Fourier transform

Background

History

In 1822, Fourier claimed (see Joseph Fourier § The Analytic Theory of Heat) that any function, whether continuous or discontinuous, can be expanded into a series of sines. That important work was corrected and expanded upon by others to provide the foundation for the various forms of the Fourier transform used since.

Complex sinusoids

In general, the coefficients ${\displaystyle \hat{f}(\xi )}$ are complex numbers, which have two equivalent forms (see Euler's formula): $${\displaystyle \hat{f}(\xi )=\underset{\text{polar coordinate form}}{\underbrace{A{e}^{i\theta }}}=\underset{\text{rectangular coordinate form}}{\underbrace{A\cos (\theta )+iA\sin (\theta )}}.}$$

The product with ${\displaystyle e^{i2\pi \xi x}}$ (Eq.2) has these forms: $${\displaystyle \begin{array}{cc}\hat{f}(\xi )\cdot e^{i2\pi \xi x}& =A{e}^{i\theta }\cdot e^{i2\pi \xi x}\\ & =\underset{\text{polar coordinate form}}{\underbrace{A{e}^{i(2\pi \xi x+\theta )}}}\\ & =\underset{\text{rectangular coordinate form}}{\underbrace{A\cos (2\pi \xi x+\theta )+iA\sin (2\pi \xi x+\theta )}},\end{array}}$$ which conveys both amplitude and phase of frequency ⁠${\displaystyle \xi }$⁠. Likewise, the intuitive interpretation of Eq.1 is that multiplying ${\displaystyle f(x)}$ by ${\displaystyle e^{-i2\pi \xi x}}$ has the effect of subtracting ${\displaystyle \xi }$ from every frequency component of function ⁠${\displaystyle f(x)}$⁠. Only the component that was at frequency ${\displaystyle \xi }$ 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).

It is noteworthy how easily the product was simplified using the polar form, and how easily the rectangular form was deduced by an application of Euler's formula.

Negative frequency

Euler's formula introduces the possibility of negative ⁠${\displaystyle \xi }$⁠. Eq.1 is defined ⁠${\displaystyle \mathrm{\forall }\xi \in \mathbb{R}}$⁠. Only certain complex-valued ${\displaystyle f(x)}$ have transforms ⁠${\displaystyle \hat{f}=0,\mathrm{\forall }\xi <0}$⁠. (See Analytic signal; a simple example is ⁠${\displaystyle e^{i2\pi {\xi }_{0}x}({\xi }_0>0)}$⁠.) But negative frequency is necessary to characterize all other complex-valued ⁠${\displaystyle f(x)}$⁠, found in signal processing, partial differential equations, radar, nonlinear optics, quantum mechanics, and others.

For a real-valued ⁠${\displaystyle f(x)}$⁠, Eq.1 has the symmetry property ${\displaystyle \hat{f}(-\xi )={\hat{f}}^{\ast }(\xi )}$ (see § Conjugation below). This redundancy enables Eq.2 to distinguish ${\displaystyle f(x)=\cos (2\pi {\xi }_{0}x)}$ from ⁠${\displaystyle e^{i2\pi {\xi }_{0}x}}$⁠. But it cannot determine the actual sign of ⁠${\displaystyle {\xi }_0}$⁠, because ${\displaystyle \cos (2\pi {\xi }_{0}x)}$ and ${\displaystyle \cos (2\pi (-{\xi }_0)x)}$ are indistinguishable on just the real numbers line.

Fourier transform for periodic functions

The Fourier transform of a periodic function cannot be defined using the integral formula directly. In order for integral in Eq.1 to be defined the function must be absolutely integrable. Instead it is common to use Fourier series. It is possible to extend the definition to include periodic functions by viewing them as tempered distributions.

This makes it possible to see a connection between the Fourier series and the Fourier transform for periodic functions that have a convergent Fourier series. If ${\displaystyle f(x)}$ is a periodic function, with period ⁠${\displaystyle P}$⁠, that has a convergent Fourier series, then: $${\displaystyle \hat{f}(\xi )=\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}{c}_n\cdot \delta \left(\xi -\frac{n}{P}\right),}$$ where ${\displaystyle c_n}$ are the Fourier series coefficients of ⁠${\displaystyle f}$⁠, and ${\displaystyle \delta }$ is the Dirac delta function. In other words, the Fourier transform is a Dirac comb function whose teeth are multiplied by the Fourier series coefficients.

Sampling the Fourier transform

The Fourier transform of an integrable function ${\displaystyle f}$ can be sampled at regular intervals of arbitrary length ⁠${\displaystyle 1/P}$⁠. These samples can be deduced from one cycle of a periodic function ⁠${\displaystyle f_P}$⁠, which has Fourier series coefficients proportional to those samples by the Poisson summation formula: $${\displaystyle f_P(x)\triangleq \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}f(x+nP)=\frac{1}{P}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}\left(\frac{k}{P}\right)e^{i2\pi \frac{k}{P}x},\mathrm{\forall }k\in \mathbb{Z}.}$$

The integrability of ${\displaystyle f}$ ensures the periodic summation converges. Therefore, the samples ${\displaystyle \hat{f}(\frac{k}{P})}$ can be determined by Fourier series analysis: $${\displaystyle \hat{f}\left(\frac{k}{P}\right)={\int }_P{f}_P(x)\cdot e^{-i2\pi \frac{k}{P}x}dx.}$$

When ${\displaystyle f(x)}$ has compact support, ${\displaystyle f_P(x)}$ has a finite number of terms within the interval of integration. When ${\displaystyle f(x)}$ does not have compact support, numerical evaluation of ${\displaystyle f_P(x)}$ requires an approximation, such as tapering ${\displaystyle f(x)}$ or truncating the number of terms.

