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. 2 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.
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. 5 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.
Let $G$ be a locally compact Hausdorff abelian topological group with invariant (= Haar) measure $\mu$. Then for each $f\in L_1(G,\mu)$, define its Fourier transform $\hat{f}$ as a function on its Pontrjagin dual group $\hat{G}$ given by
The Fourier transform of $f\in L_1(G,\mu)$ is always continuous and bounded on $\hat{G}$; the transform of the convolution of two functions is the product of the transforms of each of the functions separately.
In the classical case of Fourier series, where $G=\mathbb{Z}$ (the additive group of integers) and $\hat{G}=S^1$ (the circle group), the Fourier transform restricts to a unitary operator between the Hilbert spaces $L_2(S^1,d t)$ and $l_2(\mathbb{Z})$ and the Fourier coefficients are the numbers
for $n\in\mathbb{Z}$, where the functions $\chi_n(t)= e^{2\pi i n t}$ form an orthonormal basis of $L_2(S^1,d t)$. The Fourier transform $\hat{\chi_n}$ is then viewed as the $\mathbb{Z}$-series $\delta_n$ which in the $n$-th place has $1$ and elsewhere $0$. The Fourier transform replaces the operator of differentiation $d/d t$ by the operator of multiplication by the series $\{2\pi i n\}_{n\in\mathbb{Z}}$.
In general, if $G$ is a compact abelian group (whose Pontrjagin dual is discrete), one can normalize the invariant measure by $\mu(G)=1$ and $\hat{\mu}(X)=card(X)$ for $X\subset\hat{G}$. Then the Fourier transform restricts to a unitary operator from $L_2(X,\mu)$ to $L_2(\hat{G},\hat{\mu})$.
Throughout, let $n \in \mathbb{N}$ and write $\mathbb{R}^n$ for the Cartesian space of dimension $n$ and write $(-) \cdot (-)$ for the canonical inner product on $\mathbb{R}^n$:
In the following by a smooth function $f \in C^\infty(\mathbb{R}^n)$ on $\mathbb{R}^n$ we mean a smooth function with values in the complex numbers.
For $f \in C^\infty(\mathbb{R}^n)$, we write $f^\ast \in C^\infty(\mathbb{R}^n)$ for its pointwise complex conjugate:
(Schwartz space of functions with rapidly decreasing partial derivatives)
A complex-valued smooth function $f \in C^\infty(\mathbb{R}^n)$ is said to have rapidly decreasing partial derivatives if for all $\alpha,\beta \in \mathbb{N}^{n}$ we have
Write
for the sub-vector space on the functions with rapidly decreasing partial derivatives regarded as a topological vector space for the Frechet space struzcture induced by the seminorms
This is also called the Schwartz space.
(e.g. Hörmander 90, def. 7.1.2)
(compactly supported smooth function are functions with rapidly decreasing partial derivatives)
Every compactly supported smooth function (bump function) $b \in C^\infty_{cp}(\mathbb{R}^n)$ rapidly decreasing partial derivatives (def. \ref{SchwartzSpace):
(pointwise product and convolution product on Schwartz space)
The Schwartz space $\mathcal{S}(\mathbb{R}^n)$ (def. 1) is closed under the following operatios on smooth functions $f,g \in \mathcal{S}(\mathbb{R}^n) \hookrightarrow C^\infty(\mathbb{R}^n)$
pointwise product:
By the product law of differentiation.
(rapidly decreasing functions are integrable)
Every rapidly decreasing function $f \colon \mathbb{R}^n \to \mathbb{R}$ (def. 1) is an integrable function in that its integral exists:
In fact for each $\alpha \in \mathbb{N}^n$ the product of $f$ with the $\alpha$-power of the coordinate functions exists:
(Fourier transform of functions with rapidly decreasing partial derivatives)
The Fourier transform is the continuous linear functional
on the Schwartz space of functions with rapidly decreasing partial derivatives (def. 1), which is given by integration against the exponential plane wave functions
times the standard volume form $d^n x$:
Here the argument $k \in \mathbb{R}^n$ of the Fourier transform is also called the wave vector.
