Editing Fourier transform (section)

== 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:
<math display="block">\hat{f}(\boldsymbol{\xi}) = \mathcal{F}(f)(\boldsymbol{\xi}) = \int_{\R^n} f(\mathbf{x}) e^{-i 2\pi \boldsymbol{\xi}\cdot\mathbf{x}} \, d\mathbf{x}</math>
where {{math|'''x'''}} and {{math|'''ξ'''}} are {{mvar|n}}-dimensional [[vector (mathematics)|vectors]], and {{math|'''x''' · '''ξ'''}} is the [[dot product]] of the vectors. Alternatively, {{math|'''ξ'''}} can be viewed as belonging to the [[dual space|dual vector space]] <math>\R^{n\star}</math>, in which case the dot product becomes the [[tensor contraction|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.<ref name="Stein-Weiss-1971" />

=== Uncertainty principle ===
{{Further|Uncertainty principle}}

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 representation|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 vector space|symplectic form]].

Suppose {{math|''f''(''x'')}} is an integrable and [[square-integrable]] function. Without loss of generality, assume that {{math|''f''(''x'')}} is normalized:
<math display="block">\int_{-\infty}^\infty |f(x)|^2 \,dx=1.</math>

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'' defined by<ref>{{harvnb|Pinsky|2002|loc=chpt. 2.4.3 The Uncertainty Principle}}</ref>
<math display="block">D_0(f)=\int_{-\infty}^\infty x^2|f(x)|^2\,dx.</math>

In probability terms, this is the [[Moment (mathematics)|second moment]] of {{math|{{abs|''f''(''x'')}}<sup>2</sup>}} 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, then
<math display="block">D_0(f)D_0(\hat{f}) \geq \frac{1}{16\pi^2}.</math>

The equality is attained only in the case
<math display="block">\begin{align} 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{align} </math>
where {{math|''σ'' > 0}} is arbitrary and {{math|1=''C''<sub>1</sub> = {{sfrac|{{radic|2|4}}|{{sqrt|''σ''}}}}}} so that {{mvar|f}} is {{math|''L''<sup>2</sup>}}-normalized. In other words, where {{mvar|f}} is a (normalized) [[Gaussian function]] with variance {{math|''σ''<sup>2</sup>/2{{pi}}}}, centered at zero, and its Fourier transform is a Gaussian function with variance {{math|''σ''<sup>−2</sup>/2{{pi}}}}.  Gaussian functions are examples of [[Schwartz function]]s (see the discussion on tempered distributions below). 

In fact, this inequality implies that:
<math display="block">\left(\int_{-\infty}^\infty (x-x_0)^2|f(x)|^2\,dx\right)\left(\int_{-\infty}^\infty(\xi-\xi_0)^2\left|\hat{f}(\xi)\right|^2\,d\xi\right)\geq \frac{1}{16\pi^2}, \quad \forall x_0, \xi_0 \in \mathbb{R}.</math>
In [[quantum mechanics]], the [[momentum]] and position [[wave function]]s 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]].<ref>{{harvnb|Stein|Shakarchi|2003|loc= chpt. 5.4 The Heisenberg uncertainty principle}}</ref>

A stronger uncertainty principle is the [[Hirschman uncertainty|Hirschman uncertainty principle]], which is expressed as:
<math display="block">H\left(\left|f\right|^2\right)+H\left(\left|\hat{f}\right|^2\right)\ge \log\left(\frac{e}{2}\right)</math>
where {{math|''H''(''p'')}} is the [[differential entropy]] of the [[probability density function]] {{math|''p''(''x'')}}:
<math display="block">H(p) = -\int_{-\infty}^\infty p(x)\log\bigl(p(x)\bigr) \, dx</math>
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 ===
{{Main|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<ref>{{harvnb|Chatfield|2004|p=113}}</ref>) {{mvar|λ}} by
<math display="block">f(t) = \int_0^\infty \bigl( a(\lambda ) \cos( 2\pi \lambda t) + b(\lambda ) \sin( 2\pi \lambda t)\bigr) \, d\lambda.</math>

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):
<math display="block"> a (\lambda) = 2\int_{-\infty}^\infty f(t) \cos(2\pi\lambda t) \, dt</math>
and
<math display="block"> b (\lambda) = 2\int_{-\infty}^\infty f(t) \sin(2\pi\lambda t) \, dt. </math>

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
<math display="block"> f(t) = 2\int_0 ^{\infty} \int_{-\infty}^{\infty} f(\tau) \cos\bigl( 2\pi \lambda(\tau-t)\bigr) \, d\tau \, d\lambda.</math>
together with trigonometric identities. This is referred to as Fourier's integral formula.<ref name="Kolmogorov-Fomin-1999" /><ref>{{harvnb|Fourier|1822|p=441}}</ref><ref>{{harvnb|Poincaré|1895|p=102}}</ref><ref>{{harvnb|Whittaker|Watson|1927|p=188}}</ref>

