{{Short description|Mathematical transform that expresses a function of time as a function of frequency}}{{Fourier transforms}}File:CQT-piano-chord.png|thumb|An example application of the Fourier transform is determining the constituent pitches in a musical waveform. This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord. The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G). The remaining smaller peaks are higher-frequency overtones of the fundamental pitches. A pitch detection algorithmpitch detection algorithmIn physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.(File:Fourier transform time and frequency domains (small).gif|thumb|right|The Fourier transform relates the time domain, in red, with a function in the domain of the frequency, in blue. The component frequencies, extended for the whole frequency spectrum, are shown as peaks in the domain of the frequency.){{multiple image| total_width = 300| align = right| image1 = Sine voltage.svg| image2 = Phase shift.svg| footer =The red sinusoid can be described by peak amplitude (1), peak-to-peak (2), RMS (3), and wavelength (4). The red and blue sinusoids have a phase difference of {{mvar|θ}}.}}{{Annotated image< infty.Two measurable functions are equivalent if they are equal except on a set of measure zero. The set of all equivalence classes of integrable functions is denoted L^1(mathbb R). Then:{{math theorem|name=Definition|math_statement=The Fourier transform of a Lebesgue integrable function fin L^1(mathbb R) is defined by the formula {{EquationNote|Eq.1}}.}}The integral {{EquationNote|Eq.1}} is well-defined for all xiinmathbb R, because of the assumption |f|_1 0}}, then for any {{math|τ < −{{sfrac|a|2π}}}}.This theorem implies the Mellin inversion formula for the Laplace transformation, for any {{math|b > a}}, where {{math|F(s)}} is the Laplace transform of {{math|f(t)}}.The hypotheses can be weakened, as in the results of Carleson and Hunt, to {{math|f(t) e−at}} being {{math|L1}}, provided that {{mvar|f}} is of bounded variation in a closed neighborhood of {{mvar|t}} (cf. Dirichlet–Dini theorem), the value of {{mvar|f}} at {{mvar|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.{{harvnb|Champeney|1987|p=63}}{{math|L2}} versions of these inversion formulas are also available.{{harvnb|Widder|Wiener|1938|p=537}}

Fourier transform on Euclidean space

The Fourier transform can be defined in any arbitrary number of dimensions {{mvar|n}}. As with the one-dimensional case, there are many conventions. For an integrable function {{math|f(x)}}, this article takes the definition:hat{f}(boldsymbol{xi}) = mathcal{F}(f)(boldsymbol{xi}) = int_{R^n} f(mathbf{x}) e^{-i 2pi boldsymbol{xi}cdotmathbf{x}} , dmathbf{x}where {{math|x}} and {{math|ξ}} are {{mvar|n}}-dimensional vectors, and {{math|x · ξ}} is the dot product of the vectors. Alternatively, {{math|ξ}} can be viewed as belonging to the dual vector space R^{nstar}, in which case the dot product becomes the contraction of {{math|x}} and {{math|ξ}}, usually written as {{math|{{angbr|x, ξ}}}}.All of the basic properties listed above hold for the {{mvar|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

{{Further|Gabor limit}}Generally speaking, the more concentrated {{math|f(x)}} is, the more spread out its Fourier transform {{math|f̂(ξ)}} must be. In particular, the scaling property of the Fourier transform may be seen as saying: if we squeeze a function in {{mvar|x}}, its Fourier transform stretches out in {{mvar|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 {{math|f(x)}} is an integrable and square-integrable function. Without loss of generality, assume that {{math|f(x)}} is normalized:int_{-infty}^infty |f(x)|^2 ,dx=1.It follows from the Plancherel theorem that {{math|f̂(ξ)}} is also normalized.The spread around {{math|x {{=}} 0}} may be measured by the dispersion about zero{{harvnb|Pinsky|2002|p=131}} defined byD_0(f)=int_{-infty}^infty x^2|f(x)|^2,dx.In probability terms, this is the second moment of {{math|{{abs|f(x)}}2}} about zero.The uncertainty principle states that, if {{math|f(x)}} is absolutely continuous and the functions {{math|x·f(x)}} and {{math|f{{′}}(x)}} are square integrable, thenD_0(f)D_0(hat{f}) geq frac{1}{16pi^2}.The equality is attained only in the casebegin{align} f(x) &= C_1 , e^{-pi frac{x^2}{sigma^2} }therefore hat{f}(xi) &= sigma C_1 , e^{-pisigma^2xi^2} end{align} where {{math|σ > 0}} is arbitrary and {{math|1=C1 = {{sfrac|{{radic|2|4}}|{{sqrt|σ}}}}}} so that {{mvar|f}} is {{math|L2}}-normalized. In other words, where {{mvar|f}} is a (normalized) Gaussian function with variance {{math|σ2/2{{pi}}}}, centered at zero, and its Fourier transform is a Gaussian function with variance {{math|σ−2/2{{pi}}}}.In fact, this inequality implies that:left(int_{-infty}^infty (x-x_0)^2|f(x)|^2,dxright)left(int_{-infty}^infty(xi-xi_0)^2left|hat{f}(xi)right|^2,dxiright)geq frac{1}{16pi^2}for any {{math|x0}}, {{math|ξ0 ∈ R}}.{{harvnb|Stein|Shakarchi|2003}}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.{{harvnb|Stein|Shakarchi|2003|p=158}}A stronger uncertainty principle is the Hirschman uncertainty principle, which is expressed as:Hleft(left|fright|^2right)+Hleft(left|hat{f}right|^2right)ge logleft(frac{e}{2}right)where {{math|H(p)}} is the differential entropy of the probability density function {{math|p(x)}}:H(p) = -int_{-infty}^infty p(x)logbigl(p(x)bigr) , dxwhere 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 {{mvar|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{{harvnb|Chatfield|2004|p=113}}) {{mvar|λ}} byf(t) = int_0^infty bigl( a(lambda ) cos( 2pi lambda t) + b(lambda ) sin( 2pi lambda t)bigr) , dlambda.This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions {{mvar|a}} and {{mvar|b}} can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised): and Older literature refers to the two transform functions, the Fourier cosine transform, {{mvar|a}}, and the Fourier sine transform, {{mvar|b}}.The function {{mvar|f}} can be recovered from the sine and cosine transform using together with trigonometric identities. This is referred to as Fourier’s integral formula.{{harvnb|Fourier|1822|p=441}}{{harvnb|Poincaré|1895|p=102}}{{harvnb|Whittaker|Watson|1927|p=188}}

Spherical harmonics

Let the set of homogeneousharmonicpolynomials of degree {{mvar|k}} on {{math|Rn}} be denoted by {{math|Ak}}. The set {{math|Ak}} consists of the solid spherical harmonics of degree {{mvar|k}}. The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if {{math|1=f(x) = e−π{{abs|x}}2P(x)}} for some {{math|P(x)}} in {{math|Ak}}, then {{math|1=f̂(ξ) = i{{isup|−k}} f(ξ)}}. Let the set {{math|Hk}} be the closure in {{math|L2(Rn)}} of linear combinations of functions of the form {{math|f({{abs|x}})P(x)}} where {{math|P(x)}} is in {{math|Ak}}. The space {{math|L2(Rn)}} is then a direct sum of the spaces {{math|Hk}} and the Fourier transform maps each space {{math|Hk}} to itself and is possible to characterize the action of the Fourier transform on each space {{math|Hk}}.Let {{math|1=f(x) = f0({{abs|x}})P(x)}} (with {{math|P(x)}} in {{math|Ak}}), thenhat{f}(xi)=F_0(|xi|)P(xi)whereF_0(r) = 2pi i^{-k}r^{-frac{n+2k-2}{2}} int_0^infty f_0(s)J_frac{n+2k-2}{2}(2pi rs)s^frac{n+2k}{2},ds.Here {{math|J(n + 2k − 2)/2}} denotes the Bessel function of the first kind with order {{math|{{sfrac|n + 2k − 2|2}}}}. When {{math|k {{=}} 0}} this gives a useful formula for the Fourier transform of a radial function.{{harvnb|Grafakos|2004}} This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases {{math|n + 2}} and {{mvar|n}}{{harvnb|Grafakos|Teschl|2013}} 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 {{math|L2(Rn)}} function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in {{math|L{{isup|p}}}} for {{math|1 < p < 2}}. It is possible in some cases to define the restriction of a Fourier transform to a set {{mvar|S}}, provided {{mvar|S}} has non-zero curvature. The case when {{mvar|S}} is the unit sphere in {{math|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 {{math|Rn}} is a bounded operator on {{math|L{{isup|p}}}} provided {{math|1 ≤ p ≤ {{sfrac|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 {{math|ER}} indexed by {{math|R ∈ (0,∞)}}: such as balls of radius {{mvar|R}} centered at the origin, or cubes of side {{math|2R}}. For a given integrable function {{mvar|f}}, consider the function {{mvar|fR}} defined by:f_R(x) = int_{E_R}hat{f}(xi) e^{i 2pi xcdotxi}, dxi, quad x in mathbb{R}^n.Suppose in addition that {{math|f ∈ L{{isup|p}}(Rn)}}. For {{math|n {{=}} 1}} and {{math|1 < p < ∞}}, if one takes {{math|ER {{=}} (−R, R)}}, then {{mvar|fR}} converges to {{mvar|f}} in {{math|L{{isup|p}}}} as {{mvar|R}} tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for {{math|n > 1}}. In the case that {{mvar|ER}} is taken to be a cube with side length {{mvar|R}}, then convergence still holds. Another natural candidate is the Euclidean ball {{math|E’R {{=}} {ξ : {{abs|ξ}} < R{{)}}}}. In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in {{math|L{{isup|p}}(Rn)}}. For {{math|n ≥ 2}} it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless {{math|p {{=}} 2}}. In fact, when {{math|p ≠ 2}}, this shows that not only may {{mvar|fR}} fail to converge to {{mvar|f}} in {{math|L{{isup|p}}}}, but for some functions {{math|f ∈ L{{isup|p}}(R’n)}}, {{mvar|fR}} is not even an element of {{math|L{{isup|p’’}}}}.