Units

The frequency variable must have inverse units to the units of the original function's domain (typically named ${\displaystyle t}$ or ⁠${\displaystyle x}$⁠). For example, if ${\displaystyle t}$ is measured in seconds, ${\displaystyle \xi }$ should be in cycles per second or hertz. If the scale of time is in units of ${\displaystyle 2\pi }$ seconds, then another Greek letter ${\displaystyle \omega }$ is typically used instead to represent angular frequency (where ⁠${\displaystyle \omega =2\pi \xi }$⁠) in units of radians per second. If using ${\displaystyle x}$ for units of length, then ${\displaystyle \xi }$ must be in inverse length, e.g., wavenumbers. That is to say, there are two versions of the real line: one that is the range of ${\displaystyle t}$ and measured in units of ⁠${\displaystyle t}$⁠, and the other that is the range of ${\displaystyle \xi }$ and measured in inverse units to the units of ⁠${\displaystyle t}$⁠. 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 that have a different domain of definition.

In general, ${\displaystyle \xi }$ 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 Linear algebra for a more formal explanation and for more details.) This point of view becomes essential in generalizations 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 that 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.

In other conventions, the Fourier transform has i in the exponent instead of −i, and vice versa for the inversion formula. This convention is common in modern physics and is the default for Wolfram Alpha, and does not mean that the frequency has become negative, since there is no canonical definition of positivity for frequency of a complex wave. It simply means that ${\displaystyle \hat{f}(\xi )}$ is the amplitude of the wave ⁠${\displaystyle e^{-i2\pi \xi x}}$⁠ instead of the wave ${\displaystyle e^{i2\pi \xi x}}$ (the former, with its minus sign, is often seen in the time dependence for sinusoidal plane-wave solutions of the electromagnetic wave equation, or in the time dependence for quantum wave functions). 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. In electrical engineering the letter j is typically used for the imaginary unit instead of i because i is used for current.

When using dimensionless units, the constant factors might not be written in the transform definition. For instance, in probability theory, the characteristic function Φ of the probability density function ⁠${\displaystyle f}$⁠ of a random variable ⁠${\displaystyle X}$⁠ of continuous type is defined without a negative sign in the exponential, and since the units of ⁠${\displaystyle x}$⁠ are ignored, there is no ⁠${\displaystyle 2\pi }$⁠ either: $${\displaystyle \phi (\lambda )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{i\lambda x}dx.}$$

In probability theory and mathematical statistics, the use of the Fourier—Stieltjes transform is preferred, because 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 that 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.

Properties

Let ${\displaystyle f(x)}$ and ${\displaystyle h(x)}$ represent integrable functions Lebesgue-measurable on the real line satisfying: $${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}|f(x)|dx<\mathrm{\infty }.}$$ We denote the Fourier transforms of these functions as ${\displaystyle \hat{f}(\xi )}$ and ${\displaystyle \hat{h}(\xi )}$ respectively.

Basic properties

The Fourier transform has the following basic properties:

Linearity

$${\displaystyle af(x)+bh(x)\stackrel{\mathcal{F}}{⟺}a\hat{f}(\xi )+b\hat{h}(\xi );a,b\in \mathbb{C}}$$

Time shifting

$${\displaystyle f(x-x_0)\stackrel{\mathcal{F}}{⟺}{e}^{-i2\pi x_0\xi }\hat{f}(\xi );x_0\in \mathbb{R}}$$

Frequency shifting

$${\displaystyle e^{i2\pi {\xi }_{0}x}f(x)\stackrel{\mathcal{F}}{⟺}\hat{f}(\xi -{\xi }_0);{\xi }_0\in \mathbb{R}}$$

Time scaling

$${\displaystyle f(ax)\stackrel{\mathcal{F}}{⟺}\frac{1}{|a|}\hat{f}\left(\frac{\xi }{a}\right);a\ne 0}$$ The case ${\displaystyle a=-1}$ leads to the time-reversal property: $${\displaystyle f(-x)\stackrel{\mathcal{F}}{⟺}\hat{f}(-\xi )}$$

Symmetry

When the real and imaginary parts of a complex function are decomposed into their even and odd parts, there are four components, denoted below by the subscripts RE, RO, IE, and IO. And there is a one-to-one mapping between the four components of a complex time function and the four components of its complex frequency transform:

${\displaystyle \begin{array}{cccccccccc}\mathsf{T}\mathsf{i}\mathsf{m}\mathsf{e}\mathsf{d}\mathsf{o}\mathsf{m}\mathsf{a}\mathsf{i}\mathsf{n}& f& =& f_{{}_{\text{RE}}}& +& f_{{}_{\text{RO}}}& +& i{f}_{{}_{\text{IE}}}& +& \underbrace{i{f}_{{}_{\text{IO}}}}\\ & \Updownarrow \mathcal{F}& & \Updownarrow \mathcal{F}& & \Updownarrow \mathcal{F}& & \Updownarrow \mathcal{F}& & \Updownarrow \mathcal{F}\\ \mathsf{F}\mathsf{r}\mathsf{e}\mathsf{q}\mathsf{u}\mathsf{e}\mathsf{n}\mathsf{c}\mathsf{y}\mathsf{d}\mathsf{o}\mathsf{m}\mathsf{a}\mathsf{i}\mathsf{n}& \hat{f}& =& \hat{f}{}_{{}_{\text{RE}}}& +& \overbrace{i\hat{f}{}_{{}_{\text{IO}}}}& +& i\hat{f}{}_{{}_{\text{IE}}}& +& \hat{f}{}_{{}_{\text{RO}}}\end{array}}$

From this, various relationships are apparent, for example:

