Fourier algebra explained
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
}) 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
to be the set of Fourier transforms of functions in
}) , and it is a closed sub-algebra of
, the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call
the Fourier algebra of G.
Similarly, we write
}) for the measure algebra on Ĝ, meaning the space of all finite regular
Borel measures on Ĝ. We define
to be the set of Fourier-Stieltjes transforms of measures in
}) . It is a closed sub-algebra of
, the space of bounded continuous complex-valued functions on G with pointwise multiplication. We call
the Fourier-Stieltjes algebra of G. Equivalently,
can be defined as the linear span of the set
of continuous
positive-definite functions on G.
Since
}) is naturally included in
}) , and since the Fourier-Stieltjes transform of an
}) function is just the Fourier transform of that function, we have that
. In fact,
is a closed ideal in
.
Formal
Let
be a Fourier–Stieltjes algebra and
be a Fourier algebra such that the locally compact group
is
abelian. Let
}) be the measure algebra of finite measures on
and let
}) be the convolution algebra of integrable
functions on
, where
} is the character group of the Abelian group
.
The Fourier–Stieltjes transform of a finite measure
on
} is the function
on
defined by
\widehat{\mu}(x)=\int\widehat{G
} \overline \, d \mu(X), \quad x \in G
The space
of these functions is an algebra under pointwise multiplication is isomorphic to the measure algebra
}) . Restricted to
}) , viewed as a subspace of
}) , the Fourier–Stieltjes transform is the
Fourier transform on
}) and its image is, by definition, the Fourier algebra
. The generalized
Bochner theorem states that a measurable function on
is equal,
almost everywhere, to the Fourier–Stieltjes transform of a non-negative finite measure on
if and only if it is positive definite. Thus,
can be defined as the
linear span of the set of continuous positive-definite functions on
. This definition is still valid when
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
- "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:https://www.jstor.org/stable/2036416
- "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:https://www.jstor.org/stable/2039963
- "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: https://www.jstor.org/stable/2042723
- "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: https://www.jstor.org/stable/2037854
Notes and References
- H. Helson . J.-P. Kahane . Y. Katznelson . W. Rudin . 121739671 . The functions which operate on Fourier transforms. Acta Mathematica. 102. 1–2 . 1959. 135–157. 10.1007/bf02559571. free.