Fourier transform on function spaces

L’p spaces“>

On L’p spaces

L1“>

On L1

The definition of the Fourier transform by the integral formulahat{f}(xi) = int_{mathbb{R}^n} f(x)e^{-i 2pi xicdot x},dxis valid for Lebesgue integrable functions {{mvar|f}}; that is, {{math|f ∈ L1(Rn)}}.The Fourier transform {{math|{{mathcal|F}} : L1(Rn) → L∞(Rn)}} is a bounded operator. This follows from the observation thatleftverthat{f}(xi)rightvert 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 {{math|L1}} is a subset of the space {{math|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.L2“>

On L2

Since compactly supported smooth functions are integrable and dense in {{math|L2(Rn)}}, the Plancherel theorem allows us to extend the definition of the Fourier transform to general functions in {{math|L2(Rn)}} by continuity arguments. The Fourier transform in {{math|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 {{math|L2}} function {{mvar|f}},hat{f}(xi) = lim_{Rtoinfty}int_{|x|le R} f(x) e^{-i 2pixicdot x},dxwhere the limit is taken in the {{math|L2}} sense. (More generally, you can take a sequence of functions that are in the intersection of {{math|L1}} and {{math|L2}} and that converges to {{mvar|f}} in the {{math|L2}}-norm, and define the Fourier transform of {{mvar|f}} as the {{math|L2}} -limit of the Fourier transforms of these functions.WEB,statweb.stanford.edu/~candes/teaching/math262/Lectures/Lecture03.pdf, Applied Fourier Analysis and Elements of Modern Signal Processing Lecture 3, January 12, 2016, 2019-10-11, )Many of the properties of the Fourier transform in {{math|L1}} carry over to {{math|L2}}, by a suitable limiting argument.Furthermore, {{math|{{mathcal|F}} : L2(Rn) → L2(Rn)}} is a unitary operator.{{harvnb|Stein|Weiss|1971|loc=Thm. 2.3}} 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 {{math|f, g ∈ L2(Rn)}} we haveint_{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 {{math|L2(Rn)}} is itself under the Fourier transform.L’p“>

On other L’p

The definition of the Fourier transform can be extended to functions in {{math|L{{isup|p}}(Rn)}} for {{math|1 ≤ p ≤ 2}} by decomposing such functions into a fat tail part in {{math|L2}} plus a fat body part in {{math|L1}}. In each of these spaces, the Fourier transform of a function in {{math|L{{isup|p}}(Rn)}} is in {{math|L{{isup|q}}(Rn)}}, where {{math|1=q = {{sfrac|p|p − 1}}}} is the Hölder conjugate of {{mvar|p}} (by the Hausdorff–Young inequality). However, except for {{math|1=p = 2}}, the image is not easily characterized. Further extensions become more technical. The Fourier transform of functions in {{math|L{{isup|p}}}} for the range {{math|2 < p < ∞}} requires the study of distributions. In fact, it can be shown that there are functions in {{math|L{{isup|p}}}} with {{math|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 {{math|L1 + L2}} by considering generalized functions, or distributions. A distribution on {{math|Rn}} is a continuous linear functional on the space {{math|Cc(Rn)}} of compactly supported smooth functions, equipped with a suitable topology. The strategy is then to consider the action of the Fourier transform on {{math|Cc(Rn)}} and pass to distributions by duality. The obstruction to doing this is that the Fourier transform does not map {{math|Cc(Rn)}} to {{math|Cc(Rn)}}. In fact the Fourier transform of an element in {{math|Cc(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. 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 {{mvar|f}} and {{mvar|g}} be integrable functions, and let {{math|f̂}} and {{math|ĝ}} be their Fourier transforms respectively. Then the Fourier transform obeys the following multiplication formula,int_{mathbb{R}^n}hat{f}(x)g(x),dx=int_{mathbb{R}^n}f(x)hat{g}(x),dx.Every integrable function {{mvar|f}} defines (induces) a distribution {{mvar|Tf}} by the relationT_f(varphi)=int_{mathbb{R}^n}f(x)varphi(x),dxfor all Schwartz functions {{mvar|φ}}. So it makes sense to define Fourier transform {{math|T̂f}} of {{math|Tf}} byhat{T}_f (varphi)= T_fleft(hat{varphi}right)for all Schwartz functions {{mvar|φ}}. Extending this to all tempered distributions {{mvar|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

The Fourier transform of a finiteBorel measure {{mvar|μ}} on {{math|Rn}} is given by:{{harvnb|Pinsky|2002|p=256}}hatmu(xi)=int_{mathbb{R}^n} e^{-i 2pi x cdot xi},dmu.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. In the case that {{math|dμ {{=}} f(x) dx}}, then the formula above reduces to the usual definition for the Fourier transform of {{mvar|f}}. In the case that {{mvar|μ}} is the probability distribution associated to a random variable {{mvar|X}}, the Fourier–Stieltjes transform is closely related to the characteristic function, but the typical conventions in probability theory take {{math|eiξx}} instead of {{math|e−i2πξx}}. 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.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).

Locally compact abelian groups

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 compactHausdorff topological space so that the group operation is continuous. If {{mvar|G}} is a locally compact abelian group, it has a translation invariant measure {{mvar|μ}}, called Haar measure. For a locally compact abelian group {{mvar|G}}, the set of irreducible, i.e. one-dimensional, unitary representations are called its characters. With its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by the compact-open topology on the space of all continuous functions from G to the circle group), the set of characters {{mvar|Äœ}} is itself a locally compact abelian group, called the Pontryagin dual of {{mvar|G}}. For a function {{mvar|f}} in {{math|L1(G)}}, its Fourier transform is defined byhat{f}(xi) = int_G xi(x)f(x),dmuquad text{for any }xi in hat{G}.The Riemann–Lebesgue lemma holds in this case; {{math|fÌ‚(ξ)}} is a function vanishing at infinity on {{mvar|Äœ}}.The Fourier transform on {{nobr|{{mvar|T}} {{=}} R/Z}} is an example; here {{mvar|T}} is a locally compact abelian group, and the Haar measure {{mvar|μ}} on {{mvar|T}} can be thought of as the Lebesgue measure on [0,1). Consider the representation of {{mvar|T}} on the complex plane {{mvar|C}} that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since {{mvar|C}} is 1-dim) {e_{k}: T rightarrow GL_{1}(C) = C^{*} mid k in Z} where e_{k}(x) = e^{i 2pi kx} for xin T.The character of such representation, that is the trace of e_{k}(x) for each xin T and kin Z, is e^{i 2pi kx} itself. In the case of representation of finite group, the character table of the group {{mvar|G}} are rows of vectors such that each row is the character of one irreducible representation of {{mvar|G}}, and these vectors form an orthonormal basis of the space of class functions that map from {{mvar|G}} to {{mvar|C}} by Schur’s lemma. Now the group {{mvar|T}} is no longer finite but still compact, and it preserves the orthonormality of character table. Each row of the table is the function e_{k}(x) of xin T, and the inner product between two class functions (all functions being class functions since {{mvar|T}} is abelian) f,g in L^{2}(T, dmu) is defined as langle f, g rangle = frac{1}{|T|}int_{[0,1)}f(y)overline{g}(y)dmu(y) with the normalizing factor |T|=1. The sequence {e_{k}mid kin Z} is an orthonormal basis of the space of class functions L^{2}(T,dmu).For any representation {{mvar|V}} of a finite group {{mvar|G}}, chi_{v} can be expressed as the span sum_{i} leftlangle chi_{v},chi_{v_{i}} rightrangle chi_{v_{i}} (V_{i} are the irreps of {{mvar|G}}), such that leftlangle chi_{v}, chi_{v_{i}} rightrangle = frac{1}{|G|}sum_{gin G}chi_{v}(g)overline{chi}_{v_{i}}(g). Similarly for G = T and fin L^{2}(T, dmu), f(x) = sum_{kin Z}hat{f}(k)e_{k}. The Pontriagin dual hat{T} is {e_{k}}(kin Z) and for f in L^{2}(T, dmu), hat{f}(k) = frac{1}{|T|}int_{[0,1)}f(y)e^{-i 2pi ky}dy is its Fourier transform for e_{k} in hat{T}.

Gelfand transform

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 compactHausdorfftopological group {{mvar|G}}, as before we consider space {{math|L1(G)}}, defined using a Haar measure. With convolution as multiplication, {{math|L1(G)}} is an abelian Banach algebra. It also has an involution * given byf^*(g) = overline{fleft(g^{-1}right)}.Taking the completion with respect to the largest possibly {{math|C*}}-norm gives its enveloping {{math|C*}}-algebra, called the group {{math|C*}}-algebra {{math|C*(G)}} of {{mvar|G}}. (Any {{math|C*}}-norm on {{math|L1(G)}} is bounded by the {{math|L1}} norm, therefore their supremum exists.)Given any abelian {{math|C*}}-algebra {{mvar|A}}, the Gelfand transform gives an isomorphism between {{mvar|A}} and {{math|C0(A^)}}, where {{math|A^}} is the multiplicative linear functionals, i.e. one-dimensional representations, on {{mvar|A}} with the weak-* topology. The map is simply given bya mapsto bigl( varphi mapsto varphi(a) bigr)It turns out that the multiplicative linear functionals of {{math|C*(G)}}, after suitable identification, are exactly the characters of {{mvar|G}}, and the Gelfand transform, when restricted to the dense subset {{math|L1(G)}} is the Fourier–Pontryagin transform.