• The transform of a real-valued function (⁠${\displaystyle f_{{}_{\text{RE}}}+f_{{}_{\text{RO}}}}$⁠) is the conjugate symmetric function ⁠${\displaystyle \hat{f}{}_{{}_{\text{RE}}}+i\hat{f}{}_{{}_{\text{IO}}}}$⁠. Conversely, a conjugate symmetric transform implies a real-valued time-domain.
• The transform of an imaginary-valued function (⁠${\displaystyle i{f}_{{}_{\text{IE}}}+i{f}_{{}_{\text{IO}}}}$⁠) is the conjugate antisymmetric function ⁠${\displaystyle \hat{f}{}_{{}_{\text{RO}}}+i\hat{f}{}_{{}_{\text{IE}}}}$⁠, and the converse is true.
• The transform of a conjugate symmetric function ${\displaystyle (f_{{}_{\text{RE}}}+i{f}_{{}_{\text{IO}}})}$ is the real-valued function ⁠${\displaystyle \hat{f}{}_{{}_{\text{RE}}}+\hat{f}{}_{{}_{\text{RO}}}}$⁠, and the converse is true.
• The transform of a conjugate antisymmetric function ${\displaystyle (f_{{}_{\text{RO}}}+i{f}_{{}_{\text{IE}}})}$ is the imaginary-valued function ⁠${\displaystyle i\hat{f}{}_{{}_{\text{IE}}}+i\hat{f}{}_{{}_{\text{IO}}}}$⁠, and the converse is true.

Conjugation

$${\displaystyle (f(x){)}^{\ast }\stackrel{\mathcal{F}}{⟺}{\left(\hat{f}(-\xi )\right)}^{\ast }}$$ (Note: the ⁠${\displaystyle \ast }$⁠ denotes complex conjugation.)

In particular, if ${\displaystyle f}$ is real, then ${\displaystyle \hat{f}}$ is conjugate symmetric (a.k.a. Hermitian function): $${\displaystyle \hat{f}(-\xi )=(\hat{f}(\xi ){)}^{\ast }.}$$

If ${\displaystyle f}$ is purely imaginary, then ${\displaystyle \hat{f}}$ is odd symmetric: $${\displaystyle \hat{f}(-\xi )=-(\hat{f}(\xi ){)}^{\ast }.}$$

Real and imaginary parts

$${\displaystyle \mathrm{Re}\{f(x)\}\stackrel{\mathcal{F}}{⟺}\frac{1}{2}\left(\hat{f}(\xi )+(\hat{f}(-\xi ){)}^{\ast }\right)}$$ $${\displaystyle \mathrm{Im}\{f(x)\}\stackrel{\mathcal{F}}{⟺}\frac{1}{2i}\left(\hat{f}(\xi )-(\hat{f}(-\xi ){)}^{\ast }\right)}$$

Zero frequency component

Substituting ${\displaystyle \xi =0}$ in the definition, we obtain: $${\displaystyle \hat{f}(0)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)dx.}$$

The integral of ${\displaystyle f}$ over its domain is the total mass or DC bias of the function.

Uniform continuity and the Riemann–Lebesgue lemma

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 ${\displaystyle \hat{f}}$ of any integrable function ${\displaystyle f}$ is uniformly continuous and $${\displaystyle {\Vert \hat{f}\Vert }_{\mathrm{\infty }}\le {\Vert f\Vert }_1}$$

By the Riemann–Lebesgue lemma, $${\displaystyle \hat{f}(\xi )\to 0\text{ as }|\xi |\to \mathrm{\infty }.}$$

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 ${\displaystyle f}$ and ${\displaystyle \hat{f}}$ are integrable, the inverse equality $${\displaystyle f(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\xi )e^{i2\pi x\xi }d\xi }$$ holds for almost every x. As a result, the Fourier transform is injective on L¹(R).

Plancherel theorem and Parseval's theorem

Let ⁠${\displaystyle f(x)}$⁠ and ⁠${\displaystyle g(x)}$⁠ be integrable, and let ⁠${\displaystyle \hat{f}}$⁠ and ⁠${\displaystyle \hat{g}}$⁠ be their Fourier transforms. If ⁠${\displaystyle f(x)}$⁠ and ⁠${\displaystyle g(x)}$⁠ are also square-integrable, then the Parseval formula follows: $${\displaystyle ⟨f,g{⟩}_{L^2}={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)\overline{g(x)}dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\xi )\overline{\hat{g}(\xi )}d\xi ,}$$ where the bar denotes complex conjugation.

The Plancherel theorem, which follows from the above, states that $${\displaystyle \Vert f{\Vert }_{L^2}^2={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|f(x)\right|}^{2}dx={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{\left|\hat{f}(\xi )\right|}^{2}d\xi .}$$

Plancherel's theorem makes it possible to extend the Fourier transform, by a continuity argument, to a unitary operator on ⁠${\displaystyle L^2(\mathbb{R})}$⁠. On ⁠${\displaystyle L^1(\mathbb{R})\cap L^2(\mathbb{R})}$⁠, this extension agrees with original Fourier transform defined on ⁠${\displaystyle L^1(\mathbb{R})}$⁠, thus enlarging the domain of the Fourier transform to ⁠${\displaystyle L^1(\mathbb{R})+L^2(\mathbb{R})}$⁠ (and consequently to ⁠${\displaystyle L^p(\mathbb{R})}$⁠ for ⁠${\displaystyle 1\le p\le 2}$⁠). 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.

Convolution theorem

The Fourier transform translates between convolution and multiplication of functions. If ⁠${\displaystyle f(x)}$⁠ and ⁠${\displaystyle g(x)}$⁠ are integrable functions with Fourier transforms ⁠${\displaystyle \hat{f}}$⁠ and ⁠${\displaystyle \hat{g}(\xi )}$⁠ respectively, then the Fourier transform of the convolution is given by the product of the Fourier transforms ⁠${\displaystyle \hat{f}}$⁠ and ⁠${\displaystyle \hat{g}}$⁠ (under other conventions for the definition of the Fourier transform a constant factor may appear).

