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

All of the basic properties listed above hold for the $n$ -dimensional Fourier transform, as do Plancherel's and Parseval's theorem. When the function is integrable, the Fourier transform is still uniformly continuous and the Riemann–Lebesgue lemma holds.^[11]

Uncertainty principle

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

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

Suppose $f (x)$ is an integrable and square-integrable function. Without loss of generality, assume that $f (x)$ is normalized: $\int _{-\infty }^{\infty }|f(x)|^{2}\,dx=1.$

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

The spread around $x = 0$ may be measured by the dispersion about zero^[40] defined by $D_{0}(f)=\int _{-\infty }^{\infty }x^{2}|f(x)|^{2}\,dx.$

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

The uncertainty principle states that, if $f (x)$ is absolutely continuous and the functions $x \cdot f (x)$ and $f' (x)$ are square integrable, then^[14] $D_{0}(f)D_{0}({\hat {f}})\geq {\frac {1}{16\pi ^{2}}}.$

The equality is attained only in the case ${\begin{aligned}f(x)&=C_{1}\,e^{-\pi {\frac {x^{2}}{\sigma ^{2}}}}\\\therefore {\hat {f}}(\xi )&=\sigma C_{1}\,e^{-\pi \sigma ^{2}\xi ^{2}}\end{aligned}}$ where $σ > 0$ is arbitrary and $C 1 = ⁠ 4 \sqrt 2 / \sqrt σ ⁠$ so that $f$ is $L 2$ -normalized.^[14] In other words, where $f$ is a (normalized) Gaussian function with variance $σ 2 /2 π$ , centered at zero, and its Fourier transform is a Gaussian function with variance $σ -2 /2 π$ .

In fact, this inequality implies that: $\left(\int _{-\infty }^{\infty }(x-x_{0})^{2}|f(x)|^{2}\,dx\right)\left(\int _{-\infty }^{\infty }(\xi -\xi _{0})^{2}\left|{\hat {f}}(\xi )\right|^{2}\,d\xi \right)\geq {\frac {1}{16\pi ^{2}}}$ for any $x 0$ , $ξ 0 \in R$ .^[41]

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.^[42]

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

Sine and cosine transforms

Fourier's original formulation of the transform did not use complex numbers, but rather sines and cosines. Statisticians and others still use this form. An absolutely integrable function $f$ for which Fourier inversion holds can be expanded in terms of genuine frequencies (avoiding negative frequencies, which are sometimes considered hard to interpret physically^[43]) $λ$ by $f(t)=\int _{0}^{\infty }{\bigl (}a(\lambda )\cos(2\pi \lambda t)+b(\lambda )\sin(2\pi \lambda t){\bigr )}\,d\lambda .$

This is called an expansion as a trigonometric integral, or a Fourier integral expansion. The coefficient functions $a$ and $b$ can be found by using variants of the Fourier cosine transform and the Fourier sine transform (the normalisations are, again, not standardised): $a(\lambda )=2\int _{-\infty }^{\infty }f(t)\cos(2\pi \lambda t)\,dt$ and $b(\lambda )=2\int _{-\infty }^{\infty }f(t)\sin(2\pi \lambda t)\,dt.$

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

The function $f$ can be recovered from the sine and cosine transform using $f(t)=2\int _{0}^{\infty }\int _{-\infty }^{\infty }f(\tau )\cos {\bigl (}2\pi \lambda (\tau -t){\bigr )}\,d\tau \,d\lambda .$ together with trigonometric identities. This is referred to as Fourier's integral formula.^[36]^[44]^[45]^[46]

Spherical harmonics

Let the set of homogeneous harmonic polynomials of degree $k$ on $R n$ be denoted by $A k$ . The set $A k$ consists of the solid spherical harmonics of degree $k$ . The solid spherical harmonics play a similar role in higher dimensions to the Hermite polynomials in dimension one. Specifically, if $f (x) = e -π | x | 2 P (x)$ for some $P (x)$ in $A k$ , then $f̂ (ξ) = i - k f (ξ)$ . Let the set $H k$ be the closure in $L 2 (R n)$ of linear combinations of functions of the form $f (| x |) P (x)$ where $P (x)$ is in $A k$ . The space $L 2 (R n)$ is then a direct sum of the spaces $H k$ and the Fourier transform maps each space $H k$ to itself and is possible to characterize the action of the Fourier transform on each space $H k$ .^[11]

Let $f (x) = f 0 (| x |) P (x)$ (with $P (x)$ in $A k$ ), then ${\hat {f}}(\xi )=F_{0}(|\xi |)P(\xi )$ where $F_{0}(r)=2\pi i^{-k}r^{-{\frac {n+2k-2}{2}}}\int _{0}^{\infty }f_{0}(s)J_{\frac {n+2k-2}{2}}(2\pi rs)s^{\frac {n+2k}{2}}\,ds.$

Here $J (n + 2 k - 2)/2$ denotes the Bessel function of the first kind with order $⁠ n + 2 k - 2 / 2 ⁠$ . When $k = 0$ this gives a useful formula for the Fourier transform of a radial function.^[47] This is essentially the Hankel transform. Moreover, there is a simple recursion relating the cases $n + 2$ and $n$ ^[48] 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 $L 2 (R n)$ function cannot be defined on sets of measure 0. It is still an active area of study to understand restriction problems in $L p$ for $1 < p < 2$ . It is possible in some cases to define the restriction of a Fourier transform to a set $S$ , provided $S$ has non-zero curvature. The case when $S$ is the unit sphere in $R n$ 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 $R n$ is a bounded operator on $L p$ provided $1 \leq p \leq ⁠ 2 n + 2 / n + 3 ⁠$ .

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 $E R$ indexed by $R \in (0,\infty)$ : such as balls of radius $R$ centered at the origin, or cubes of side $2 R$ . For a given integrable function $f$ , consider the function $f R$ defined by: $f_{R}(x)=\int _{E_{R}}{\hat {f}}(\xi )e^{i2\pi x\cdot \xi }\,d\xi ,\quad x\in \mathbb {R} ^{n}.$

Suppose in addition that $f \in L p (R n)$ . For $n = 1$ and $1 < p < \infty$ , if one takes $E R = (- R, R)$ , then $f R$ converges to $f$ in $L p$ as $R$ tends to infinity, by the boundedness of the Hilbert transform. Naively one may hope the same holds true for $n > 1$ . In the case that $E R$ is taken to be a cube with side length $R$ , then convergence still holds. Another natural candidate is the Euclidean ball $E R = {ξ : | ξ | < R}$ . In order for this partial sum operator to converge, it is necessary that the multiplier for the unit ball be bounded in $L p (R n)$ . For $n \geq 2$ it is a celebrated theorem of Charles Fefferman that the multiplier for the unit ball is never bounded unless $p = 2$ .^[23] In fact, when $p \neq 2$ , this shows that not only may $f R$ fail to converge to $f$ in $L p$ , but for some functions $f \in L p (R n)$ , $f R$ is not even an element of $L p$ .