Compact non-abelian groups

The Fourier transform can also be defined for functions on a non-abelian group, provided that the group is compact. Removing the assumption that the underlying group is abelian, irreducible unitary representations need not always be one-dimensional. This means the Fourier transform on a non-abelian group takes values as Hilbert space operators.{{harvnb|Hewitt|Ross|1970|loc=Chapter 8}} The Fourier transform on compact groups is a major tool in representation theory{{harvnb|Knapp|2001}} and non-commutative harmonic analysis.Let {{mvar|G}} be a compact Hausdorfftopological group. Let {{math|Σ}} denote the collection of all isomorphism classes of finite-dimensional irreducible unitary representations, along with a definite choice of representation {{math|U{{isup|(σ)}}}} on the Hilbert space {{math|Hσ}} of finite dimension {{math|dσ}} for each {{math|σ ∈ Σ}}. If {{mvar|μ}} is a finite Borel measure on {{mvar|G}}, then the Fourier–Stieltjes transform of {{mvar|μ}} is the operator on {{math|Hσ}} defined byleftlangle hat{mu}xi,etarightrangle_{H_sigma} = int_G leftlangle overline{U}^{(sigma)}_gxi,etarightrangle,dmu(g)where {{math|{{overline|U}}{{isup|(σ)}}}} is the complex-conjugate representation of {{math|U(σ)}} acting on {{math|Hσ}}. If {{mvar|μ}} is absolutely continuous with respect to the left-invariant probability measure {{mvar|λ}} on {{mvar|G}}, represented asdmu = f , dlambdafor some {{math|f ∈ L1(λ)}}, one identifies the Fourier transform of {{mvar|f}} with the Fourier–Stieltjes transform of {{mvar|μ}}.The mappingmumapstohat{mu}defines an isomorphism between the Banach space {{math|M(G)}} of finite Borel measures (see rca space) and a closed subspace of the Banach space {{math|C∞(Σ)}} consisting of all sequences {{math|E {{=}} (Eσ)}} indexed by {{math|Σ}} of (bounded) linear operators {{math|Eσ : Hσ → Hσ}} for which the norm|| f(x),|begin{align} &widehat{f}(xi) triangleq widehat {f_1}(xi) &= int_{-infty}^infty f(x) e^{-i 2pi xi x}, dx end{align}|begin{align} &widehat{f}(omega) triangleq widehat {f_2}(omega) &= frac{1}{sqrt{2 pi}} int_{-infty}^infty f(x) e^{-i omega x}, dx end{align}|begin{align} &widehat{f}(omega) triangleq widehat {f_3}(omega) &= int_{-infty}^infty f(x) e^{-i omega x}, dx end{align}|Definitions| 101| a, f(x) + b, g(x),| a, widehat{f}(xi) + b, widehat{g}(xi),| a, widehat{f}(omega) + b, widehat{g}(omega),| a, widehat{f}(omega) + b, widehat{g}(omega),|Linearity| 102| f(x - a),| e^{-i 2pi xi a} widehat{f}(xi),| e^{- i a omega} widehat{f}(omega),| e^{- i a omega} widehat{f}(omega),|Shift in time domain| 103| f(x)e^{iax},| widehat{f} left(xi - frac{a}{2pi}right),| widehat{f}(omega - a),| widehat{f}(omega - a),|Shift in frequency domain, dual of 102| 104| f(a x),| 105| widehat {f_n}(x),| widehat {f_1}(x) stackrel{mathcal{F}_1}{longleftrightarrow} f(-xi),| widehat {f_2}(x) stackrel{mathcal{F}_2}{longleftrightarrow} f(-omega),| widehat {f_3}(x) stackrel{mathcal{F}_3}{longleftrightarrow} 2pi f(-omega),|The same transform is applied twice, but x replaces the frequency variable (ξ or ω) after the first transform.| 106| frac{d^n f(x)}{dx^n},| (i 2pi xi)^n widehat{f}(xi),| (iomega)^n widehat{f}(omega),| (iomega)^n widehat{f}(omega),|106.5|int_{-infty}^{x} f(tau) d tau|frac{widehat{f}(xi)}{i 2 pi xi} + C , delta(xi)|frac{1}{sqrt{2 pi}}, widehat{f_1} ! left(frac{omega}{2 pi}right)|widehat{f_1} left(frac{omega}{2 pi}right)| 107| x^n f(x),| left (frac{i}{2pi}right)^n frac{d^n widehat{f}(xi)}{dxi^n},| i^n frac{d^n widehat{f}(omega)}{domega^n}| i^n frac{d^n widehat{f}(omega)}{domega^n}|This is the dual of 106| 108| (f * g)(x),| widehat{f}(xi) widehat{g}(xi),| sqrt{2pi} widehat{f}(omega) widehat{g}(omega),| widehat{f}(omega) widehat{g}(omega),| 109| f(x) g(x),| left(widehat{f} * widehat{g}right)(xi),| frac{1}sqrt{2pi}left(widehat{f} * widehat{g}right)(omega),| frac{1}{2pi}left(widehat{f} * widehat{g}right)(omega),|This is the dual of 108| 110| 113| 114|115| f(x) cos (a x)| frac{ widehat{f}left(xi - frac{a}{2pi}right)+widehat{f}left(xi+frac{a}{2pi}right)}{2}| frac{widehat{f}(omega-a)+widehat{f}(omega+a)}{2},| frac{widehat{f}(omega-a)+widehat{f}(omega+a)}{2}|This follows from rules 101 and 103 using Euler’s formula:{{br}}cos(a x) = frac{e^{i a x} + e^{-i a x}}{2}.|116| f(x)sin( ax)| frac{widehat{f}left(xi-frac{a}{2pi}right)-widehat{f}left(xi+frac{a}{2pi}right)}{2i}| frac{widehat{f}(omega-a)-widehat{f}(omega+a)}{2i}| frac{widehat{f}(omega-a)-widehat{f}(omega+a)}{2i}|This follows from 101 and 103 using Euler’s formula:{{br}}sin(a x) = frac{e^{i a x} - e^{-i a x}}{2i}.