(e.g. Hörmander, lemma 7.1.3)
The Fourier transform $\widehat{(-)}$ (def. 2) on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ (def. 1) is an isomorphism, with inverse function the inverse Fourier transform
given by
Hence in the language of harmonic analysis the function $\widecheck g \colon \mathbb{R}^n \to \mathbb{C}$ is the superposition of plane waves in which the plane wave with wave vector $k\in \mathbb{R}^n$ appears with amplitude $g(k)$.
(e.g. Hörmander, theorem 7.1.5)
(basic properties of the Fourier transform)
The Fourier transform $\widehat{(-)}$ (def. 2) on the Schwartz space $\mathcal{S}(\mathbb{R}^n)$ (def. 1) satisfies the following properties, for all $f,g \in \mathcal{S}(\mathbb{R}^n)$:
(interchanging coordinate multiplication with partial derivatives)
(interchanging pointwise multiplication with convolution product, remark 1):
(unitarity, Parseval's theorem)
(e.g Hörmander 90, lemma 7.1.3, theorem 7.1.6)
The Schwartz space of functions with rapidly decreasing partial derivatives (def. 1) serves the purpose to support the Fourier transform (def. 2) together with its inverse (prop. 3), 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:
A tempered distribution is a continuous linear functional
on the Schwartz space (def. 1) of functions with rapidly decaying partial derivatives. The vector space of all tempered distributions is canonically a topological vector space as the dual space to the Schwartz space, denoted
e.g. (Hörmander 90, def. 7.1.7)
(some non-singular tempered distributions)
Every function with rapidly decreasing partial derivatives $f \in \mathcal{S}(\mathbb{R}^n)$ (def. 1) induces a tempered distribution $u_f \in \mathcal{S}'(\mathbb{R}^n)$ (def. 4) by integrating against it:
This construction is a linear inclusion
of the Schwartz space into its dual space of tempered distributions. This is a dense subspace inclusion.
In fact already the restriction of this inclusion to the compactly supported smooth functions (example 1) is a dense subspace inclusion:
This means that every tempered distribution is a limit of a sequence of ordinary functions with rapidly decreasing partial derivatives, and in fact even the limit of a sequence of compactly supported smooth functions (bump functions).
It is in this sense that tempered distributions are “generalized functions”.
(e.g. Hörmander 90, lemma 7.1.8)
(compactly supported distributions are tempered distributions)
Every compactly supported distribution is a tempered distribution (def. 4), hence there is a linear inclusion
Write
for the distribution given by point evaluation of functions at the origin of $\mathbb{R}^n$:
This is clearly a compactly supported distribution; hence a tempered distribution by example 3.
We write just “$\delta(-)$” (without the subscript) for the corresponding generalized function (example 2), so that
(square integrable functions induce tempered distributions)
Let $f \in L^p(\mathbb{R}^n)$ be a function in the $p$th Lebesgue space, e.g. for $p = 2$ this means that $f$ is a square integrable function. Then the operation of integration against the measure $f dvol$
is a tempered distribution (def. 4).
(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:
(Fourier transform of distributions on tempered distributions)
The Fourier transform of distributions of a tempered distribution $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. 4) is the tempered distribution $\widehat u$ defined on a smooth function $f \in \mathcal{S}(\mathbb{R}^n)$ in the Schwartz space (def. 1) by
where on the right $\widehat f \in \mathcal{S}(\mathbb{R}^n)$ is the Fourier transform of functions from def. 2.
