Fourier algebra

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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 algebra of a locally compact group were introduced by Pierre Eymard in 1964.

Contents 1 Definition 1.1 Informal 1.2 Formal 2 References

[edit] Definition

[edit] Informal

Let G be a locally compact abelian group, and Ĝ the dual group of G. Then the Fourier transform of functions in Failed to parse (Cannot write to or create math output directory): L_1(\widehat{\mathit{G}}) , the group algebra of Failed to parse (Cannot write to or create math output directory): (\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 Failed to parse (Cannot write to or create math output directory): M(\widehat{\mathit{G}}) , the measure algebra of Failed to parse (Cannot write to or create math output directory): (\widehat{\mathit{G}}) , also a subalgebra of CB(G), called the Fourier-Stieltjes algebra of G.

[edit] Formal

Let Failed to parse (Cannot write to or create math output directory): B(\mathit{G})

be a Fourier–Stieltjes algebra and Failed to parse (Cannot write to or create math output directory):  A(\mathit{G}) 
be a Fourier algebra such that the locally compact group Failed to parse (Cannot write to or create math output directory):  \mathit{G} 
is abelian. Let Failed to parse (Cannot write to or create math output directory):  M(\widehat{\mathit{G}}) 
be the measure algebra of finite measures on Failed to parse (Cannot write to or create math output directory):  \widehat{G} 
and let Failed to parse (Cannot write to or create math output directory):  L_1(\widehat{\mathit{G}}) 
be the convolution algebra of integrable functions on Failed to parse (Cannot write to or create math output directory):  \widehat{G} 

, where Failed to parse (Cannot write to or create math output directory): \widehat{\mathit{G}}

is the character group of the Abelian group Failed to parse (Cannot write to or create math output directory):  \mathit{G} 

.

The Fourier–Stieltjes transform of a finite measure Failed to parse (Cannot write to or create math output directory): \mu

on Failed to parse (Cannot write to or create math output directory):  \widehat{\mathit{G}} 
is the function Failed to parse (Cannot write to or create math output directory):  \widehat{\mu} 
on Failed to parse (Cannot write to or create math output directory):  \mathit{G} 
defined by

Failed to parse (Cannot write to or create math output directory): \widehat{\mu}(x) = \int_{\widehat{G}} \overline{X(x)} \, d \mu(X), \quad x \in G

The space Failed to parse (Cannot write to or create math output directory): B(\mathit{G})

of these functions is an algebra under pointwise multiplication is isomorphic to the measure algebra Failed to parse (Cannot write to or create math output directory):  M(\widehat{\mathit{G}}) 

. Restricted to Failed to parse (Cannot write to or create math output directory): L_1(\widehat{\mathit{G}}) , viewed as a subspace of Failed to parse (Cannot write to or create math output directory): M(\widehat{\mathit{G}}) , the Fourier–Stieltjes transform is the Fourier transform on Failed to parse (Cannot write to or create math output directory): L_1(\widehat{\mathit{G}})

and its image is, by definition, the Fourier algebra Failed to parse (Cannot write to or create math output directory):  A(\mathit{G}) 

. The generalized Bochner theorem states that a measurable function on Failed to parse (Cannot write to or create math output directory): \mathit{G}

is equal, almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on Failed to parse (Cannot write to or create math output directory):  \widehat{G} 
if and only if it is positive definite. Thus, Failed to parse (Cannot write to or create math output directory):  B(\mathit{G}) 
can be defined as the linear span of the set of continuous positive-definite functions on Failed to parse (Cannot write to or create math output directory):  \mathit{G} 

. This definition is still valid when Failed to parse (Cannot write to or create math output directory): \mathit{G}

is not Abelian.

[edit] References

1. Encyclopaedia of Mathematics — ISBN 1402006098 [1]

2. "Functions that Operate in the Fourier Algebra of a Compact Group" Charles F. Dunkl Proceedings of the American Mathematical Society, Vol. 21, No. 3. (Jun., 1969), pp. 540–544. Stable URL:[2]

3. "Functions which Operate in the Fourier Algebra of a Discrete Group" Leonede de Michele; Paolo M. Soardi, Proceedings of the American Mathematical Society, Vol. 45, No. 3. (Sep., 1974), pp. 389–392. Stable URL:[3]

4. "Uniform Closures of Fourier-Stieltjes Algebras", Ching Chou, Proceedings of the American Mathematical Society, Vol. 77, No. 1. (Oct., 1979), pp. 99–102. Stable URL: [4]

5. "Centralizers of the Fourier Algebra of an Amenable Group", P. F. Renaud, Proceedings of the American Mathematical Society, Vol. 32, No. 2. (Apr., 1972), pp. 539–542. Stable URL: [5]

6.summary of concept