unit pulse as a function of time ({{math>f(t)}}) and its Fourier transform as a function of frequency ({{math	fÌ‚(ω)}}). The bottom row shows a delayed unit pulse as a function of time ({{math>g(t)}}) and its Fourier transform as a function of frequency ({{math	ĝ(ω)}}). Translation (geometry)>Translation (i.e. delay) in the time domain is interpreted as complex phase shifts in the frequency domain. The Fourier transform decomposes a function into eigenfunctions for the group of translations. The imaginary part of {{math|ĝ(ω)}} is negated because a negative sign exponent has been used in the Fourier transform, which is the default as derived from the Fourier series, but the sign does not matter for a transform that is not going to be reversed.| image=Fourier_unit_pulse.svg| image-width = 300| annotations ={{Annotation|20|40|scriptstyle f(t)}}{{Annotation|170|40|scriptstyle widehat{f}(omega)}}{{Annotation|20|140|scriptstyle g(t)}}{{Annotation|170|140|scriptstyle widehat{g}(omega)}}{{Annotation|130|80|scriptstyle t}}{{Annotation|280|85|scriptstyle omega}}{{Annotation|130|192|scriptstyle t}}{{Annotation|280|180|scriptstyle omega}}}}Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced the transform in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.The Fourier transform can be formally defined as an improper Riemann integral, making it an integral transform, although this definition is not suitable for many applications requiring a more sophisticated integration theory.Depending on the application a Lebesgue integral, distributional, or other approach may be most appropriate. For example, many relatively simple applications use the Dirac delta function, which can be treated formally as if it were a function, but the justification requires a mathematically more sophisticated viewpoint.{{harvtxt|Vretblad|2000}} provides solid justification for these formal procedures without going too deeply into functional analysis or the theory of distributions.The Fourier transform can also be generalized to functions of several variables on Euclidean space, sending a function of {{nowrap|3-dimensional}} ‘position space’ to a function of {{nowrap|3-dimensional}} momentum (or a function of space and time to a function of 4-momentum). This idea makes the spatial Fourier transform very natural in the study of waves, as well as in quantum mechanics, where it is important to be able to represent wave solutions as functions of either position or momentum and sometimes both. In general, functions to which Fourier methods are applicable are complex-valued, and possibly vector-valued.In relativistic quantum mechanics one encounters vector-valued Fourier transforms of multi-component wave functions. In quantum field theory, operator-valued Fourier transforms of operator-valued functions of spacetime are in frequent use, see for example {{harvtxt|Greiner|Reinhardt|1996}}. Still further generalization is possible to functions on groups, which, besides the original Fourier transform on {{math|R}} or {{math|Rn}}, notably includes the discrete-time Fourier transform (DTFT, group = {{math|Z}}), the discrete Fourier transform (DFT, group = {{math|Z mod N}}) and the Fourier series or circular Fourier transform (group = {{math|S1}}, the unit circle ≈ closed finite interval with endpoints identified). The latter is routinely employed to handle periodic functions. The fast Fourier transform (FFT) is an algorithm for computing the DFT. Definition The Fourier transform is an analysis process, decomposing a complex-valued function textstyle f(x) into its constituent frequencies and their amplitudes. The inverse process is synthesis, which recreates textstyle f(x) from its transform.We can start with an analogy, the Fourier series, which analyzes textstyle f(x) on a bounded interval textstyle x in [-P/2, P/2], for some positive real number P. The constituent frequencies are a discrete set of harmonics at frequencies tfrac{n}{P}, n in mathbb Z, whose amplitude and phase are given by the analysis formula:c_n = tfrac{1}{P} int_{-P/2}^{P/2} f(x) , e^{-i 2pi frac{n}{P}x} , dx.The actual Fourier series is the synthesis formula:f(x) = sum_{n=-infty}^infty c_n, e^{i 2pi tfrac{n}{P}x},quad textstyle x in [-P/2, P/2].The analogy for a function textstyle f(x) can be obtained formally from the analysis formula by taking the limit as Ptoinfty, while at the same time taking n so that tfrac{n}{P} to xi in mathbb R.{{harvnb|Khare|Butola|Rajora|2023|pp=13–14}}{{harvnb|Kaiser|1994|p=29}}{{harvnb|Rahman|2011|p=11}} Formally carrying this out, we obtain, for rapidly decreasing f:For this article, a rapidly decreasing function is a function f(x) on the reals that tends to zero together with all derivatives as xtopminfty: lim_{xtoinfty} f^{(n)}(x) = 0, n=0,1,2,dots. See Schwartz function.{{harvnb|Dym|McKean|1985}}{{Equation box 1|indent =:|title =Fourier transform		{{EquationRef|Eq.1}}}}|cellpadding= 6|border|border colour = #0073CF|background colour=#F5FFFA}}It is easy to see, assuming the hypothesis of rapid decreasing, that the integral {{EquationNote|Eq.1}} converges for all real xi, and (using the Riemann–Lebesgue lemma) that the transformed function widehat f is also rapidly decreasing. The validity of this definition for classes of functions f that are not necessarily rapidly decreasing is discussed later in this section.Evaluating {{EquationNote|Eq.1}} for all values of xi produces the frequency-domain function. The complex number widehat{f}(xi), in polar coordinates, conveys both amplitude and phase of frequency xi. The intuitive interpretation of {{EquationNote|Eq.1}} is that the effect of multiplying f(x) by e^{-i 2pi xi x} is to subtract xi from every frequency component of function f(x).A possible source of confusion is the frequency-shifting property; i.e. the transform of function f(x)e^{-i 2pi xi_0 x} is widehat{f}(xi+xi_0).  The value of this function at  xi=0  is widehat{f}(xi_0), meaning that a frequency xi_0 has been shifted to zero (also see Negative frequency). Only the component that was at frequency 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 {{slink||Example}})The corresponding synthesis formula for such a function is:{{Equation box 1|indent =:|title = Inverse transform		{{EquationRef|Eq.2}}}}|cellpadding= 6|border|border colour = #0073CF|background colour=#F5FFFA}}{{EquationNote|Eq.2}} is a representation of f(x) as a weighted summation of complex exponential functions.This is also known as the Fourier inversion theorem, and was first introduced in Fourier’s Analytical Theory of Heat.{{harvnb|Fourier|1822|p=525}}{{harvnb|Fourier|1878|p=408}}{{harvtxt|Jordan|1883}} proves on pp. 216–226 the Fourier integral theorem before studying Fourier series.{{harvnb|Titchmarsh|1986|p=1}}The functions f and widehat{f} are referred to as a Fourier transform pair.{{harvnb|Rahman|2011|p=10}}.  A common notation for designating transform pairs is:{{harvnb|Oppenheim|Schafer|Buck|1999|p=58}}f(x) stackrel{mathcal{F}}{longleftrightarrow} widehat f(xi),   for example   operatorname{rect}(x) stackrel{mathcal{F}}{longleftrightarrow} operatorname{sinc}(xi). Definition for Lebesgue integrable functions Until now, we have been dealing with Schwartz functions, which decay rapidly at infinity, with all derivatives. This excludes many functions of practical importance from the definition, such as the rect function. A measurable function f:mathbb Rtomathbb C is called (Lebesgue) integrable if the Lebesgue integral of its absolute value is finite:	_1 = int_{mathbb R}	,dx	= sup_{sigmainSigma}left	is finite. The “convolution theorem” asserts that, furthermore, this isomorphism of Banach spaces is in fact an isometric isomorphism of C*-algebras into a subspace of {{math|C∞(Σ)}}. Multiplication on {{math|M(G)}} is given by convolution of measures and the involution * defined byf^*(g) = overline{fleft(g^{-1}right)},and {{math|C∞(Σ)}} has a natural {{math|C*}}-algebra structure as Hilbert space operators.The Peter–Weyl theorem holds, and a version of the Fourier inversion formula (Plancherel’s theorem) follows: if {{math|f ∈ L2(G)}}, thenf(g) = sum_{sigmainSigma} d_sigma operatorname{tr}left(hat{f}(sigma)U^{(sigma)}_gright)where the summation is understood as convergent in the {{math|L2}} sense.The generalization of the Fourier transform to the noncommutative situation has also in part contributed to the development of noncommutative geometry.{{Citation needed|date=May 2009}} In this context, a categorical generalization of the Fourier transform to noncommutative groups is Tannaka–Krein duality, which replaces the group of characters with the category of representations. However, this loses the connection with harmonic functions. Alternatives In signal processing terms, a function (of time) is a representation of a signal with perfect time resolution, but no frequency information, while the Fourier transform has perfect frequency resolution, but no time information: the magnitude of the Fourier transform at a point is how much frequency content there is, but location is only given by phase (argument of the Fourier transform at a point), and standing waves are not localized in time – a sine wave continues out to infinity, without decaying. This limits the usefulness of the Fourier transform for analyzing signals that are localized in time, notably transients, or any signal of finite extent.As alternatives to the Fourier transform, in time–frequency analysis, one uses time–frequency transforms or time–frequency distributions to represent signals in a form that has some time information and some frequency information – by the uncertainty principle, there is a trade-off between these. These can be generalizations of the Fourier transform, such as the short-time Fourier transform, fractional Fourier transform, Synchrosqueezing Fourier transform,JOURNAL, Correia, L. B., Justo, J. F., Angélico, B. A., Polynomial Adaptive Synchrosqueezing Fourier Transform: A method to optimize multiresolution, Digital Signal Processing, 2024, 150, 104526, 10.1016/j.dsp.2024.104526, or other functions to represent signals, as in wavelet transforms and chirplet transforms, with the wavelet analog of the (continuous) Fourier transform being the continuous wavelet transform. Applications {{see also|Spectral density#Applications}}(File:Commutative diagram illustrating problem solving via the Fourier transform.svg|thumb|400px|Some problems, such as certain differential equations, become easier to solve when the Fourier transform is applied. In that case the solution to the original problem is recovered using the inverse Fourier transform.)Linear operations performed in one domain (time or frequency) have corresponding operations in the other domain, which are sometimes easier to perform. The operation of differentiation in the time domain corresponds to multiplication by the frequency,Up to an imaginary constant factor whose magnitude depends on what Fourier transform convention is used. so some differential equations are easier to analyze in the frequency domain. Also, convolution in the time domain corresponds to ordinary multiplication in the frequency domain (see Convolution theorem). After performing the desired operations, transformation of the result can be made back to the time domain. Harmonic analysis is the systematic study of the relationship between the frequency and time domains, including the kinds of functions or operations that are “simpler” in one or the other, and has deep connections to many areas of modern mathematics. Analysis of differential equations Perhaps the most important use of the Fourier transformation is to solve partial differential equations.Many of the equations of the mathematical physics of the nineteenth century can be treated this way. Fourier studied the heat equation, which in one dimension and in dimensionless units isfrac{partial^2 y(x, t)}{partial^2 x} = frac{partial y(x, t)}{partial t}.The example we will give, a slightly more difficult one, is the wave equation in one dimension,frac{partial^2y(x, t)}{partial^2 x} = frac{partial^2y(x, t)}{partial^2t}.As usual, the problem is not to find a solution: there are infinitely many. The problem is that of the so-called “boundary problem”: find a solution which satisfies the “boundary conditions“y(x, 0) = f(x),qquad frac{partial y(x, 0)}{partial t} = g(x).Here, {{mvar|f}} and {{mvar|g}} are given functions. For the heat equation, only one boundary condition can be required (usually the first one). But for the wave equation, there are