This means that if: $${\displaystyle h(x)=(f\ast g)(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(y)g(x-y)dy,}$$ where ∗ denotes the convolution operation, then: $${\displaystyle \hat{h}(\xi )=\hat{f}(\xi )\hat{g}(\xi ).}$$

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

Conversely, if ⁠${\displaystyle f(x)}$⁠ can be decomposed as the product of two square integrable functions ⁠${\displaystyle p(x)}$⁠ and ⁠${\displaystyle q(x)}$⁠, then the Fourier transform of ⁠${\displaystyle f(x)}$⁠ is given by the convolution of the respective Fourier transforms ⁠${\displaystyle \hat{p}(\xi )}$⁠ and ⁠${\displaystyle \hat{q}(\xi )}$⁠.

Cross-correlation theorem

In an analogous manner, it can be shown that if ⁠${\displaystyle h(x)}$⁠ is the cross-correlation of ⁠${\displaystyle f(x)}$⁠ and ⁠${\displaystyle g(x)}$⁠: $${\displaystyle h(x)=(f\star g)(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\overline{f(y)}g(x+y)dy}$$ then the Fourier transform of ⁠${\displaystyle h(x)}$⁠ is: $${\displaystyle \hat{h}(\xi )=\overline{\hat{f}(\xi )}\hat{g}(\xi ).}$$

As a special case, the autocorrelation of function ⁠${\displaystyle f(x)}$⁠ is: $${\displaystyle h(x)=(f\star f)(x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\overline{f(y)}f(x+y)dy}$$ for which $${\displaystyle \hat{h}(\xi )=\overline{\hat{f}(\xi )}\hat{f}(\xi )={\left|\hat{f}(\xi )\right|}^2.}$$

Differentiation

Suppose f(x) is differentiable almost everywhere, and both ⁠${\displaystyle f}$⁠ and its derivative ⁠${\displaystyle f^{\prime }}$⁠ are integrable (in ⁠${\displaystyle L^1(\mathbb{R})}$⁠). Then the Fourier transform of the derivative is given by $${\displaystyle \widehat{f^{\prime }}(\xi )=\mathcal{F}\left\{\frac{d}{dx}f(x)\right\}=i2\pi \xi \hat{f}(\xi ).}$$ More generally, the Fourier transformation of the ⁠${\displaystyle n}$⁠th derivative ⁠${\displaystyle f^{(n)}}$⁠ is given by $${\displaystyle \widehat{f^{(n)}}(\xi )=\mathcal{F}\left\{\frac{d^n}{d{x}^n}f(x)\right\}=(i2\pi \xi {)}^n\hat{f}(\xi ).}$$

Analogously, ⁠${\displaystyle {\mathcal{F}}^{-1}\left\{\frac{d^n}{d{\xi }^n}\hat{f}(\xi )\right\}=(-i2\pi x{)}^{n}f(x)}$⁠, so ⁠${\displaystyle \mathcal{F}\left\{x^{n}f(x)\right\}={\left(\frac{i}{2\pi }\right)}^n\frac{d^n}{d{\xi }^n}\hat{f}(\xi )}$⁠.

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 "⁠${\displaystyle f(x)}$⁠ is smooth if and only if ⁠${\displaystyle \hat{f}(\xi )}$⁠ quickly falls to ⁠${\displaystyle 0}$⁠ for ⁠${\displaystyle |\xi |\to \mathrm{\infty }}$⁠". By using the analogous rules for the inverse Fourier transform, one can also say "⁠${\displaystyle f(x)}$⁠ quickly falls to ⁠${\displaystyle 0}$⁠ for ⁠${\displaystyle |x|\to \mathrm{\infty }}$⁠ if and only if ⁠${\displaystyle \hat{f}(\xi )}$⁠ is smooth".

Eigenfunctions

The Fourier transform is a linear transform that has eigenfunctions obeying ⁠${\displaystyle \mathcal{F}[\psi ]=\lambda \psi }$⁠, with ⁠${\displaystyle \lambda \in \mathbb{C}}$⁠.

A set of eigenfunctions is found by noting that the homogeneous differential equation $${\displaystyle \left[U\left(\frac{1}{2\pi }\frac{d}{dx}\right)+U(x)\right]\psi (x)=0}$$ leads to eigenfunctions ${\displaystyle \psi (x)}$ of the Fourier transform ${\displaystyle \mathcal{F}}$ as long as the form of the equation remains invariant under Fourier transform. In other words, every solution ${\displaystyle \psi (x)}$ and its Fourier transform ${\displaystyle \hat{\psi }(\xi )}$ obey the same equation. Assuming uniqueness of the solutions, every solution ${\displaystyle \psi (x)}$ must therefore be an eigenfunction of the Fourier transform. The form of the equation remains unchanged under Fourier transform if ${\displaystyle U(x)}$ can be expanded in a power series in which for all terms the same factor of either one of ⁠${\displaystyle \pm 1}$⁠, ⁠${\displaystyle \pm i}$⁠ arises from the factors ${\displaystyle i^n}$ introduced by the differentiation rules upon Fourier transforming the homogeneous differential equation because this factor may then be cancelled. The simplest allowable ${\displaystyle U(x)=x}$ leads to the standard normal distribution.