=== Spherical harmonics ===
Let the set of [[Homogeneous polynomial|homogeneous]] [[Harmonic function|harmonic]] [[polynomial]]s of degree {{mvar|k}} on {{math|'''R'''<sup>''n''</sup>}} be denoted by {{math|'''A'''<sub>''k''</sub>}}. The set {{math|'''A'''<sub>''k''</sub>}} 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''<sup>−π{{abs|''x''}}<sup>2</sup></sup>''P''(''x'')}} for some {{math|''P''(''x'')}} in {{math|'''A'''<sub>''k''</sub>}}, then {{math|1=''f̂''(''ξ'') = ''i''{{isup|−''k''}} ''f''(''ξ'')}}. Let the set {{math|'''H'''<sub>''k''</sub>}} be the closure in {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} of linear combinations of functions of the form {{math|''f''({{abs|''x''}})''P''(''x'')}} where {{math|''P''(''x'')}} is in {{math|'''A'''<sub>''k''</sub>}}. The space {{math|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} is then a direct sum of the spaces {{math|'''H'''<sub>''k''</sub>}} and the Fourier transform maps each space {{math|'''H'''<sub>''k''</sub>}} to itself and is possible to characterize the action of the Fourier transform on each space {{math|'''H'''<sub>''k''</sub>}}.<ref name="Stein-Weiss-1971" />

Let {{math|1=''f''(''x'') = ''f''<sub>0</sub>({{abs|''x''}})''P''(''x'')}} (with {{math|''P''(''x'')}} in {{math|'''A'''<sub>''k''</sub>}}), then
<math display="block">\hat{f}(\xi)=F_0(|\xi|)P(\xi)</math>
where
<math display="block">F_0(r) = 2\pi i^{-k}r^{-\frac{n+2k-2}{2}} \int_0^\infty f_0(s)J_\frac{n+2k-2}{2}(2\pi rs)s^\frac{n+2k}{2}\,ds.</math>

Here {{math|''J''<sub>(''n'' + 2''k'' − 2)/2</sub>}} denotes the [[Bessel function]] of the first kind with order {{math|{{sfrac|''n'' + 2''k'' − 2|2}}}}. When {{math|''k'' {{=}} 0}} this gives a useful formula for the Fourier transform of a radial function.<ref>{{harvnb|Grafakos|2004}}</ref> This is essentially the [[Hankel transform]]. Moreover, there is a simple recursion relating the cases {{math|''n'' + 2}} and {{mvar|n}}<ref>{{harvnb|Grafakos|Teschl|2013}}</ref> 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|''L''<sup>2</sup>('''R'''<sup>''n''</sup>)}} 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|'''R'''<sup>''n''</sup>}} is of particular interest. In this case the Tomas–[[Elias Stein|Stein]] restriction theorem states that the restriction of the Fourier transform to the unit sphere in {{math|'''R'''<sup>''n''</sup>}} is a bounded operator on {{math|''L''{{isup|''p''}}}} provided {{math|1 ≤ ''p'' ≤ {{sfrac|2''n'' + 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|''E''<sub>''R''</sub>}} indexed by {{math|''R'' ∈ (0,∞)}}: such as balls of radius {{mvar|R}} centered at the origin, or cubes of side {{math|2''R''}}. For a given integrable function {{mvar|f}}, consider the function {{mvar|f<sub>R</sub>}} defined by:
<math display="block">f_R(x) = \int_{E_R}\hat{f}(\xi) e^{i 2\pi x\cdot\xi}\, d\xi, \quad x \in \mathbb{R}^n.</math>

Suppose in addition that {{math|''f'' ∈ ''L''{{isup|''p''}}('''R'''<sup>''n''</sup>)}}. For {{math|''n'' {{=}} 1}} and {{math|1 < ''p'' < ∞}}, if one takes {{math|''E<sub>R</sub>'' {{=}} (−''R'', ''R'')}}, then {{mvar|f<sub>R</sub>}} 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|E<sub>R</sub>}} is taken to be a cube with side length {{mvar|R}}, then convergence still holds. Another natural candidate is the Euclidean ball {{math|''E''<sub>''R''</sub> {{=}} {''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''}}('''R'''<sup>''n''</sup>)}}. 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}}.<ref name="Duoandikoetxea-2001" /> In fact, when {{math|''p'' ≠ 2}}, this shows that not only may {{mvar|f<sub>R</sub>}} fail to converge to {{mvar|f}} in {{math|''L''{{isup|''p''}}}}, but for some functions {{math|''f'' ∈ ''L''{{isup|''p''}}('''R'''<sup>''n''</sup>)}}, {{mvar|f<sub>R</sub>}} is not even an element of {{math|''L''{{isup|''p''}}}}.