still infinitely many solutions {{mvar|y}} which satisfy the first boundary condition. But when one imposes both conditions, there is only one possible solution.It is easier to find the Fourier transform {{mvar|Å·}} of the solution than to find the solution directly. This is because the Fourier transformation takes differentiation into multiplication by the Fourier-dual variable, and so a partial differential equation applied to the original function is transformed into multiplication by polynomial functions of the dual variables applied to the transformed function. After {{mvar|Å·}} is determined, we can apply the inverse Fourier transformation to find {{mvar|y}}.Fourier’s method is as follows. First, note that any function of the forms cosbigl(2pixi(xpm t)bigr) text{ or } sinbigl(2pixi(x pm t)bigr) satisfies the wave equation. These are called the elementary solutions.Second, note that therefore any integralbegin{align} y(x, t) = int_{0}^{infty} dxi Bigl[ &a_+(xi)cosbigl(2pixi(x + t)bigr) + a_-(xi)cosbigl(2pixi(x - t)bigr) +{} &b_+(xi)sinbigl(2pixi(x + t)bigr) + b_-(xi)sinleft(2pixi(x - t)right) Bigr] end{align}satisfies the wave equation for arbitrary {{math|a+, a−, b+, b−}}. This integral may be interpreted as a continuous linear combination of solutions for the linear equation.Now this resembles the formula for the Fourier synthesis of a function. In fact, this is the real inverse Fourier transform of {{math|a±}} and {{math|b±}} in the variable {{mvar|x}}.The third step is to examine how to find the specific unknown coefficient functions {{math|a±}} and {{math|b±}} that will lead to {{mvar|y}} satisfying the boundary conditions. We are interested in the values of these solutions at {{math|1=t = 0}}. So we will set {{math|1=t = 0}}. Assuming that the conditions needed for Fourier inversion are satisfied, we can then find the Fourier sine and cosine transforms (in the variable {{mvar|x}}) of both sides and obtain 2int_{-infty}^infty y(x,0) cos(2pixi x) , dx = a_+ + a_- and2int_{-infty}^infty y(x,0) sin(2pixi x) , dx = b_+ + b_-.Similarly, taking the derivative of {{mvar|y}} with respect to {{mvar|t}} and then applying the Fourier sine and cosine transformations yields2int_{-infty}^infty frac{partial y(u,0)}{partial t} sin (2pixi x) , dx = (2pixi)left(-a_+ + a_-right)and2int_{-infty}^infty frac{partial y(u,0)}{partial t} cos (2pixi x) , dx = (2pixi)left(b_+ - b_-right).These are four linear equations for the four unknowns {{math|a±}} and {{math|b±}}, in terms of the Fourier sine and cosine transforms of the boundary conditions, which are easily solved by elementary algebra, provided that these transforms can be found.In summary, we chose a set of elementary solutions, parametrized by {{mvar|ξ}}, of which the general solution would be a (continuous) linear combination in the form of an integral over the parameter {{mvar|ξ}}. But this integral was in the form of a Fourier integral. The next step was to express the boundary conditions in terms of these integrals, and set them equal to the given functions {{mvar|f}} and {{mvar|g}}. But these expressions also took the form of a Fourier integral because of the properties of the Fourier transform of a derivative. The last step was to exploit Fourier inversion by applying the Fourier transformation to both sides, thus obtaining expressions for the coefficient functions {{math|a±}} and {{math|b±}} in terms of the given boundary conditions {{mvar|f}} and {{mvar|g}}.From a higher point of view, Fourier’s procedure can be reformulated more conceptually. Since there are two variables, we will use the Fourier transformation in both {{mvar|x}} and {{mvar|t}} rather than operate as Fourier did, who only transformed in the spatial variables. Note that {{mvar|Å·}} must be considered in the sense of a distribution since {{math|y(x, t)}} is not going to be {{math|L1}}: as a wave, it will persist through time and thus is not a transient phenomenon. But it will be bounded and so its Fourier transform can be defined as a distribution. The operational properties of the Fourier transformation that are relevant to this equation are that it takes differentiation in {{mvar|x}} to multiplication by {{math|i2πξ}} and differentiation with respect to {{mvar|t}} to multiplication by {{math|i2Ï€f}} where {{mvar|f}} is the frequency. Then the wave equation becomes an algebraic equation in {{mvar|Å·}}:xi^2 hat y (xi, f) = f^2 hat y (xi, f).This is equivalent to requiring {{math|1=Å·(ξ, f) = 0}} unless {{math|1=ξ = ±f}}. Right away, this explains why the choice of elementary solutions we made earlier worked so well: obviously {{math|1=fÌ‚ = δ(ξ ± f)}} will be solutions. Applying Fourier inversion to these delta functions, we obtain the elementary solutions we picked earlier. But from the higher point of view, one does not pick elementary solutions, but rather considers the space of all distributions which are supported on the (degenerate) conic {{math|1=ξ{{isup|2}} − f{{isup|2}} = 0}}.We may as well consider the distributions supported on the conic that are given by distributions of one variable on the line {{math|1=ξ = f}} plus distributions on the line {{math|ξ {{=}} −f}} as follows: if {{mvar|Φ}} is any test function,iint hat y phi(xi,f) , dxi , df = int s_+ phi(xi,xi) , dxi + int s_- phi(xi,-xi) , dxi,where {{math|s+}}, and {{math|s−}}, are distributions of one variable.Then Fourier inversion gives, for the boundary conditions, something very similar to what we had more concretely above (put {{math|1=Φ(ξ, f) = ei2Ï€(xξ+tf)}}, which is clearly of polynomial growth): y(x,0) = intbigl{s_+(xi) + s_-(xi)bigr} e^{i 2pi xi x+0} , dxi and frac{partial y(x,0)}{partial t} = intbigl{s_+(xi) - s_-(xi)bigr} i 2pi xi e^{i 2pixi x+0} , dxi. Now, as before, applying the one-variable Fourier transformation in the variable {{mvar|x}} to these functions of {{mvar|x}} yields two equations in the two unknown distributions {{math|s±}} (which can be taken to be ordinary functions if the boundary conditions are {{math|L1}} or {{math|L2}}).From a calculational point of view, the drawback of course is that one must first calculate the Fourier transforms of the boundary conditions, then assemble the solution from these, and then calculate an inverse Fourier transform. Closed form formulas are rare, except when there is some geometric symmetry that can be exploited, and the numerical calculations are difficult because of the oscillatory nature of the integrals, which makes convergence slow and hard to estimate. For practical calculations, other methods are often used.The twentieth century has seen the extension of these methods to all linear partial differential equations with polynomial coefficients, and by extending the notion of Fourier transformation to include Fourier integral operators, some non-linear equations as well. Fourier-transform spectroscopy The Fourier transform is also used in nuclear magnetic resonance (NMR) and in other kinds of spectroscopy, e.g. infrared (FTIR). In NMR an exponentially shaped free induction decay (FID) signal is acquired in the time domain and Fourier-transformed to a Lorentzian line-shape in the frequency domain. The Fourier transform is also used in magnetic resonance imaging (MRI) and mass spectrometry. Quantum mechanics The Fourier transform is useful in quantum mechanics in at least two different ways. To begin with, the basic conceptual structure of quantum mechanics postulates the existence of pairs of complementary variables, connected by the Heisenberg uncertainty principle. For example, in one dimension, the spatial variable {{mvar|q}} of, say, a particle, can only be measured by the quantum mechanical “position operator” at the cost of losing information about the momentum {{mvar|p}} of the particle. Therefore, the physical state of the particle can either be described by a function, called “the wave function”, of {{mvar|q}} or by a function of {{mvar|p}} but not by a function of both variables. The variable {{mvar|p}} is called the conjugate variable to {{mvar|q}}. In classical mechanics, the physical state of a particle (existing in one dimension, for simplicity of exposition) would be given by assigning definite values to both {{mvar|p}} and {{mvar|q}} simultaneously. Thus, the set of all possible physical states is the two-dimensional real vector space with a {{mvar|p}}-axis and a {{mvar|q}}-axis called the phase space.In contrast, quantum mechanics chooses a polarisation of this space in the sense that it picks a subspace of one-half the dimension, for example, the {{mvar|q}}-axis alone, but instead of considering only points, takes the set of all complex-valued “wave functions” on this axis. Nevertheless, choosing the {{mvar|p}}-axis is an equally valid polarisation, yielding a different representation of the set of possible physical states of the particle. Both representations of the wavefunction are related by a Fourier transform, such that phi(p) = int dq, psi (q) e^{-i pq/h} ,or, equivalently, psi(q) = int dp , phi (p) e^{i pq/h}.Physically realisable states are {{math|L2}}, and so by the Plancherel theorem, their Fourier transforms are also {{math|L2}}. (Note that since {{mvar|q}} is in units of distance and {{mvar|p}} is in units of momentum, the presence of the Planck constant in the exponent makes the exponent dimensionless, as it should be.)Therefore, the Fourier transform can be used to pass from one way of representing the state of the particle, by a wave function of position, to another way of representing the state of the particle: by a wave function of momentum. Infinitely many different polarisations are possible, and all are equally valid. Being able to transform states from one representation to another by the Fourier transform is not only convenient but also the underlying reason of the Heisenberg uncertainty principle.The other use of the Fourier transform in both quantum mechanics and quantum field theory is to solve the applicable wave equation. In non-relativistic quantum mechanics, Schrödinger’s equation for a time-varying wave function in one-dimension, not subject to external forces, is-frac{partial^2}{partial x^2} psi(x,t) = i frac h{2pi} frac{partial}{partial t} psi(x,t).This is the same as the heat equation except for the presence of the imaginary unit {{mvar|i}}. Fourier methods can be used to solve this equation.In the presence of a potential, given by the potential energy function {{math|V(x)}}, the equation becomes-frac{partial^2}{partial x^2} psi(x,t) + V(x)psi(x,t) = i frac h{2pi} frac{partial}{partial t} psi(x,t).The “elementary solutions”, as we referred to them above, are the so-called “stationary states” of the particle, and Fourier’s algorithm, as described above, can still be used to solve the boundary value problem of the future evolution of {{mvar|ψ}} given its values for {{math|t {{=}} 0}}. Neither of these approaches is of much practical use in quantum mechanics. Boundary value problems and the time-evolution of the wave function is not of much practical interest: it is the stationary states that are most important.In relativistic quantum mechanics, Schrödinger’s equation becomes a wave equation as was usual in classical physics, except that complex-valued waves are considered. A simple example, in the absence of interactions with other particles or fields, is the free one-dimensional Klein–Gordon–Schrödinger–Fock equation, this time in dimensionless units,left (frac{partial^2}{partial x^2} +1 right) psi(x,t) = frac{partial^2}{partial t^2} psi(x,t).This is, from the mathematical point of view, the same as the wave equation of classical physics solved above (but with a complex-valued wave, which makes no difference in the methods). This is of great use in quantum field theory: each separate Fourier component of a wave can be treated as a separate harmonic oscillator and then quantized, a procedure known as “second quantization”. Fourier methods have been adapted to also deal with non-trivial interactions.Finally, the number operator of the quantum harmonic oscillator can be interpreted, for example via the Mehler kernel, as the generator of the Fourier transform mathcal{F}. Signal processing The Fourier transform is used for the spectral analysis of time-series. The subject of statistical signal processing does not, however, usually apply the Fourier transformation to the signal itself. Even if a real signal is indeed transient, it has been found in practice advisable to model a signal by a function (or, alternatively, a stochastic process) which