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Fourier transform in nLab

Contents

Context

Harmonic analysis

Contents

Idea

Basic idea

Generally, a Fourier transform is an isomorphism between the algebra of complex-valued functions on a suitable topological group and a convolution product-algebra structure on the Pontrjagin dual group. The study of Fourier transforms is also called Fourier analysis.

Typically, such as in the case over Cartesian space (def. below) this means to decompose any suitable function as a superposition of complex plane waves, which may be thought of as the “harmonics” of the given function. Therefore one speaks of harmonic analysis.

Generalizations

The concept of Fourier transforms of functions generalizes in a variety of ways. Core part of the subject of Fourier analysis is the generalization to Fourier transform of distributions (def. below). The asymptotic growth of the Fourier transform of distributions reflects the singularity structure of the distributions, in dependence of the direction of the wave vector (the “wave front set”). The study of this behaviour is called microlocal analysis.

If the role of complex plane waves in the Fourier transform are replaced by wavelets?, one speaks of the wavelet transform?.

For noncommutative topological groups, instead of continuous characters one should consider irreducible unitary representations, which makes the subject much more difficult. There are also generalizations in noncommutative geometry, see quantum group Fourier transform.

General definition

Let GG be a locally compact Hausdorff abelian topological group with invariant (= Haar) measure μ\mu. Then for each f∈L 1(G,μ)f\in L_1(G,\mu), define its Fourier transform f^\hat{f} as a function on its Pontrjagin dual group G^\hat{G} given by

f^(χ)=∫ Gf(x)χ(x)¯dμ(x),χ∈G^. \hat{f}(\chi) = \int_G f(x) \widebar{\chi(x)} d\mu(x),\,\,\,\chi\in\hat{G}.

The Fourier transform of f∈L 1(G,μ)f\in L_1(G,\mu) is always continuous and bounded on G^\hat{G}; the transform of the convolution of two functions is the product of the transforms of each of the functions separately.

Over the circle and the integers

In the classical case of Fourier series, where G=ℤG=\mathbb{Z} (the additive group of integers) and G^=S 1\hat{G}=S^1 (the circle group), the Fourier transform restricts to a unitary operator between the Hilbert spaces L 2(S 1,dt)L_2(S^1,d t) and l 2(ℤ)l_2(\mathbb{Z}) and the Fourier coefficients are the numbers

c n≔f^(χ n)=∫ 0 1f(t)e −2πintdt, c_n \;\coloneqq\; \hat{f}(\chi_n) \;=\; \int_0^1 f(t) e^{-2\pi \mathrm{i} n t} \mathrm{d} t \,,

for n∈ℤn\in\mathbb{Z}, where the functions χ n(t)=e 2πint\chi_n(t)= e^{2\pi i n t} form an orthonormal basis of L 2(S 1,dt)L_2(S^1,d t). The Fourier transform χ n^\hat{\chi_n} is then viewed as the ℤ\mathbb{Z}-series δ n\delta_n which in the nn-th place has 11 and elsewhere 00. The Fourier transform replaces the operator of differentiation d/dtd/d t by the operator of multiplication by the series {2πin} n∈ℤ\{2\pi i n\}_{n\in\mathbb{Z}}.

Over compact abelian groups and discrete groups

In general, if GG is a compact abelian group (whose Pontrjagin dual is discrete), one can normalize the invariant measure by μ(G)=1\mu(G)=1 and μ^(X)=card(X)\hat{\mu}(X)=card(X) for X⊂G^X\subset\hat{G}. Then the Fourier transform restricts to a unitary operator from L 2(X,μ)L_2(X,\mu) to L 2(G^,μ^)L_2(\hat{G},\hat{\mu}).

Over cyclic groups (the discretized circle)

Over Cartesian spaces

Throughout, let n∈ℕn \in \mathbb{N} and write ℝ n\mathbb{R}^n for the Cartesian space of dimension nn and write (−)⋅(−)(-) \cdot (-) for the canonical inner product on ℝ n\mathbb{R}^n:

k⋅x≔∑a=1nk nx n. k \cdot x \;\coloneqq\; \underoverset{a = 1}{n}{\sum} k_n x^n \,.

In the following by a smooth function f∈C ∞(ℝ n)f \in C^\infty(\mathbb{R}^n) on ℝ n\mathbb{R}^n we mean a smooth function with values in the complex numbers.

