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.

Definition

Informal

Let G be a locally compact abelian group, and Ĝ the dual group of G. Then the Fourier transform of functions in $L_1(\widehat{\mathit{G}})$, the group algebra of $(\widehat{\mathit{G}})$, 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 $M(\widehat{\mathit{G}})$, the measure algebra of $(\widehat{\mathit{G}})$, also a subalgebra of CB(G), called the Fourier-Stieltjes algebra of G.

Formal

Let $B(\mathit{G})$ be a Fourier–Stieltjes algebra and $A(\mathit{G})$ be a Fourier algebra such that the locally compact group $\mathit{G}$ is abelian. Let $M(\widehat{\mathit{G}})$ be the measure algebra of finite measures on $\widehat{G}$ and let $L_1(\widehat{\mathit{G}})$ be the convolution algebra of integrable functions on $\widehat{G}$, where $\widehat{\mathit{G}}$ is the character group of the Abelian group $\mathit{G}$.

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

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

The space $B(\mathit{G})$ of these functions is an algebra under pointwise multiplication is isomorphic to the measure algebra $M(\widehat{\mathit{G}})$. Restricted to $L_1(\widehat{\mathit{G}})$, viewed as a subspace of $M(\widehat{\mathit{G}})$, the Fourier–Stieltjes transform is the Fourier transform on $L_1(\widehat{\mathit{G}})$ and its image is, by definition, the Fourier algebra $A(\mathit{G})$. The generalized Bochner theorem states that a measurable function on $\mathit{G}$ is equal, almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on $\widehat{G}$ if and only if it is positive definite. Thus, $B(\mathit{G})$ can be defined as the linear span of the set of continuous positive-definite functions on $\mathit{G}$. This definition is still valid when $\mathit{G}$ 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.^[1] In 1969 Dunkl proved the result holds when G is compact and contains an infinite abelian subgroup.

References

1. H. Helson, J.-P. Kahane, Y. Katznelson, W. Rudin (1959). "The functions which operate on Fourier transforms" (PDF). Acta Mathematica 102: 135–157. doi:10.1007/bf02559571.

• Renault, Jean (2001), "Fourier-algebra(2)", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• "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: