Editing Fourier transform (section)

=== Locally compact abelian groups ===
{{Main|Pontryagin duality}}
The Fourier transform may be generalized to any [[locally compact abelian group]], i.e., an [[abelian group]] that is also a [[locally compact Hausdorff space]] such that the group operation is continuous. If {{mvar|G}} is a locally compact abelian group, it has a translation invariant measure {{mvar|μ}}, called [[Haar measure]]. For a locally compact abelian group {{mvar|G}}, the set of irreducible, i.e. one-dimensional, unitary representations are called its [[character group|characters]]. With its natural group structure and the topology of uniform convergence on compact sets (that is, the topology induced by the [[compact-open topology]] on the space of all continuous functions from <math>G</math> to the [[circle group]]), the set of characters {{mvar|Ĝ}} is itself a locally compact abelian group, called the ''Pontryagin dual'' of {{mvar|G}}. For a function {{mvar|f}} in {{math|''L''<sup>1</sup>(''G'')}}, its Fourier transform is defined by{{sfn|Katznelson|2004}}
<math display="block">\hat{f}(\xi) = \int_G \xi(x)f(x)\,d\mu\quad \text{for any }\xi \in \hat{G}.</math>

The Riemann–Lebesgue lemma holds in this case; {{math|''f̂''(''ξ'')}} is a function vanishing at infinity on {{mvar|Ĝ}}.

The Fourier transform on {{nobr|{{mvar|T}} {{=}} R/Z}} is an example; here {{mvar|T}} is a locally compact abelian group, and the Haar measure {{mvar|μ}} on {{mvar|T}} can be thought of as the Lebesgue measure on [0,1). Consider the representation of {{mvar|T}} on the complex plane {{mvar|C}} that is a 1-dimensional complex vector space. There are a group of representations (which are irreducible since {{mvar|C}} is 1-dim) <math>\{e_{k}: T \rightarrow GL_{1}(C) = C^{*} \mid k \in Z\}</math> where <math>e_{k}(x) = e^{i 2\pi kx}</math> for <math>x\in T</math>.

The character of such representation, that is the trace of <math>e_{k}(x)</math> for each <math>x\in T</math> and <math>k\in Z</math>, is <math>e^{i 2\pi kx}</math> itself. In the case of representation of finite group, the character table of the group {{mvar|G}} are rows of vectors such that each row is the character of one irreducible representation of {{mvar|G}}, and these vectors form an orthonormal basis of the space of class functions that map from {{mvar|G}} to {{mvar|C}} by Schur's lemma. Now the group {{mvar|T}} is no longer finite but still compact, and it preserves the orthonormality of character table. Each row of the table is the function <math>e_{k}(x)</math> of <math>x\in T,</math> and the inner product between two class functions (all functions being class functions since {{mvar|T}} is abelian) <math>f,g \in L^{2}(T, d\mu)</math> is defined as <math display="inline">\langle f, g \rangle = \frac{1}{|T|}\int_{[0,1)}f(y)\overline{g}(y)d\mu(y)</math> with the normalizing factor <math>|T|=1</math>. The sequence <math>\{e_{k}\mid k\in Z\}</math> is an orthonormal basis of the space of class functions <math>L^{2}(T,d\mu)</math>.

For any representation {{mvar|V}} of a finite group {{mvar|G}}, <math>\chi_{v}</math> can be expressed as the span <math display="inline">\sum_{i} \left\langle \chi_{v},\chi_{v_{i}} \right\rangle \chi_{v_{i}}</math> (<math>V_{i}</math> are the irreps of {{mvar|G}}), such that <math display="inline">\left\langle \chi_{v}, \chi_{v_{i}} \right\rangle = \frac{1}{|G|}\sum_{g\in G}\chi_{v}(g)\overline{\chi}_{v_{i}}(g)</math>. Similarly for <math>G = T</math> and <math>f\in L^{2}(T, d\mu)</math>, <math display="inline">f(x) = \sum_{k\in Z}\hat{f}(k)e_{k}</math>. The Pontriagin dual <math>\hat{T}</math> is <math>\{e_{k}\}(k\in Z)</math> and for <math>f \in L^{2}(T, d\mu)</math>, <math display="inline">\hat{f}(k) = \frac{1}{|T|}\int_{[0,1)}f(y)e^{-i 2\pi ky}dy</math> is its Fourier transform for <math>e_{k} \in \hat{T}</math>.