For f∈C ∞(ℝ n)f \in C^\infty(\mathbb{R}^n), we write f *∈C ∞(ℝ n)f^\ast \in C^\infty(\mathbb{R}^n) for its pointwise complex conjugate:

f *(x)≔(f(x)) *. f^\ast(x) \coloneqq (f(x))^\ast \,.

On functions with rapidly decreasing partial derivatives

(e.g. Hörmander 90, def. 7.1.2)

Proposition

(pointwise product and convolution product on Schwartz space)

The Schwartz space 𝒮(ℝ n)\mathcal{S}(\mathbb{R}^n) (def. ) is closed under the following operations on smooth functions f,g∈𝒮(ℝ n)↪C ∞(ℝ n)f,g \in \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)

  1. pointwise product:

    (f⋅g)(x)≔f(x)⋅g(x) (f \cdot g)(x) \coloneqq f(x) \cdot g(x)

  2. convolution product:

    (f⋆g)(x)≔∫y∈ℝ nf(y)⋅g(x−y)dvol(y). (f \star g)(x) \coloneqq \underset{y \in \mathbb{R}^n}{\int} f(y)\cdot g(x-y) \, dvol(y) \,.

(e.g. Hörmander, lemma 7.1.3)

(e.g. Hörmander, theorem 7.1.5)

Proposition

(basic properties of the Fourier transform)

The Fourier transform (−)^\widehat{(-)} (def. ) on the Schwartz space 𝒮(ℝ n)\mathcal{S}(\mathbb{R}^n) (def. ) satisfies the following properties, for all f,g∈𝒮(ℝ n)f,g \in \mathcal{S}(\mathbb{R}^n):

  1. (interchanging coordinate multiplication with partial derivatives)

    (2)x af^=+i∂ af^AAAAA−i∂ af^=k af^ \widehat{ x^a f } = + i \partial_a \widehat f \phantom{AAAAA} \widehat{ - i\partial_a f} = k_a \widehat f

  2. (interchanging pointwise multiplication with convolution product, remark ):

    (3)(f⋆g)^=f^⋅g^AAAAf⋅g^=(2π) −nf^⋆g^ \widehat {(f \star g)} = \widehat{f} \cdot \widehat{g} \phantom{AAAA} \widehat{ f \cdot g } = (2\pi)^{-n} \widehat{f} \star \widehat{g}

  3. (unitarity, Parseval's theorem)

    ∫x∈ℝ nf(x)g *(x)d nx=∫k∈ℝ nf^(k)g^ *(k)d nk \underset{x \in \mathbb{R}^n}{\int} f(x) g^\ast(x)\, d^n x \;=\; \underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \widehat{g}^\ast(k) \, d^n k

  4. (4)∫k∈ℝ nf^(k)⋅g(k)d nk=∫x∈ℝ nf(x)⋅g^(x)d nx \underset{k \in \mathbb{R}^n}{\int} \widehat{f}(k) \cdot g(k) \, d^n k \;=\; \underset{x \in \mathbb{R}^n}{\int} f(x) \cdot \widehat{g}(x) \, d^n x

(e.g Hörmander 90, lemma 7.1.3, theorem 7.1.6)

On tempered distributions

The Schwartz space of functions with rapidly decreasing partial derivatives (def. ) serves the purpose to support the Fourier transform (def. ) together with its inverse (prop. ), but for many applications one needs to apply the Fourier transform to more general functions, and in fact to generalized functions in the sense of distributions (via this prop.). But with the Schwartz space in hand, this generalization is readily obtained by formal duality:

e.g. (Hörmander 90, def. 7.1.7)

(e.g. Hörmander 90, lemma 7.1.8)

Example

(delta distribution)

Write

δ 0(−)∈ℰ′(ℝ n) \delta_0(-) \;\in\; \mathcal{E}'(\mathbb{R}^n)

for the distribution given by point evaluation of functions at the origin of ℝ n\mathbb{R}^n:

δ 0(−):f↦f(0). \delta_0(-) \;\colon\; f \mapsto f(0) \,.

This is clearly a compactly supported distribution; hence a tempered distribution by example .

We write just “δ(−)\delta(-)” (without the subscript) for the corresponding generalized function (example ), so that

∫x∈ℝ nδ(x)f(x)d nx≔f(0). \underset{x \in \mathbb{R}^n}{\int} \delta(x) f(x) \, d^n x \;\coloneqq\; f(0) \,.

