) for 1 ≤ p ≤ 2 by decomposing such functions into a fat tail part in L² plus a fat body part in L¹. In each of these spaces, the Fourier transform of a function in L^p(Rⁿ) is in L^q(Rⁿ), where q = p/p − 1 is the Hölder conjugate of p (by the Hausdorff–Young inequality). However, except for p = 2, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in L^p for the range 2 < p < ∞ requires the study of distributions.[15] In fact, it can be shown that there are functions in L^p with p > 2 so that the Fourier transform is not defined as a function.[16]

One might consider enlarging the domain of the Fourier transform from L¹ + L² by considering generalized functions, or distributions. A distribution on Rⁿ is a continuous linear functional on the space C_c(Rⁿ) of compactly supported smooth functions, equipped with a suitable topology. The strategy is then to consider the action of the Fourier transform on C_c(Rⁿ) and pass to distributions by duality. The obstruction to doing this is that the Fourier transform does not map C_c(Rⁿ) to C_c(Rⁿ). In fact the Fourier transform of an element in C_c(Rⁿ) can not vanish on an open set; see the above discussion on the uncertainty principle. The right space here is the slightly larger space of Schwartz functions. The Fourier transform is an automorphism on the Schwartz space, as a topological vector space, and thus induces an automorphism on its dual, the space of tempered distributions.[16] The tempered distributions include all the integrable functions mentioned above, as well as well-behaved functions of polynomial growth and distributions of compact support.

For the definition of the Fourier transform of a tempered distribution, let f and g be integrable functions, and let f̂ and ĝ be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula,[16]

${\displaystyle \int _{\mathbb {R} ^{n}}{\hat {f}}(x)g(x)\,dx=\int _{\mathbb {R} ^{n}}f(x){\hat {g}}(x)\,dx.}$

Every integrable function f defines (induces) a distribution T_f by the relation

${\displaystyle T_{f}(\varphi )=\int _{\mathbb {R} ^{n}}f(x)\varphi (x)\,dx}$

for all Schwartz functions φ. So it makes sense to define Fourier transform T̂_f of T_f by

${\displaystyle {\hat {T}}_{f}(\varphi )=T_{f}\left({\hat {\varphi }}\right)}$

for all Schwartz functions φ. Extending this to all tempered distributions T gives the general definition of the Fourier transform.

Distributions can be differentiated and the above-mentioned compatibility of the Fourier transform with differentiation and convolution remains true for tempered distributions.

The Fourier transform of a finite Borel measure μ on Rⁿ is given by:[43]

${\displaystyle {\hat {\mu }}(\xi )=\int _{\mathbb {R} ^{n}}e^{-2\pi ix\cdot \xi }\,d\mu .}$

This transform continues to enjoy many of the properties of the Fourier transform of integrable functions. One notable difference is that the Riemann–Lebesgue lemma fails for measures.[15] In the case that dμ = f(x) dx, then the formula above reduces to the usual definition for the Fourier transform of f. In the case that μ is the probability distribution associated to a random variable X, the Fourier–Stieltjes transform is closely related to the characteristic function, but the typical conventions in probability theory take e^ixξ instead of e^−2πixξ.[14] In the case when the distribution has a probability density function this definition reduces to the Fourier transform applied to the probability density function, again with a different choice of constants.

The Fourier transform may be used to give a characterization of measures. Bochner's theorem characterizes which functions may arise as the Fourier–Stieltjes transform of a positive measure on the circle.[15]

Furthermore, the Dirac delta function, although not a function, is a finite Borel measure. Its Fourier transform is a constant function (whose specific value depends upon the form of the Fourier transform used).

The Kaniadakis κ-Fourier transform is a κ-deformation of the Fourier transform, associated with the Kaniadakis statistics, which is defined as:[44]

${\displaystyle {\cal {F}}_{\kappa }[f(x)](\omega )={1 \over {\sqrt {2\,\pi }}}\int \limits _{-\infty }\limits ^{+\infty }f(x)\,{\exp(-i\,x_{\{\kappa \}}\,\omega _{\{\kappa \}}) \over {\sqrt {1+\kappa ^{2}\,x^{2}}}}\,dx}$

where ${\displaystyle z_{\{\kappa \}}={\frac {1}{\kappa }}\,{\rm {arcsinh}}\,(\kappa \,z)}$ is a κ-number and ${\displaystyle 0\leq |\kappa |<1}$ is the entropic index linked with the Kaniadakis entropy.