(e.g. Hörmander 90, def. 1.7.9)
(Fourier transform of distributions indeed generalizes Fourier transform of functions with rapidly decreasing partial derivatives)
Let $u_f \in \mathcal{S}'(\mathbb{R}^n)$ be a non-singular tempered distribution induced, via example 2, from a function with rapidly decreasing partial derivatives $f \in \mathcal{S}(\mathbb{R}^n)$.
Then its Fourier transform of distributions (def. 5) is the non-singular distribution induced from the Fourier transform of $f$:
Let $g \in \mathcal{S}(\mathbb{R}^n)$. Then
Here all equalities hold by definition, except for the third: this is property (4) from prop. 3.
(Fourier transform of compactly supported distributions)
Under the identification of smooth functions of bounded growth with non-singular tempered distributions (example 2), the Fourier transform of distributions (def. 5) of a tempered distribution that happens to be compactly supported (example 3)
is simply
(Hörmander 90, theorem 7.1.14)
(Fourier transform of the delta-distribution)
The Fourier transform (def. 5) of the delta distribution (def. 4), via example 7, is the constant function on 1:
This implies by the Fourier inversion theorem (prop. 5) that the delta distribution itself has equivalently the following expression as a generalized function
in the sense that for every function with rapidly decreasing partial derivatives $f \in \mathcal{S}(\mathbb{R}^n)$ (def. 1) we have
which is the statement of the Fourier inversion theorem for smooth functions (prop. 3).
(Here in the last step we used change of integration variables $k \mapsto -k$ which introduces one sign $(-1)^{n}$ for the new volume form, but another sign $(-1)^n$ 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.
(Paley-Wiener-Schwartz theorem)
Let $u \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ be a compactly supported distribution regarded as a tempered distribution by example 3. Then its Fourier transform of distributions (def. 5) is a non-singular distribution induced from a smooth function that grows at most exponentially.
(Fourier inversion theorem for Fourier transform of distributions)
The operation of forming the Fourier transform of distributions $\widehat{u}$ (def. 5) tempered distributions $u \in \mathcal{S}'(\mathbb{R}^n)$ (def. 4) is an isomorphism, with inverse given by
where on the right $\widecheck{g}$ is the ordinary inverse Fourier transform of $g$ according to prop. 3.
By def. 5 this follows immediately from the Fourier inversion theorem for smooth functions (prop. 3).
We have the following distributional generalization of the basic property (3) from prop. 3:
(Fourier transform of distributions interchanges convolution of distributions with pointwise product)
Let
be a tempered distribution (def. 4) and
be a compactly supported distribution, regarded as a tempered distribution via example 3.
Observe here that the Paley-Wiener-Schwartz theorem (prop. 4) implies that the Fourier transform of distributions of $u_1$ is a non-singular distribution $\widehat{u_1} \in C^\infty(\mathbb{R}^n)$ so that the product $\widehat{u_1} \cdot \widehat{u_2}$ is always defined.
Then the Fourier transform of distributions of the convolution product of distributions is the product of the Fourier transform of distributions:
(e.g. Hörmander 90, theorem 7.1.15)
(product of distributions via Fourier transform of distributions)
Prop. 6 together with the Fourier inversion theorem (prop. 5) suggests to define the product of distributions $u_1 \cdot u_2$ for compactly supported distributions $u_1, u_2 \in \mathcal{E}'(\mathbb{R}^n) \hookrightarrow \mathcal{S}'(\mathbb{R}^n)$ by the formula
which would complete the generalization of of property (3) from prop. 3.
For this to make sense, the convolution product of the smooth functions on the right needs to exist, which is not guaranteed (prop. 1 does not apply here!). The condition that this exists is the Hörmander-condition on the wave front set of $u_1$ and $u_2$. See at product of distributions for more.
wavelet transform?
Lecture notes include
John Peacock, Fourier analysis 2013 (part 1 pdf, part 2 pdf, part 3 pdf, part 4 pdf, part 5 pdf)
Gerald B. Folland, A course in abstract harmonic analysis, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1995. x+276 pp. gBooks
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)