More generally, a set of eigenfunctions is also found by noting that the differentiation rules imply that the ordinary differential equation $${\displaystyle \left[W\left(\frac{i}{2\pi }\frac{d}{dx}\right)+W(x)\right]\psi (x)=C\psi (x)}$$ with ${\displaystyle C}$ constant and ${\displaystyle W(x)}$ being a non-constant even function remains invariant in form when applying the Fourier transform ${\displaystyle \mathcal{F}}$ to both sides of the equation. The simplest example is provided by ⁠${\displaystyle W(x)=x^2}$⁠, which is equivalent to considering the Schrödinger equation for the quantum harmonic oscillator. The corresponding solutions provide an important choice of an orthonormal basis for L²(R) and are given by the "physicist's" Hermite functions. Equivalently one may use $${\displaystyle {\psi }_n(x)=\frac{\sqrt[4]{2}}{\sqrt{n!}}{e}^{-\pi x^2}{\mathrm{H}\mathrm{e}}_n\left(2x\sqrt{\pi }\right),}$$ where ⁠${\displaystyle {\mathrm{H}\mathrm{e}}_n(x)}$⁠ are the "probabilist's" Hermite polynomials, defined as $${\displaystyle {\mathrm{H}\mathrm{e}}_n(x)=(-1{)}^n{e}^{\frac{1}{2}{x}^2}{\left(\frac{d}{dx}\right)}^n{e}^{-\frac{1}{2}{x}^2}.}$$

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 ⁠${\displaystyle L^2(\mathbb{R})}$⁠. However, this choice of eigenfunctions is not unique. Because of ${\displaystyle {\mathcal{F}}^4=\mathrm{i}\mathrm{d}}$ there are only four different eigenvalues of the Fourier transform (the fourth roots of unity ⁠${\displaystyle \pm 1}$⁠ and ⁠${\displaystyle \pm i}$⁠) 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 H_k simply by multiplication by i^k.

Since the complete set of Hermite functions ψ_n provides a resolution of the identity they diagonalize the Fourier operator, i.e. the Fourier transform can be represented by such a sum of terms weighted by the above eigenvalues, and these sums can be explicitly summed: $${\displaystyle \mathcal{F}[f](\xi )=\int dxf(x)\sum _{n\ge 0}(-i{)}^n{\psi }_n(x){\psi }_n(\xi ).}$$

This approach to define the Fourier transform was first proposed by Norbert Wiener. 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. In physics, this transform was introduced by Edward Condon. This change of basis becomes possible because the Fourier transform is a unitary transform when using the right conventions. Consequently, under the proper conditions it may be expected to result from a self-adjoint generator ${\displaystyle N}$ via $${\displaystyle \mathcal{F}[\psi ]=e^{-itN}\psi .}$$

The operator ${\displaystyle N}$ is the number operator of the quantum harmonic oscillator written as $${\displaystyle N\equiv \frac{1}{4\pi }\left(2\pi x-\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right)\left(2\pi x+\frac{\mathrm{\partial }}{\mathrm{\partial }x}\right)=-\frac{1}{4\pi }\frac{{\mathrm{\partial }}^2}{\mathrm{\partial }{x}^2}+\pi x^2-\frac{1}{2}.}$$

It can be interpreted as the generator of fractional Fourier transforms for arbitrary values of t, and of the conventional continuous Fourier transform ${\displaystyle \mathcal{F}}$ for the particular value ⁠${\displaystyle t=\pi /2}$⁠, with the Mehler kernel implementing the corresponding active transform. The eigenfunctions of ${\displaystyle N}$ are the Hermite functions ⁠${\displaystyle {\psi }_n(x)}$⁠, which are therefore also eigenfunctions of ⁠${\displaystyle \mathcal{F}}$⁠.

Upon extending the Fourier transform to distributions the Dirac comb is also an eigenfunction of the Fourier transform.

Inversion and periodicity

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: ⁠${\displaystyle \left({\mathcal{F}}^{2}f\right)(x)=f(-x)}$⁠, 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: ⁠${\displaystyle {\mathcal{F}}^3\left(\hat{f}\right)=f}$⁠. 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: $${\displaystyle \begin{array}{cc}{\mathcal{F}}^0& =\mathrm{i}\mathrm{d},\\ {\mathcal{F}}^1& =\mathcal{F},\\ {\mathcal{F}}^2& =\mathcal{P},\\ {\mathcal{F}}^3& ={\mathcal{F}}^{-1}=\mathcal{P}\circ \mathcal{F}=\mathcal{F}\circ \mathcal{P},\\ {\mathcal{F}}^4& =\mathrm{i}\mathrm{d}\end{array}}$$ 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 ⁠${\displaystyle x}$⁠-axis and frequency as the ⁠${\displaystyle y}$⁠-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.

Connection with the Heisenberg group

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^i2πξx, (M_ξ f)(x) = e^i2πξx f (x). These operators do not commute, as their (group) commutator is $${\displaystyle \left(M_{\xi }^{-1}{T}_y^{-1}{M}_{\xi }{T}_{y}f\right)(x)=e^{i2\pi \xi y}f(x),}$$ which is multiplication by the constant (independent of x) e^i2πξy ∈ 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 $${\displaystyle \left(x_1,{\xi }_1,t_1\right)\cdot \left(x_2,{\xi }_2,t_2\right)=\left(x_1+x_2,{\xi }_1+{\xi }_2,t_1{t}_2{e}^{-2i\pi x_1{\xi }_2}\right).}$$

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 $${\displaystyle J\left(\begin{array}{c}x\\ \xi \end{array}\right)=\left(\begin{array}{c}-\xi \\ x\end{array}\right)}$$ so that J² = −I. This J can be extended to a unique automorphism of H₁: $${\displaystyle j\left(x,\xi ,t\right)=\left(-\xi ,x,t{e}^{-i2\pi \xi x}\right).}$$

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 $${\displaystyle \rho \circ j=W\rho W^{\ast }.}$$ This operator W is the Fourier transform.

Many of the standard properties of the Fourier transform are immediate consequences of this more general framework. 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.

Complex domain

The integral for the Fourier transform $${\displaystyle \hat{f}(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-2i\pi t\xi }f(t)dt}$$ can be studied for complex values of its argument ξ. Depending on the properties of f, this might not converge off the real axis at all, or it might converge to a complex analytic function for all values of ξ = σ + iτ, or something in between.