The κ-Fourier transform is based on the κ-Fourier series,[45] in which the classical Fourier series and Fourier transform are particular case in the ${\displaystyle \kappa \rightarrow 0}$ limiting case. This transform imposes an asymptotically log-periodic behavior (or κ-deformed phase by deforminh ${\displaystyle \omega }$) and a damping factor following a wavelet-like behavior (${\displaystyle {\sqrt {1+\kappa ^{2}\,x^{2}}}}$).

The Fourier transform may be generalized to any locally compact abelian group. A locally compact abelian group is an abelian group that is at the same time a locally compact Hausdorff topological space so that the group operation is continuous. If G is a locally compact abelian group, it has a translation invariant measure μ, called Haar measure. For a locally compact abelian group G, the set of irreducible, i.e. one-dimensional, unitary representations are called its characters. With its natural group structure and the topology of pointwise convergence, the set of characters Ĝ is itself a locally compact abelian group, called the Pontryagin dual of G. For a function f in L¹(G), its Fourier transform is defined by[15]

${\displaystyle {\hat {f}}(\xi )=\int _{G}\xi (x)f(x)\,d\mu \qquad {\text{for any }}\xi \in {\hat {G}}.}$

The Riemann–Lebesgue lemma holds in this case; f̂(ξ) is a function vanishing at infinity on Ĝ.

The Fourier transform on T=R/Z is an example; here T is a locally compact abelian group, and the Haar measure μ on T can be thought of as the Lebesgue measure on [0,1). Consider the representation of T on the complex plane C that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since C is 1-dim) ${\displaystyle \{e_{k}:T\rightarrow GL_{1}(C)=C^{*}\mid k\in Z\}}$ where ${\displaystyle e_{k}(x)=e^{2\pi ikx}}$ for ${\displaystyle x\in T}$.

The character of such representation, that is the trace of ${\displaystyle e_{k}(x)}$ for each ${\displaystyle x\in T}$ and ${\displaystyle k\in Z}$, is ${\displaystyle e^{2\pi ikx}}$ itself. In the case of representation of finite group, the character table of the group G are rows of vectors such that each row is the character of one irreducible representation of G, and these vectors form an orthonormal basis of the space of class functions that map from G to C by Schur's lemma. Now the group T is no longer finite but still compact, and it preserves the orthonormality of character table. Each row of the table is the function ${\displaystyle e_{k}(x)}$ of ${\displaystyle x\in T,}$ and the inner product between two class functions (all functions being class functions since T is abelian) ${\displaystyle f,g\in L^{2}(T,d\mu )}$ is defined as ${\textstyle \langle f,g\rangle ={\frac {1}{|T|}}\int _{[0,1)}f(y){\overline {g}}(y)d\mu (y)}$ with the normalizing factor ${\displaystyle |T|=1}$. The sequence ${\displaystyle \{e_{k}\mid k\in Z\}}$ is an orthonormal basis of the space of class functions ${\displaystyle L^{2}(T,d\mu )}$.

For any representation V of a finite group G, ${\displaystyle \chi _{v}}$ can be expressed as the span ${\textstyle \sum _{i}\left\langle \chi _{v},\chi _{v_{i}}\right\rangle \chi _{v_{i}}}$ (${\displaystyle V_{i}}$ are the irreps of G), such that ${\textstyle \left\langle \chi _{v},\chi _{v_{i}}\right\rangle ={\frac {1}{|G|}}\sum _{g\in G}\chi _{v}(g){\overline {\chi }}_{v_{i}}(g)}$. Similarly for ${\displaystyle G=T}$ and ${\displaystyle f\in L^{2}(T,d\mu )}$, ${\textstyle f(x)=\sum _{k\in Z}{\hat {f}}(k)e_{k}}$. The Pontriagin dual ${\displaystyle {\hat {T}}}$ is ${\displaystyle \{e_{k}\}(k\in Z)}$ and for ${\displaystyle f\in L^{2}(T,d\mu )}$, ${\textstyle {\hat {f}}(k)={\frac {1}{|T|}}\int _{[0,1)}f(y)e^{-2\pi iky}dy}$ is its Fourier transform for ${\displaystyle e_{k}\in {\hat {T}}}$.

The Fourier transform is also a special case of Gelfand transform. In this particular context, it is closely related to the Pontryagin duality map defined above.

Given an abelian locally compact Hausdorff topological group G, as before we consider space L¹(G), defined using a Haar measure. With convolution as multiplication, L¹(G) is an abelian Banach algebra. It also has an involution * given by

${\displaystyle f^{*}(g)={\overline {f\left(g^{-1}\right)}}.}$