(e.g. Hörmander 90, below lemma 7.1.8)

Property (4) of the ordinary Fourier transform on functions with rapidly decreasing partial derivatives motivates and justifies the fullowing generalization:

(e.g. Hörmander 90, def. 1.7.9)

Proof

Let g∈𝒮(ℝ n)g \in \mathcal{S}(\mathbb{R}^n). Then

u f^(g) ≔u f(g^) =∫x∈ℝ nf(x)g^(x)d nx =∫x∈ℝ nf^(x)g(x)d nx =u f^(g) \begin{aligned} \widehat{u_f}(g) & \coloneqq u_f\left( \widehat{g}\right) \\ & = \underset{x \in \mathbb{R}^n}{\int} f(x) \hat g(x)\, d^n x \\ & = \underset{x \in \mathbb{R}^n}{\int} \hat f(x) g(x) \, d^n x \\ & = u_{\hat f}(g) \end{aligned}

Here all equalities hold by definition, except for the third: this is property (4) from prop. .

(Hörmander 90, theorem 7.1.14)

Example

(Fourier transform of the delta-distribution)

The Fourier transform (def. ) of the delta distribution (def. ), via example , is the constant function on 1:

δ^(k) =∫x∈ℝ nδ(x)e −ikxdx =1 \begin{aligned} \widehat {\delta}(k) & = \underset{x \in \mathbb{R}^n}{\int} \delta(x) e^{- i k x} \, d x \\ & = 1 \end{aligned}

This implies by the Fourier inversion theorem (prop. ) that the delta distribution itself has equivalently the following expression as a generalized function

δ(x) =δ 0^ˇ(x) =∫k∈ℝ ne ik⋅xd nk(2π) n \begin{aligned} \delta(x) & = \widecheck{\widehat {\delta_0}}(x) \\ & = \underset{k \in \mathbb{R}^n}{\int} e^{i k \cdot x} \, \frac{d^n k}{ (2\pi)^n } \end{aligned}

in the sense that for every function with rapidly decreasing partial derivatives f∈𝒮(ℝ n)f \in \mathcal{S}(\mathbb{R}^n) (def. ) we have

f(x) =∫y∈ℝ nf(y)δ(y−x)d ny =∫y∈ℝ n∫k∈ℝ nf(y)e ik⋅(y−x)d nk(2π) nd ny =∫k∈ℝ ne −ik⋅x∫y∈ℝ nf(y)e ik⋅yd ny⏟=f^(−k)d nk(2π) n =+∫k∈ℝ ne ik⋅x∫y∈ℝ nf(y)e −ik⋅yd ny⏟=f^(k)d nk(2π) n =f^ˇ(x) \begin{aligned} f(x) & = \underset{y \in \mathbb{R}^n}{\int} f(y) \delta(y-x) \, d^n y \\ & = \underset{y \in \mathbb{R}^n}{\int} \underset{k \in \mathbb{R}^n}{\int} f(y) e^{i k \cdot (y-x)} \, \frac{d^n k}{(2\pi)^n} \, d^n y \\ & = \underset{k \in \mathbb{R}^n}{\int} e^{- i k \cdot x} \underset{= \widehat{f}(-k) }{ \underbrace{ \underset{y \in \mathbb{R}^n}{\int} f(y) e^{i k \cdot y} \, d^n y } } \,\, \frac{d^n k}{(2\pi)^n} \\ & = + \underset{k \in \mathbb{R}^n}{\int} e^{i k \cdot x} \underset{= \widehat{f}(k) }{ \underbrace{ \underset{y \in \mathbb{R}^n}{\int} f(y) e^{- i k \cdot y} \, d^n y } } \,\, \frac{d^n k}{(2\pi)^n} \\ & = \widecheck{\widehat{f}}(x) \end{aligned}

which is the statement of the Fourier inversion theorem for smooth functions (prop. ).

(Here in the last step we used change of integration variables k↦−kk \mapsto -k which introduces one sign (−1) n(-1)^{n} for the new volume form, but another sign (−1) n(-1)^n from the re-orientation of the integration domain. )

Equivalently, the above computation shows that the delta distribution is the neutral element for the convolution product of distributions.

We have the following distributional generalization of the basic property (3) from prop. :

(e.g. Hörmander 90, theorem 7.1.15)

References

Textbook account in mathematical physics:

Lecture notes:

Discussion in the broader context of functional analysis and distribution theory:

  • Lars Hörmander, chapter 7 of The analysis of linear partial differential operators, vol. I, Springer 1983, 1990

  • Sergiu Klainerman, chapter 5 of of Lecture notes in analysis, 2011 (pdf)

Last revised on February 15, 2025 at 17:16:12. See the history of this page for a list of all contributions to it.