is stationary in the sense that its characteristic properties are constant over all time. The Fourier transform of such a function does not exist in the usual sense, and it has been found more useful for the analysis of signals to instead take the Fourier transform of its autocorrelation function.The autocorrelation function {{mvar|R}} of a function {{mvar|f}} is defined byR_f (tau) = lim_{Trightarrow infty} frac{1}{2T} int_{-T}^T f(t) f(t+tau) , dt. This function is a function of the time-lag {{mvar|Ï„}} elapsing between the values of {{mvar|f}} to be correlated.For most functions {{mvar|f}} that occur in practice, {{mvar|R}} is a bounded even function of the time-lag {{mvar|Ï„}} and for typical noisy signals it turns out to be uniformly continuous with a maximum at {{math|Ï„ {{=}} 0}}.The autocorrelation function, more properly called the autocovariance function unless it is normalized in some appropriate fashion, measures the strength of the correlation between the values of {{mvar|f}} separated by a time lag. This is a way of searching for the correlation of {{mvar|f}} with its own past. It is useful even for other statistical tasks besides the analysis of signals. For example, if {{math|f(t)}} represents the temperature at time {{mvar|t}}, one expects a strong correlation with the temperature at a time lag of 24 hours.It possesses a Fourier transform, P_f(xi) = int_{-infty}^infty R_f (tau) e^{-i 2pi xitau} , dtau. This Fourier transform is called the power spectral density function of {{mvar|f}}. (Unless all periodic components are first filtered out from {{mvar|f}}, this integral will diverge, but it is easy to filter out such periodicities.)The power spectrum, as indicated by this density function {{mvar|P}}, measures the amount of variance contributed to the data by the frequency {{mvar|ξ}}. In electrical signals, the variance is proportional to the average power (energy per unit time), and so the power spectrum describes how much the different frequencies contribute to the average power of the signal. This process is called the spectral analysis of time-series and is analogous to the usual analysis of variance of data that is not a time-series (ANOVA).Knowledge of which frequencies are “important” in this sense is crucial for the proper design of filters and for the proper evaluation of measuring apparatuses. It can also be useful for the scientific analysis of the phenomena responsible for producing the data.The power spectrum of a signal can also be approximately measured directly by measuring the average power that remains in a signal after all the frequencies outside a narrow band have been filtered out.Spectral analysis is carried out for visual signals as well. The power spectrum ignores all phase relations, which is good enough for many purposes, but for video signals other types of spectral analysis must also be employed, still using the Fourier transform as a tool. Other notations Other common notations for hat f(xi) include:tilde{f}(xi), F(xi), mathcal{F}left(fright)(xi), left(mathcal{F}fright)(xi), mathcal{F}(f), mathcal{F}{f}, mathcal{F} bigl(f(t)bigr), mathcal{F} bigl{f(t)bigr}.In the sciences and engineering it is also common to make substitutions like these:xi rightarrow f, quad x rightarrow t, quad f rightarrow x,quad hat f rightarrow X. So the transform pair f(x) stackrel{mathcal{F}}{Longleftrightarrow} hat{f}(xi) can become x(t) stackrel{mathcal{F}}{Longleftrightarrow} X(f)A disadvantage of the capital letter notation is when expressing a transform such as widehat{fcdot g} or widehat{f’}, which become the more awkward mathcal{F}{fcdot g} and mathcal{F} { f’ } . In some contexts such as particle physics, the same symbol f may be used for both for a function as well as it Fourier transform, with the two only distinguished by their argument I.e. f(k_1 + k_2) would refer to the Fourier transform because of the momentum argument, while f(x_0 + pi vec r) would refer to the original function because of the positional argument. Although tildes may be used as in tilde{f} to indicate Fourier transforms, tildes may also be used to indicate a modification of a quantity with a more Lorentz invariant form, such as tilde{dk} = frac{dk}{(2pi)^32omega}, so care must be taken. Similarly, hat f often denotes the Hilbert transform of f.The interpretation of the complex function {{math|fÌ‚(ξ)}} may be aided by expressing it in polar coordinate formhat f(xi) = A(xi) e^{ivarphi(xi)}in terms of the two real functions {{math|A(ξ)}} and {{math|φ(ξ)}} where:A(xi) = left|hat f(xi)right|,is the amplitude andvarphi (xi) = arg left( hat f(xi) right), is the phase (see arg function).Then the inverse transform can be written:f(x) = int _{-infty}^infty A(xi) e^{ ibigl(2pi xi x +varphi (xi)bigr)},dxi,which is a recombination of all the frequency components of {{math|f(x)}}. Each component is a complex sinusoid of the form {{math|e2Ï€ixξ}} whose amplitude is {{math|A(ξ)}} and whose initial phase angle (at {{math|1=x = 0}}) is {{math|φ(ξ)}}.The Fourier transform may be thought of as a mapping on function spaces. This mapping is here denoted {{mathcal|F}} and {{math|{{mathcal|F}}(f)}} is used to denote the Fourier transform of the function {{mvar|f}}. This mapping is linear, which means that {{mathcal|F}} can also be seen as a linear transformation on the function space and implies that the standard notation in linear algebra of applying a linear transformation to a vector (here the function {{math|f}}) can be used to write {{math|{{mathcal|F}} f}} instead of {{math|{{mathcal|F}}(f)}}. Since the result of applying the Fourier transform is again a function, we can be interested in the value of this function evaluated at the value {{mvar|ξ}} for its variable, and this is denoted either as {{math|{{mathcal|F}} f(ξ)}} or as {{math|({{mathcal|F}} f)(ξ)}}. Notice that in the former case, it is implicitly understood that {{mathcal|F}} is applied first to {{mvar|f}} and then the resulting function is evaluated at {{mvar|ξ}}, not the other way around.In mathematics and various applied sciences, it is often necessary to distinguish between a function {{mvar|f}} and the value of {{mvar|f}} when its variable equals {{mvar|x}}, denoted {{math|f(x)}}. This means that a notation like {{math|{{mathcal|F}}(f(x))}} formally can be interpreted as the Fourier transform of the values of {{mvar|f}} at {{mvar|x}}. Despite this flaw, the previous notation appears frequently, often when a particular function or a function of a particular variable is to be transformed. For example,mathcal Fbigl( operatorname{rect}(x) bigr) = operatorname{sinc}(xi)is sometimes used to express that the Fourier transform of a rectangular function is a sinc function, ormathcal Fbigl(f(x + x_0)bigr) = mathcal Fbigl(f(x)bigr), e^{i 2pi x_0 xi}is used to express the shift property of the Fourier transform.Notice, that the last example is only correct under the assumption that the transformed function is a function of {{mvar|x}}, not of {{math|x0}}.As discussed above, the characteristic function of a random variable is the same as the Fourier–Stieltjes transform of its distribution measure, but in this context it is typical to take a different convention for the constants. Typically characteristic function is definedEleft(e^{itcdot X}right)=int e^{itcdot x} , dmu_X(x).As in the case of the “non-unitary angular frequency” convention above, the factor of 2{{pi}} appears in neither the normalizing constant nor the exponent. Unlike any of the conventions appearing above, this convention takes the opposite sign in the exponent. Computation methods The appropriate computation method largely depends how the original mathematical function is represented and the desired form of the output function. In this section we consider both functions of a continuous variable, f(x), and functions of a discrete variable (i.e. ordered pairs of x and f values). For discrete-valued x, the transform integral becomes a summation of sinusoids, which is still a continuous function of frequency (xi or omega). When the sinusoids are harmonically-related (i.e. when the x-values are spaced at integer multiples of an interval), the transform is called discrete-time Fourier transform (DTFT). Discrete Fourier transforms and fast Fourier transforms Sampling the DTFT at equally-spaced values of frequency is the most common modern method of computation. Efficient procedures, depending on the frequency resolution needed, are described at {{slink|Discrete-time Fourier transform|Sampling the DTFT|nopage=n}}. The discrete Fourier transform (DFT), used there, is usually computed by a fast Fourier transform (FFT) algorithm. Analytic integration of closed-form functions Tables of closed-form Fourier transforms, such as {{slink||Square-integrable functions, one-dimensional}} and {{slink|Discrete-time Fourier transform|Table of discrete-time Fourier transforms|nopage=y}}, are created by mathematically evaluating the Fourier analysis integral (or summation) into another closed-form function of frequency (xi or omega).{{harvnb|Gradshteyn|Ryzhik|Geronimus|Tseytlin|2015}} When mathematically possible, this provides a transform for a continuum of frequency values.Many computer algebra systems such as Matlab and Mathematica that are capable of symbolic integration are capable of computing Fourier transforms analytically. For example, to compute the Fourier transform of {{math|1=cos(6Ï€t) e−πt2}} one might enter the command {{code|integrate cos(6*pi*t) exp(−pi*t^2) exp(-i*2*pi*f*t) from -inf to inf}} into Wolfram Alpha.The direct command {{code|fourier transform of cos(6*pi*t) exp(−pi*t^2)}} would also work for Wolfram Alpha, although the options for the convention (see {{Section link|2=Other_conventions}}) must be changed away from the default option, which is actually equivalent to {{code|integrate cos(6*pi*t) exp(−pi*t^2) exp(i*omega*t) /sqrt(2*pi) from -inf to inf}}. Numerical integration of closed-form continuous functions Discrete sampling of the Fourier transform can also be done by numerical integration of the definition at each value of frequency for which transform is desired.{{harvnb|Press|Flannery|Teukolsky|Vetterling|1992}}{{harvnb|Bailey|Swarztrauber|1994}}{{harvnb|Lado|1971}} The numerical integration approach works on a much broader class of functions than the analytic approach. Numerical integration of a series of ordered pairs If the input function is a series of ordered pairs, numerical integration reduces to just a summation over the set of data pairs.{{harvnb|Simonen|Olkkonen|1985}} The DTFT is a common subcase of this more general situation. Tables of important Fourier transforms The following tables record some closed-form Fourier transforms. For functions {{math|f(x)}} and {{math|g(x)}} denote their Fourier transforms by {{math|fÌ‚}} and {{math|ĝ}}. Only the three most common conventions are included. It may be useful to notice that entry 105 gives a relationship between the Fourier transform of a function and the original function, which can be seen as relating the Fourier transform and its inverse. Functional relationships, one-dimensional The Fourier transforms in this table may be found in {{harvtxt|Erdélyi|1954}} or {{harvtxt|Kammler|2000|loc=appendix}}.{| class=“wikitable“! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !!Remarks
a|} widehat{f}left( frac{xi}{a} right),	a|} widehat{f}left( frac{omega}{a} right),	a|} widehat{f}left( frac{omega}{a} right),	{{abs	a}}}} is large, then {{math>f(ax)}} is concentrated around 0 and{{br}} frac{1}{	}hat{f} left( frac{omega}{a} right),{{br}}spreads out and flattens.
th}}-order derivative.As {{math|f}} is a Schwartz function
	TITLE=THE INTEGRATION PROPERTY OF THE FOURIER TRANSFORM	URL-STATUS=LIVE	ARCHIVE-DATE=2022-01-26	WEBSITE=THE FOURIER TRANSFORM .COM, Note: delta is the Dirac delta function and C is the average (DC) value of f(x) such that int_{-infty}^infty (f(x) - C) , dx = 0
f ∗ g}} denotes the convolution of {{mvar	g}} — this rule is the convolution theorem
f(x)}} purely real| widehat{f}(-xi) = overline{widehat{f}(xi)},| widehat{f}(-omega) = overline{widehat{f}(omega)},| widehat{f}(-omega) = overline{widehat{f}(omega)},	{{overline|z}}}} indicates the complex conjugate.
f(x)}} purely imaginary| widehat{f}(-xi) = -overline{widehat{f}(xi)},| widehat{f}(-omega) = -overline{widehat{f}(omega)},| widehat{f}(-omega) = -overline{widehat{f}(omega)},	{{overline|z}}}} indicates the complex conjugate.
				Complex conjugate>Complex conjugation, generalization of 110 and 113

