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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–Stieltjes transform on the Fourier algebra of a locally compact group were introduced by Pierre Eymard in 1964.

Definition

Informal

Let G be a locally compact abelian group, and Ĝ the dual group of G. Then ${\displaystyle L_{1}({\hat {\mathit {G}}})}$ is the space of all functions on Ĝ which are integrable with respect to the Haar measure on Ĝ, and it has a Banach algebra structure where the product of two functions is convolution. We define ${\displaystyle A(G)}$ to be the set of Fourier transforms of functions in ${\displaystyle L_{1}({\hat {\mathit {G}}})}$, and it is a closed sub-algebra of ${\displaystyle CB(G)}$, the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call ${\displaystyle A(G)}$ the Fourier algebra of G.

Similarly, we write ${\displaystyle M({\hat {\mathit {G}}})}$ for the measure algebra on Ĝ, meaning the space of all finite regular Borel measures on Ĝ. We define ${\displaystyle B(G)}$ to be the set of Fourier-Stieltjes transforms of measures in ${\displaystyle M({\hat {\mathit {G}}})}$. It is a closed sub-algebra of ${\displaystyle CB(G)}$, the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call ${\displaystyle B(G)}$ the Fourier-Stieltjes algebra of G. Equivalently, ${\displaystyle B(G)}$ can be defined as the linear span of the set ${\displaystyle P(G)}$ of continuous positive-definite functions on G.

Since ${\displaystyle L_{1}({\hat {\mathit {G}}})}$ is naturally included in ${\displaystyle M({\hat {\mathit {G}}})}$, and since the Fourier-Stieltjes transform of an ${\displaystyle L_{1}({\hat {\mathit {G}}})}$ function is just the Fourier transform of that function, we have that ${\displaystyle A(G)\subset B(G)}$. In fact, ${\displaystyle A(G)}$ is a closed ideal in ${\displaystyle B(G)}$.

Formal

Let ${\displaystyle B({\mathit {G}})}$ be a Fourier–Stieltjes algebra and ${\displaystyle A({\mathit {G}})}$ be a Fourier algebra such that the locally compact group ${\displaystyle {\mathit {G}}}$ is abelian. Let ${\displaystyle M({\widehat {\mathit {G}}})}$ be the measure algebra of finite measures on ${\displaystyle {\widehat {G}}}$ and let ${\displaystyle L_{1}({\widehat {\mathit {G}}})}$ be the convolution algebra of integrable functions on ${\displaystyle {\widehat {G}}}$, where ${\displaystyle {\widehat {\mathit {G}}}}$ is the character group of the Abelian group ${\displaystyle {\mathit {G}}}$.

The Fourier–Stieltjes transform of a finite measure ${\displaystyle \mu }$ on ${\displaystyle {\widehat {\mathit {G}}}}$ is the function ${\displaystyle {\widehat {\mu }}}$ on ${\displaystyle {\mathit {G}}}$ defined by

${\displaystyle {\widehat {\mu }}(x)=\int _{\widehat {G}}{\overline {X(x)}}\,d\mu (X),\quad x\in G}$

The space ${\displaystyle B({\mathit {G}})}$ of these functions is an algebra under pointwise multiplication is isomorphic to the measure algebra ${\displaystyle M({\widehat {\mathit {G}}})}$. Restricted to ${\displaystyle L_{1}({\widehat {\mathit {G}}})}$, viewed as a subspace of ${\displaystyle M({\widehat {\mathit {G}}})}$, the Fourier–Stieltjes transform is the Fourier transform on ${\displaystyle L_{1}({\widehat {\mathit {G}}})}$ and its image is, by definition, the Fourier algebra ${\displaystyle A({\mathit {G}})}$. The generalized Bochner theorem states that a measurable function on ${\displaystyle {\mathit {G}}}$ is equal, almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on ${\displaystyle {\widehat {G}}}$ if and only if it is positive definite. Thus, ${\displaystyle B({\mathit {G}})}$ can be defined as the linear span of the set of continuous positive-definite functions on ${\displaystyle {\mathit {G}}}$. This definition is still valid when ${\displaystyle {\mathit {G}}}$ is not Abelian.

Helson–Kahane–Katznelson–Rudin theorem

Let A(G) be the Fourier algebra of a compact group G. Building upon the work of Wiener, Lévy, Gelfand, and Beurling, in 1959 Helson, Kahane, Katznelson, and Rudin proved that, when G is compact and abelian, a function f defined on a closed convex subset of the plane operates in A(G) if and only if f is real analytic. In 1969 Dunkl proved the result holds when G is compact and contains an infinite abelian subgroup.

References

1. Renault, Jean (2001) , "Fourier-algebra(2)", Encyclopedia of Mathematics, EMS Press
2. H. Helson; J.-P. Kahane; Y. Katznelson; W. Rudin (1959). "The functions which operate on Fourier transforms" (PDF). Acta Mathematica. 102 (1–2): 135–157. doi:10.1007/bf02559571. S2CID 121739671.

• "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:
• "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:
• "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:
• "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:

• Harmonic analysis
• Algebras