nLab
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 $L_2(S^1,dt)$ and $l_2(\mathbb{Z})$ and the Fourier coefficients are the numbers

for $n\in \mathbb{Z}$, where the functions ${\chi }_n(t)=e^{2\pi int}$ form an orthonormal basis of $L_2(S^1,dt)$. The Fourier transform $\widehat{{\chi }_n}$ 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 $d/dt$ by the operator of multiplication by the series $\{2\pi in{\}}_{n\in \mathbb{Z}}$.

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 $\hat{\mu }(X)=\mathrm{card}(X)$ for $X\subset \hat{G}$. Then the Fourier transform restricts to a unitary operator from $L_2(X,\mu )$ to $L_2(\hat{G},\hat{\mu })$.

A Fourier transform of a function on the real line $ℝ$ is called its Fourier integral:

It is usually defined as a linear automorphism of the Schwarz space? $S(ℝ)\to S(ℝ)$; there is also an appropriate extension to the space of distributions $S\prime (ℝ)$ by $⟨\hat{f},\varphi ⟩:=⟨f,\hat{\varphi }⟩$ where $f\in S\prime (ℝ)$ and $\varphi \in S(ℝ)$. The Fourier transform and the inverse Fourier transform are continuous, mutually inverse operators $S\prime (ℝ)\to S\prime (ℝ)$. There is also a unitary operator on $L_2(ℝ,dx)$ which when restricted to $L_2(ℝ,dx)\cap L_1(ℝ,\mathrm{dx})$ agrees with the Fourier transform.

The study of the Fourier transform and its generalizations is the main subject of harmonic analysis?. 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.