The Paley–Wiener theorem says that f is smooth (i.e., n-times differentiable for all positive integers n) and compactly supported if and only if f̂ (σ + iτ) is a holomorphic function for which there exists a constant a > 0 such that for any integer n ≥ 0, $${\displaystyle \left|{\xi }^n\hat{f}(\xi )\right|\le C_n{e}^{2\pi a|\tau |}}$$ for some constant C_n. (In this case, f is supported on [−a, a].) This can be expressed by saying that f̂ is an entire function that is rapidly decreasing in σ (for fixed τ) and of exponential growth in τ (uniformly in σ).

(If f is not smooth, but only L², a corresponding version holds with the rapid-decrease condition replaced by the appropriate L² condition.) The space of such functions of a complex variable is called the Paley–Wiener space. This theorem has been generalised to semisimple Lie groups.

If f is supported on the half-line t ≥ 0, then f is said to be "causal" because the impulse response function of a physically realisable filter must have this property, as no effect can precede its cause. Paley and Wiener showed that then, under suitable integrability hypotheses, f̂ extends to a holomorphic function on the complex lower half-plane τ < 0 that tends to zero as τ goes to −∞. A simple converse in this form is false; precise converses require additional growth or Hardy-space hypotheses.

Laplace transform

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 is causal and of exponential growth, i.e., $${\displaystyle f(t)=0(t<0),|f(t)|<C{e}^{at}(t\ge 0)}$$ for some constants C, a ≥ 0, then $${\displaystyle \hat{f}(i\tau )={\int }_0^{\mathrm{\infty }}{e}^{2\pi \tau t}f(t)dt,}$$ convergent for all 2πτ < −a, is a one-sided Laplace transform of f.

The usual one-sided version of the Laplace transform is $${\displaystyle F(s)={\int }_0^{\mathrm{\infty }}f(t)e^{-st}dt.}$$

If f is causal and the integrals converge, 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 = i2πξ.

From another, perhaps more classical viewpoint, the Laplace transform by its form involves an additional exponential regulating term that lets it converge outside of the imaginary line where the Fourier transform is defined. As such it can converge for functions and integrals with at most exponential growth in the regulated direction, 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 that 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 compatible 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 of Fourier methods. Both of them are subsumed by the far more general, and more abstract, idea of harmonic analysis.

Inversion

Still with ⁠${\displaystyle \xi =\sigma +i\tau }$⁠, if ${\displaystyle \hat{f}}$ is complex analytic for a ≤ τ ≤ b and has sufficient decay in horizontal strips, then $${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\sigma +ia)e^{i2\pi (\sigma +ia)t}d\sigma ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\sigma +ib)e^{i2\pi (\sigma +ib)t}d\sigma }$$ by Cauchy's integral theorem. Therefore, the Fourier inversion formula can use integration along different lines, parallel to the real axis.

Theorem: If f(t) = 0 for t < 0, and |f(t)| < Ce^at for some constants C, a > 0 and t ≥ 0, then $${\displaystyle f(t)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\hat{f}(\sigma +i\tau )e^{i2\pi (\sigma +i\tau )t}d\sigma ,}$$ for any τ < −⁠a/2π⁠, under the usual hypotheses for Fourier inversion.

This theorem implies the Mellin inversion formula for the Laplace transformation, $${\displaystyle f(t)=\frac{1}{i2\pi }{\int }_{b-i\mathrm{\infty }}^{b+i\mathrm{\infty }}F(s)e^{st}ds}$$ for any b > a, where F(s) is the Laplace transform of f(t).

The hypotheses can be weakened, as in standard Fourier inversion results, to f(t) e^−at being L¹, provided that f be of bounded variation in a closed neighborhood of t (cf. Dini test), the value of f at t be taken to be the arithmetic mean of the left and right limits, and that the integrals be taken in the sense of Cauchy principal values.

L² versions of these inversion formulas are also available.

Fourier transform on Euclidean space

The Fourier transform can be defined in any arbitrary number of dimensions n. As with the one-dimensional case, there are many conventions. For an integrable function f(x), this article takes the definition: $${\displaystyle \hat{f}(\mathit{\xi })=\mathcal{F}(f)(\mathit{\xi })={\int }_{{\mathbb{R}}^n}f(\mathbf{x})e^{-i2\pi \mathit{\xi }\cdot \mathbf{x}}d\mathbf{x}}$$ where x and ξ are n-dimensional vectors, and x · ξ is the dot product of the vectors. Alternatively, ξ can be viewed as belonging to the dual vector space ⁠${\displaystyle {\mathbb{R}}^{n\star }}$⁠, in which case the dot product becomes the contraction of x and ξ, usually written as ⟨x, ξ⟩.

All of the basic properties listed above hold for the n-dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds.

Uncertainty principle

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

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 defined by $${\displaystyle D_0(f)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{x}^2|f(x){|}^{2}dx.}$$

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 $${\displaystyle D_0(f)D_0(\hat{f})\ge \frac{1}{16{\pi }^2}.}$$

The equality is attained only in the case $${\displaystyle \begin{array}{cc}f(x)& =C_1{e}^{-\pi \frac{x^2}{{\sigma }^2}}\\ \therefore \hat{f}(\xi )& =\sigma C_1{e}^{-\pi {\sigma }^2{\xi }^2}\end{array}}$$ where σ > 0 is arbitrary and C₁ = ⁠⁴√2/√σ⁠ so that f is L²-normalized. In other words, where f is a (normalized) Gaussian function with variance σ²/2π, centered at zero, and its Fourier transform is a Gaussian function with variance σ⁻²/2π. Gaussian functions are examples of Schwartz functions (see the discussion on tempered distributions below).

