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 formula_1 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 formula_2 to be the set of Fourier transforms of functions in formula_1, and it is a closed sub-algebra of formula_4, the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call formula_2 the Fourier algebra of G.
Similarly, we write formula_6 for the measure algebra on Ĝ, meaning the space of all finite regular Borel measures on Ĝ. We define formula_7 to be the set of Fourier-Stieltjes transforms of measures in formula_6. It is a closed sub-algebra of formula_4, the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call formula_7 the Fourier-Stieltjes algebra of G. Equivalently, formula_7 can be defined as the linear span of the set formula_12 of continuous positive-definite functions on G.
Since formula_1 is naturally included in formula_6, and since the Fourier-Stieltjes transform of an formula_1 function is just the Fourier transform of that function, we have that formula_16. In fact, formula_2 is a closed ideal in formula_7.
Formal.
Let formula_19 be a Fourier–Stieltjes algebra and formula_20 be a Fourier algebra such that the locally compact group formula_21 is abelian. Let formula_22 be the measure algebra of finite measures on formula_23 and let formula_24 be the convolution algebra of integrable functions on formula_23, where formula_26 is the character group of the Abelian group formula_21.
The Fourier–Stieltjes transform of a finite measure formula_28 on formula_26 is the function formula_30 on formula_21 defined by
The space formula_19 of these functions is an algebra under pointwise multiplication is isomorphic to the measure algebra formula_22. Restricted to formula_24, viewed as a subspace of formula_22, the Fourier–Stieltjes transform is the Fourier transform on formula_24 and its image is, by definition, the Fourier algebra formula_20. The generalized Bochner theorem states that a measurable function on formula_21 is equal, almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on formula_23 if and only if it is positive definite. Thus, formula_19 can be defined as the linear span of the set of continuous positive-definite functions on formula_21. This definition is still valid when formula_21 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.