Fourier algebra

Fourier and related algebras occur naturally in the harmonic analysis of locally compact groups. They play an important role in the duality theories of these groups. The Fourier–Stieltjes algebra and the Fourier–Stieltjes transform on the Fourier algebra of a locally compact group were introduced by Pierre Eymard in 1964.

Let G be a locally compact abelian group, and Ĝ the dual group of G. Then the Fourier transform of functions in ${\displaystyle L_{1}({\widehat {\mathit {G}}})}$, the group algebra of ${\displaystyle ({\widehat {\mathit {G}}})}$, is a sub-algebra A(G) of CB(G), the space of bounded continuous complex-valued functions on G with pointwise multiplication called the Fourier algebra of G, and the Fourier-Stieltjes transform of measures in ${\displaystyle M({\widehat {\mathit {G}}})}$, the measure algebra of ${\displaystyle ({\widehat {\mathit {G}}})}$, also a subalgebra of CB(G), called the Fourier-Stieltjes algebra of G.

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