In fact, this inequality implies that: $${\displaystyle \left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(x-x_0{)}^2|f(x){|}^{2}dx\right)\left({\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(\xi -{\xi }_0{)}^2{\left|\hat{f}(\xi )\right|}^{2}d\xi \right)\ge \frac{1}{16{\pi }^2},\mathrm{\forall }{x}_0,{\xi }_0\in \mathbb{R}.}$$ In quantum mechanics, the momentum and position wave functions are Fourier transform pairs, up to a factor of the Planck constant. With this constant properly taken into account, the inequality above becomes the statement of the Heisenberg uncertainty principle.

A stronger uncertainty principle is the Hirschman uncertainty principle, which is expressed as: $${\displaystyle H\left({\left|f\right|}^2\right)+H\left({\left|\hat{f}\right|}^2\right)\ge \log \left(\frac{e}{2}\right)}$$ where H(p) is the differential entropy of the probability density function p(x): $${\displaystyle H(p)=-{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}p(x)\log (p(x))dx}$$ where the logarithms may be in any base that is consistent. The equality is attained for a Gaussian, as in the previous case.

Sine and cosine transforms

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) λ by $${\displaystyle f(t)={\int }_0^{\mathrm{\infty }}(a(\lambda )\cos (2\pi \lambda t)+b(\lambda )\sin (2\pi \lambda t))d\lambda .}$$

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): $${\displaystyle a(\lambda )=2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\cos (2\pi \lambda t)dt}$$ and $${\displaystyle b(\lambda )=2{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(t)\sin (2\pi \lambda t)dt.}$$

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 $${\displaystyle f(t)=2{\int }_0^{\mathrm{\infty }}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(\tau )\cos (2\pi \lambda (\tau -t))d\tau d\lambda .}$$ together with trigonometric identities. This is referred to as Fourier's integral formula.

Spherical harmonics

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 ⁠${\displaystyle \hat{f}(\xi )=i^{-k}f(\xi )}$⁠. 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 it is possible to characterize the action of the Fourier transform on each space H_k.

Let f(x) = f₀(|x|)P(x) (with P(x) in A_k), then $${\displaystyle \hat{f}(\xi )=F_0(|\xi |)P(\xi )}$$ where $${\displaystyle F_0(r)=2\pi i^{-k}{r}^{-\frac{n+2k-2}{2}}{\int }_0^{\mathrm{\infty }}{f}_0(s)J_{\frac{n+2k-2}{2}}(2\pi rs)s^{\frac{n+2k}{2}}ds.}$$

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. This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases n + 2 and n allowing to compute, e.g., the three-dimensional Fourier transform of a radial function from the one-dimensional one.

Restriction problems

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. 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: $${\displaystyle f_R(x)={\int }_{E_R}\hat{f}(\xi )e^{i2\pi x\cdot \xi }d\xi ,x\in {\mathbb{R}}^n.}$$

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. 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.

Fourier transform on function spaces

The definition of the Fourier transform naturally extends from ${\displaystyle L^1(\mathbb{R})}$ to ⁠${\displaystyle L^1({\mathbb{R}}^n)}$⁠. That is, if ${\displaystyle f\in L^1({\mathbb{R}}^n)}$ then the Fourier transform ${\displaystyle \mathcal{F}:L^1({\mathbb{R}}^n)\to L^{\mathrm{\infty }}({\mathbb{R}}^n)}$ is given by $${\displaystyle f(x)\mapsto \hat{f}(\xi )={\int }_{{\mathbb{R}}^n}f(x)e^{-i2\pi \xi \cdot x}dx,\mathrm{\forall }\xi \in {\mathbb{R}}^n.}$$ This operator is bounded as $${\displaystyle \underset{\xi \in {\mathbb{R}}^n}{sup}\left|\hat{f}(\xi )\right|\le {\int }_{{\mathbb{R}}^n}|f(x)|dx,}$$ which shows that its operator norm is bounded by 1. The Riemann–Lebesgue lemma shows that if ${\displaystyle f\in L^1({\mathbb{R}}^n)}$ then its Fourier transform actually belongs to the space of continuous functions that vanish at infinity, i.e., ⁠${\displaystyle \hat{f}\in C_0({\mathbb{R}}^n)\subset L^{\mathrm{\infty }}({\mathbb{R}}^n)}$⁠. Furthermore, the image of ${\displaystyle L^1}$ under ${\displaystyle \mathcal{F}}$ is a strict subset of ⁠${\displaystyle C_0({\mathbb{R}}^n)}$⁠.

Similarly to the case of one variable, the Fourier transform can be defined on ⁠${\displaystyle L^2({\mathbb{R}}^n)}$⁠. The Fourier transform in ${\displaystyle L^2({\mathbb{R}}^n)}$ is no longer given by an ordinary Lebesgue integral, although it can be computed by an improper integral, i.e., $${\displaystyle \hat{f}(\xi )=\underset{R\to \mathrm{\infty }}{lim}{\int }_{|x|\le R}f(x)e^{-i2\pi \xi \cdot x}dx}$$ where the limit is taken in the L² sense.

Furthermore, ${\displaystyle \mathcal{F}:L^2({\mathbb{R}}^n)\to L^2({\mathbb{R}}^n)}$ is a unitary operator. For an operator to be unitary it is sufficient to show that it is bijective and preserves the inner product. The Fourier inversion theorem implies that the transform is bijective. Also, for any f, g ∈ L²(Rⁿ) we have $${\displaystyle {\int }_{{\mathbb{R}}^n}f(x)\mathcal{F}g(x)dx={\int }_{{\mathbb{R}}^n}\mathcal{F}f(x)g(x)dx.}$$ So $${\displaystyle \begin{array}{cc}{\int }_{{\mathbb{R}}^n}\overline{\mathcal{F}f(x)}\mathcal{F}g(x)dx& ={\int }_{{\mathbb{R}}^n}{\mathcal{F}}^{-1}\overline{f(x)}\mathcal{F}g(x)dx\\ & ={\int }_{{\mathbb{R}}^n}\mathcal{F}{\mathcal{F}}^{-1}\overline{f(x)}g(x)dx={\int }_{{\mathbb{R}}^n}\overline{f(x)}g(x)dx\end{array}}$$

So the transform preserves the inner product.