Square-integrable functions, one-dimensional

The Fourier transforms in this table may be found in {{harvtxt|Campbell|Foster|1948}}, {{harvtxt|Erdélyi|1954}}, or {{harvtxt|Kammler|2000|loc=appendix}}.{| class=“wikitable“! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks|| f(x),|begin{align} &hat{f}(xi) triangleq hat f_1(xi) &= int_{-infty}^infty f(x) e^{-i 2pi xi x}, dx end{align}|begin{align} &hat{f}(omega) triangleq hat f_2(omega) &= frac{1}{sqrt{2 pi}} int_{-infty}^infty f(x) e^{-i omega x}, dx end{align}|begin{align} &hat{f}(omega) triangleq hat f_3(omega) &= int_{-infty}^infty f(x) e^{-i omega x}, dx end{align}|Definitionsrect}} 201| operatorname{rect}(a x) ,a|}, operatorname{sinc}left(frac{xi}{a}right)| frac{1}{sqrt{2 pi a^2}}, operatorname{sinc}left(frac{omega}{2pi a}right)a|}, operatorname{sinc}left(frac{omega}{2pi a}right)rectangular function>rectangular pulse and the normalized sinc function, here defined as {{mathx) = {{sfrac>sin(Ï€x)|Ï€x}}}}| 202| operatorname{sinc}(a x),a|}, operatorname{rect}left(frac{xi}{a} right),| frac{1}{sqrt{2pi a^2}}, operatorname{rect}left(frac{omega}{2 pi a}right)a|}, operatorname{rect}left(frac{omega}{2 pi a}right)rectangular function is an ideal low-pass filter, and the sinc function is the Anticausal system>non-causal impulse response of such a filter. The sinc function is defined here as {{mathx) = {{sfrac>sin(Ï€x)|Ï€x}}}}| 203| operatorname{sinc}^2 (a x)a|}, operatorname{tri} left( frac{xi}{a} right) | frac{1}{sqrt{2pi a^2}}, operatorname{tri} left( frac{omega}{2pi a} right) a|}, operatorname{tri} left( frac{omega}{2pi a} right) tri(x)}} is the triangular function| 204| operatorname{tri} (a x)a|}, operatorname{sinc}^2 left( frac{xi}{a} right) ,| frac{1}{sqrt{2pi a^2}} , operatorname{sinc}^2 left( frac{omega}{2pi a} right) a|} , operatorname{sinc}^2 left( frac{omega}{2pi a} right) | Dual of rule 203.| 205| e^{- a x} u(x) ,| frac{1}{a + i 2pi xi}| frac{1}{sqrt{2 pi} (a + i omega)}| frac{1}{a + i omega}u(x)}} is the Heaviside step function and {{math>a > 0}}.| 206| e^{-alpha x^2},| sqrt{frac{pi}{alpha}}, e^{-frac{(pi xi)^2}{alpha}}| frac{1}{sqrt{2 alpha}}, e^{-frac{omega^2}{4 alpha}}| sqrt{frac{pi}{alpha}}, e^{-frac{omega^2}{4 alpha}}Gaussian function {{math>e−αx2}} is its own Fourier transform for some choice of {{mvarRe(α) > 0}}.| 208x|} ,| frac{2 a}{a^2 + 4 pi^2 xi^2} | sqrt{frac{2}{pi}} , frac{a}{a^2 + omega^2} | frac{2a}{a^2 + omega^{2}} Re(a) > 0}}. That is, the Fourier transform of a two-sided decaying exponential function is a Lorentzian function.| 209| operatorname{sech}(a x) ,| frac{pi}{a} operatorname{sech} left( frac{pi^2}{ a} xi right)| frac{1}{a}sqrt{frac{pi}{2}} operatorname{sech}left( frac{pi}{2 a} omega right)| frac{pi}{a}operatorname{sech}left( frac{pi}{2 a} omega right)Hyperbolic function>Hyperbolic secant is its own Fourier transform| 210| e^{-frac{a^2 x^2}2} H_n(a x),| frac{sqrt{2pi}(-i)^n}{a} e^{-frac{2pi^2xi^2}{a^2}} H_nleft(frac{2pixi}aright)| frac{(-i)^n}{a} e^{-frac{omega^2}{2 a^2}} H_nleft(frac omega aright)| frac{(-i)^n sqrt{2pi}}{a} e^{-frac{omega^2}{2 a^2}} H_nleft(frac omega a right)Hn}} is the {{mvarHermite polynomial. If {{math>a {{=}} 1}} then the Gauss–Hermite functions are eigenfunctions of the Fourier transform operator. For a derivation, see Hermite polynomials#Hermite functions as eigenfunctions of the Fourier transform. The formula reduces to 206 for {{math>n {{=}} 0}}.

Distributions, one-dimensional

The Fourier transforms in this table may be found in {{harvtxt|Erdélyi|1954}} or {{harvtxt|Kammler|2000|loc=appendix}}.{| class=“wikitable“! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks|| f(x),|begin{align} &hat{f}(xi) triangleq hat f_1(xi) &= int_{-infty}^infty f(x) e^{-i 2pi xi x}, dx end{align}|begin{align} &hat{f}(omega) triangleq hat f_2(omega) &= frac{1}{sqrt{2 pi}} int_{-infty}^infty f(x) e^{-i omega x}, dx end{align}|begin{align} &hat{f}(omega) triangleq hat f_3(omega) &= int_{-infty}^infty f(x) e^{-i omega x}, dx end{align}|Definitions| 301| 1| delta(xi)| sqrt{2pi}, delta(omega)| 2pidelta(omega)δ(ξ)}} denotes the Dirac delta function.| 302| delta(x),| 1| frac{1}{sqrt{2pi}},| 1|Dual of rule 301.| 303| e^{i a x}| deltaleft(xi - frac{a}{2pi}right)| sqrt{2 pi}, delta(omega - a)| 2 pidelta(omega - a)|This follows from 103 and 301.| 304| cos (a x)| frac{ deltaleft(xi - frac{a}{2pi}right)+deltaleft(xi+frac{a}{2pi}right)}{2}| sqrt{2 pi},frac{delta(omega-a)+delta(omega+a)}{2}| pileft(delta(omega-a)+delta(omega+a)right)|This follows from rules 101 and 303 using Euler’s formula:{{br}}cos(a x) = frac{e^{i a x} + e^{-i a x}}{2}.| 305| sin( ax)| frac{deltaleft(xi-frac{a}{2pi}right)-deltaleft(xi+frac{a}{2pi}right)}{2i}| sqrt{2 pi},frac{delta(omega-a)-delta(omega+a)}{2i}| -ipibigl(delta(omega-a)-delta(omega+a)bigr)|This follows from 101 and 303 using{{br}}sin(a x) = frac{e^{i a x} - e^{-i a x}}{2i}.| 306| cos left( a x^2 right) | sqrt{frac{pi}{a}} cos left( frac{pi^2 xi^2}{a} - frac{pi}{4} right) | frac{1}{sqrt{2 a}} cos left( frac{omega^2}{4 a} - frac{pi}{4} right) | sqrt{frac{pi}{a}} cos left( frac{omega^2}{4a} - frac{pi}{4} right) |This follows from 101 and 207 using{{br}}cos(a x^2) = frac{e^{i a x^2} + e^{-i a x^2}}{2}.| 307| sin left( a x^2 right) | - sqrt{frac{pi}{a}} sin left( frac{pi^2 xi^2}{a} - frac{pi}{4} right) | frac{-1}{sqrt{2 a}} sin left( frac{omega^2}{4 a} - frac{pi}{4} right) | -sqrt{frac{pi}{a}}sin left( frac{omega^2}{4a} - frac{pi}{4} right)|This follows from 101 and 207 using{{br}}sin(a x^2) = frac{e^{i a x^2} - e^{-i a x^2}}{2i}.|308| e^{-pi ialpha x^2},| frac{1}{sqrt{alpha}}, e^{-ifrac{pi}{4}} e^{ifrac{pi xi^2}{alpha}}| frac{1}{sqrt{2pi alpha}}, e^{-ifrac{pi}{4}} e^{ifrac{omega^2}{4pi alpha}}| frac{1}{sqrt{alpha}}, e^{-ifrac{pi}{4}} e^{ifrac{omega^2}{4pi alpha}} |Here it is assumed alpha is real. For the case that alpha is complex see table entry 206 above.| 309| x^n,| left(frac{i}{2pi}right)^n delta^{(n)} (xi)| i^n sqrt{2pi} delta^{(n)} (omega)| 2pi i^ndelta^{(n)} (omega)n}} is a natural number and {{mathδ{{isup>(n)}}(ξ)}} is the {{mvar|n}}th distribution derivative of the Dirac delta function. This rule follows from rules 107 and 301. Combining this rule with 101, we can transform all polynomials.| 310| delta^{(n)}(x)| (i 2pi xi)^n| frac{(iomega)^n}{sqrt{2pi}} | (iomega)^nδ{{isupn)}}(ξ)}} is the {{mvar>n}}th distribution derivative of the Dirac delta function. This rule follows from 106 and 302.| 311| frac{1}{x}| -ipisgn(xi)| -isqrt{frac{pi}{2}}sgn(omega)| -ipisgn(omega)sgn(ξ)}} is the sign function. Note that {{math1|x}}}} is not a distribution. It is necessary to use the Cauchy principal value when testing against Schwartz functions. This rule is useful in studying the Hilbert transform.| 312|begin{align}&frac{1}{x^n} &:= frac{(-1)^{n-1}}{(n-1)!}frac{d^n}{dx^n}log |x|end{align}| -ipi frac{(-i 2pi xi)^{n-1}}{(n-1)!} sgn(xi)| -isqrt{frac{pi}{2}}, frac{(-iomega)^{n-1}}{(n-1)!}sgn(omega)| -ipi frac{(-iomega)^{n-1}}{(n-1)!}sgn(omega){{sfracxn}}}} is the homogeneous distribution defined by the distributional derivative{{br}}frac{(-1)^{n-1}}{(n-1)!}frac{d^n}{dx^n}log| 313x|^alpha2pixi|^{alpha+1}}omega|^{alpha+1}} omega|^{alpha+1}} 0 > α > −1}}. For {{mathα > 0}} some singular terms arise at the origin that can be found by differentiating 320. If {{math>Re α > −1}}, then {{mathx}}α}} is a locally integrable function, and so a tempered distribution. The function {{mathα ↦ {{abs>x}}α}} is a holomorphic function from the right half-plane to the space of tempered distributions. It admits a unique meromorphic extension to a tempered distribution, also denoted {{mathx}}α}} for {{math|α ≠ −1, −3, ...}} (See homogeneous distribution.)| x|}} xi|}} omega|}}omega|}} | Special case of 313.| 314| sgn(x)| frac{1}{ipi xi}| sqrt{frac{2}{pi}} frac{1}{iomega } | frac{2}{iomega }|The dual of rule 311. This time the Fourier transforms need to be considered as a Cauchy principal value.| 315| u(x)| frac{1}{2}left(frac{1}{i pi xi} + delta(xi)right)| sqrt{frac{pi}{2}} left( frac{1}{i pi omega} + delta(omega)right)| pileft( frac{1}{i pi omega} + delta(omega)right)u(x)}} is the Heaviside unit step function; this follows from rules 101, 301, and 314.| 316| sum_{n=-infty}^{infty} delta (x - n T)| frac{1}{T} sum_{k=-infty}^{infty} delta left( xi -frac{k }{T}right)| frac{sqrt{2pi }}{T}sum_{k=-infty}^{infty} delta left( omega -frac{2pi k}{T}right)| frac{2pi}{T}sum_{k=-infty}^{infty} delta left( omega -frac{2pi k}{T}right)|This function is known as the Dirac comb function. This result can be derived from 302 and 102, together with the fact that{{br}}begin{align}& sum_{n=-infty}^{infty} e^{inx}

