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.^[1]

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.^[2] In 1969 Dunkl proved the result holds when G is compact and contains an infinite abelian subgroup.