On other Lp

For ⁠${\displaystyle 1<p<2}$⁠, the Fourier transform can be defined on ${\displaystyle L^p(\mathbb{R})}$ by Riesz–Thorin interpolation, which amounts to decomposing such functions into a fat tail part ${\displaystyle |f|\le 1}$ in L² plus a fat body part ${\displaystyle |f|>1}$ 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. 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.

Tempered distributions

One might consider enlarging the domain of the Fourier transform from ${\displaystyle L^1+L^2}$ by considering generalized functions, or distributions. A distribution on ${\displaystyle {\mathbb{R}}^n}$ is a continuous linear functional on the space ${\displaystyle C_c^{\mathrm{\infty }}({\mathbb{R}}^n)}$ of compactly supported smooth functions (i.e. bump functions), equipped with a suitable topology. Since ${\displaystyle C_c^{\mathrm{\infty }}({\mathbb{R}}^n)}$ is dense in ⁠${\displaystyle L^2({\mathbb{R}}^n)}$⁠, the Plancherel theorem allows one to extend the definition of the Fourier transform to general functions in ${\displaystyle L^2({\mathbb{R}}^n)}$ by continuity arguments. The strategy is then to consider the action of the Fourier transform on ${\displaystyle C_c^{\mathrm{\infty }}({\mathbb{R}}^n)}$ and pass to distributions by duality. The obstruction to doing this is that the Fourier transform does not map ${\displaystyle C_c^{\mathrm{\infty }}({\mathbb{R}}^n)}$ to ⁠${\displaystyle C_c^{\mathrm{\infty }}({\mathbb{R}}^n)}$⁠. In fact the Fourier transform of an element in ${\displaystyle C_c^{\mathrm{\infty }}({\mathbb{R}}^n)}$ can not vanish on an open set; see the above discussion on the uncertainty principle.

The Fourier transform can also be defined for tempered distributions ⁠${\displaystyle {\mathcal{S}}^{\prime }({\mathbb{R}}^n)}$⁠, dual to the space of Schwartz functions ⁠${\displaystyle \mathcal{S}({\mathbb{R}}^n)}$⁠. A Schwartz function is a smooth function that decays at infinity, along with all of its derivatives, hence ${\displaystyle C_c^{\mathrm{\infty }}({\mathbb{R}}^n)\subset \mathcal{S}({\mathbb{R}}^n)}$ and: $${\displaystyle \mathcal{F}:C_c^{\mathrm{\infty }}({\mathbb{R}}^n)\to \mathcal{S}({\mathbb{R}}^n)\setminus C_c^{\mathrm{\infty }}({\mathbb{R}}^n).}$$ The Fourier transform is an automorphism of the Schwartz space and, by duality, also an automorphism of the space of tempered distributions. The tempered distributions include well-behaved functions of polynomial growth, distributions of compact support as well as all the integrable functions mentioned above.

For the definition of the Fourier transform of a tempered distribution, let ${\displaystyle f}$ and ${\displaystyle g}$ be integrable functions, and let ${\displaystyle \hat{f}}$ and ${\displaystyle \hat{g}}$ be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula, $${\displaystyle {\int }_{{\mathbb{R}}^n}\hat{f}(x)g(x)dx={\int }_{{\mathbb{R}}^n}f(x)\hat{g}(x)dx.}$$

Every integrable function ${\displaystyle f}$ defines (induces) a distribution ${\displaystyle T_f}$ by the relation $${\displaystyle T_f(\phi )={\int }_{{\mathbb{R}}^n}f(x)\phi (x)dx,\mathrm{\forall }\phi \in \mathcal{S}({\mathbb{R}}^n).}$$ So it makes sense to define the Fourier transform of a tempered distribution ${\displaystyle T_f\in {\mathcal{S}}^{\prime }(\mathbb{R})}$ by the duality: $${\displaystyle ⟨{\hat{T}}_f,\phi ⟩=⟨T_f,\hat{\phi }⟩,\mathrm{\forall }\phi \in \mathcal{S}({\mathbb{R}}^n).}$$ Extending this to all tempered distributions ${\displaystyle T}$ 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.

Generalizations

Fourier–Stieltjes transform on measurable spaces

The Fourier transform of a finite Borel measure μ on Rⁿ, given by the bounded, uniformly continuous function: $${\displaystyle \hat{\mu }(\xi )={\int }_{{\mathbb{R}}^n}{e}^{-i2\pi x\cdot \xi }d\mu ,}$$ is called the Fourier–Stieltjes transform due to its connection with the Riemann-Stieltjes integral representation of (Radon) measures. If ${\displaystyle \mu }$ is the probability distribution of a random variable ${\displaystyle X}$ then its Fourier–Stieltjes transform is, by definition, a characteristic function. If, in addition, the probability distribution has a probability density function, this definition is subject to the usual Fourier transform. Stated more generally, when ${\displaystyle \mu }$ is absolutely continuous with respect to the Lebesgue measure, i.e., $${\displaystyle d\mu =f(x)dx,}$$ then $${\displaystyle \hat{\mu }(\xi )=\hat{f}(\xi ),}$$ and the Fourier-Stieltjes transform reduces to the usual definition of the Fourier transform. That is, the notable difference with the Fourier transform of integrable functions is that the Fourier-Stieltjes transform need not vanish at infinity, i.e., the Riemann–Lebesgue lemma fails for measures.