{}& 2pisum_{k-infty}^{infty} delta(x+2pi k)

end{align}{{br}}as distributions.| 317| J_0 (x)| frac{2, operatorname{rect}(pixi)}{sqrt{1 - 4 pi^2 xi^2}} | sqrt{frac{2}{pi}} , frac{operatorname{rect}left( frac{omega}{2} right)}{sqrt{1 - omega^2}} | frac{2,operatorname{rect}left(frac{omega}{2} right)}{sqrt{1 - omega^2}}J0(x)}} is the zeroth order Bessel function of first kind.| 318| J_n (x)| frac{2 (-i)^n T_n (2 pi xi) operatorname{rect}(pi xi)}{sqrt{1 - 4 pi^2 xi^2}} | sqrt{frac{2}{pi}} frac{ (-i)^n T_n (omega) operatorname{rect} left( frac{omega}{2} right)}{sqrt{1 - omega^2}} | frac{2(-i)^n T_n (omega) operatorname{rect} left( frac{omega}{2} right)}{sqrt{1 - omega^2}} Jn(x)}} is the {{mvarBessel function of first kind. The function {{math>Tn(x)}} is the Chebyshev polynomial of the first kind.| 319 x right| xi right|} - gamma delta left( xi right) omega right|} - sqrt{2 pi} gamma delta left( omega right) omega right|} - 2 pi gamma delta left( omega right) γ}} is the Euler–Mascheroni constant. It is necessary to use a finite part integral when testing {{math1ξ}}}}}} or {{math1ω}}}}}}against Schwartz functions. The details of this might change the coefficient of the delta function. | 320| left( mp ix right)^{-alpha}| frac{left(2piright)^alpha}{Gammaleft(alpharight)}uleft(pm xi right)left(pm xi right)^{alpha-1} | frac{sqrt{2pi}}{Gammaleft(alpharight)}uleft(pmomegaright)left(pmomegaright)^{alpha-1} | frac{2pi}{Gammaleft(alpharight)}uleft(pmomegaright)left(pmomegaright)^{alpha-1} 1 > α > 0}}. Use differentiation to derive formula for higher exponents. {{mvar|u}} is the Heaviside function.

Two-dimensional functions {| class“wikitable”

! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks|400| f(x,y)|begin{align}& hat{f}(xi_x, xi_y)triangleq & iint f(x,y) e^{-i 2pi(xi_x x+xi_y y)},dx,dy end{align}|begin{align}& hat{f}(omega_x,omega_y)triangleq & frac{1}{2 pi} iint f(x,y) e^{-i (omega_x x +omega_y y)}, dx,dy end{align}|begin{align}& hat{f}(omega_x,omega_y)triangleq & iint f(x,y) e^{-i(omega_x x+omega_y y)}, dx,dy end{align}ξx}}, {{mvarωx}}, {{mvar|ωy}} are real numbers. The integrals are taken over the entire plane.|401| e^{-pileft(a^2x^2+b^2y^2right)}ab|} e^{-pileft(frac{xi_x^2}{a^2} + frac{xi_y^2}{b^2}right)}ab|} e^{-frac{1}{4pi}left(frac{omega_x^2}{a^2} + frac{omega_y^2}{b^2}right)}ab|} e^{-frac{1}{4pi}left(frac{omega_x^2}{a^2} + frac{omega_y^2}{b^2}right)}|Both functions are Gaussians, which may not have unit volume.|402| operatorname{circ}left(sqrt{x^2+y^2}right)| frac{J_1left(2 pi sqrt{xi_x^2+xi_y^2}right)}{sqrt{xi_x^2+xi_y^2}}| frac{J_1left(sqrt{omega_x^2+omega_y^2}right)}{sqrt{omega_x^2+omega_y^2}}| frac{2pi J_1left(sqrt{omega_x^2+omega_y^2}right)}{sqrt{omega_x^2+omega_y^2}}1=circ(r) = 1}} for {{mathr ≤ 1}}, and is 0 otherwise. The result is the amplitude distribution of the Airy disk, and is expressed using {{math>J1}} (the order-1 Bessel function of the first kind).{{harvnbWeissloc=Thm. IV.3.3}}|403| frac{1}{sqrt{x^2+y^2}}| frac{1}{sqrt{xi_x^2+xi_y^2}}| frac{1}{sqrt{omega_x^2+omega_y^2}}| frac{2pi}{sqrt{omega_x^2+omega_y^2}}Hankel transform of {{math>1=r−1}}, a 2-D Fourier “self-transform”.{{harvnb2010}}|404| frac{i}{x+i y}| frac{1}{xi_x+ixi_y}| frac{1}{omega_x+iomega_y}| frac{2pi}{omega_x+iomega_y}|n}}-dimensional functions{| class“wikitable“”>

Formulas for general {{math|n}}-dimensional functions{| class“wikitable”

! !! Function !! Fourier transform {{br}} unitary, ordinary frequency !! Fourier transform {{br}} unitary, angular frequency !! Fourier transform {{br}} non-unitary, angular frequency !! Remarks|500| f(mathbf x),|begin{align} &hat{f_1}(boldsymbol xi) triangleq &int_{mathbb{R}^n}f(mathbf x) e^{-i 2pi boldsymbol xi cdot mathbf x }, d mathbf x end{align}|begin{align} &hat{f_2}(boldsymbol omega) triangleq &frac{1}{{(2 pi)}^frac{n}{2}} int_{mathbb{R}^n} f(mathbf x) e^{-i boldsymbol omega cdot mathbf x}, d mathbf x end{align}|begin{align} &hat{f_3}(boldsymbol omega) triangleq &int_{mathbb{R}^n}f(mathbf x) e^{-i boldsymbol omega cdot mathbf x}, d mathbf x end{align}||501mathbf xmathbf x|^2right)^deltaboldsymbol xiboldsymbol xi|)boldsymbol omegarightboldsymbol omega|)frac{boldsymbol omega}{2pi}rightboldsymbol omega|!)χ[0, 1]}} is the indicator function of the interval {{mathΓ(x)}} is the gamma function. The function {{mathJ{{sfrac>nδ}} is a Bessel function of the first kind, with order {{math>{{sfracn>2}} + δ}}. Taking {{mathn = 2}} and {{math>1=δ = 0}} produces 402.{{harvnbWeissloc=Thm. 4.15}}|502mathbf x|^{-alpha}, quad 0 < operatorname{Re} alpha < n.boldsymbol xi|^{-(n - alpha)}boldsymbol omega|^{-(n - alpha)}boldsymbol omega|^{-(n - alpha)}Riesz potential where the constant is given by{{br}}c_{n, alpha} = pi^frac{n}{2} 2^alpha frac{Gammaleft(frac{alpha}{2}right)}{Gammaleft(frac{n - alpha}{2}right)}.{{br}}The formula also holds for all {{math>α ≠ n, n + 2, ...}} by analytic continuation, but then the function and its Fourier transforms need to be understood as suitably regularized tempered distributions. See homogeneous distribution.In {{harvnbShilovp=363}}, with the non-unitary conventions of this table, the transform of ^lambda is given to be{{br}} 2^{lambda+n}pi^{tfrac12 n}frac{Gammaleft(frac{lambda+n}{2}right)}{Gammaleft(-frac{lambda}{2}right)}^{-lambda-n}{{br}}from which this follows, with lambda=-alpha.|503boldsymbol sigmaright|left(2piright)^frac{n}{2}} e^{-frac{1}{2} mathbf x^{mathrm T} boldsymbol sigma^{-mathrm T} boldsymbol sigma^{-1} mathbf x}| e^{-2pi^2 boldsymbol xi^{mathrm T} boldsymbol sigma boldsymbol sigma^{mathrm T} boldsymbol xi} | (2pi)^{-frac{n}{2}} e^{-frac{1}{2} boldsymbol omega^{mathrm T} boldsymbol sigma boldsymbol sigma^{mathrm T} boldsymbol omega} | e^{-frac{1}{2} boldsymbol omega^{mathrm T} boldsymbol sigma boldsymbol sigma^{mathrm T} boldsymbol omega} multivariate normal distribution normalized to 1 with a mean of 0. Bold variables are vectors or matrices. Following the notation of the aforementioned page, {{math>Σ {{=}} σ σT}} and {{math|Σ−1 {{=}} σ−T σ−1}}|504mathbf x|}boldsymbol{xi}|^2right)^frac{n+1}{2}}boldsymbol{omega}|^2right)^frac{n+1}{2}}boldsymbol{omega}|^2right)^frac{n+1}{2}}Stein1971Re(α) > 0}}

See also

{{div col|colwidth=22em}}

• Analog signal processing
• Beevers–Lipson strip
• Constant-Q transform
• Discrete Fourier transform
• DFT matrix
• Fast Fourier transform
• Fourier integral operator
• Fourier inversion theorem
• Fourier multiplier
• Fourier series
• Fourier sine transform
• Fourier–Deligne transform
• Fourier–Mukai transform
• Fractional Fourier transform
• Indirect Fourier transform
• Integral transform
  • Hankel transform
  • Hartley transform
• Laplace transform
• Least-squares spectral analysis
• Linear canonical transform
• Mellin transform
• Multidimensional transform
• NGC 4622, especially the image NGC 4622 Fourier transform {{math|1=m = 2}}.
• Nonlocal operator
• Quantum Fourier transform
• Quadratic Fourier transform
• Short-time Fourier transform
• Spectral density
  • Spectral density estimation
• Symbolic integration
• Time stretch dispersive Fourier transform
• Transform (mathematics)

{{div col end}}

Notes

{{reflist|group=note}}

Citations

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References

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External links

• {{Commons category-inline}}
• Encyclopedia of Mathematics
• {{MathWorld | urlname = FourierTransform | title = Fourier Transform}}
• Fourier Transform in Crystallography

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