Wiener algebra

0.0.1 Definition and classification of the Wiener algebra

Let W be the space of all complex functions on [0,2π[ whose Fourier series converges absolutely, that is, all functions f:[0,2π[ whose Fourier series


is such that n|f^(n)|< .

Under pointwise operations and the norm f=n|f^(n)|, W is a commutative Banach algebra of continuous functionsMathworldPlanetmathPlanetmath, with an identity elementMathworldPlanetmath. W is usually called the Wiener algebraMathworldPlanetmath.

Theorem - W is isometrically isomorphic to the Banach algebraMathworldPlanetmath 1() with the convolution productPlanetmathPlanetmath. The isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath is given by:


0.0.2 Wiener’s Theorem

Theorem (Wiener) - If fW has no zeros then 1/fW, that is, 1/f has an absolutely convergent Fourier series.

Proof : We want to prove that f is invertible in W. As W is commutativePlanetmathPlanetmathPlanetmathPlanetmath, that is the same as proving that f does not belong to any maximal ideal of W. Therefore we only need to show that f is not in the kernel of any multiplicative linear functional of W.

Let ϕ be a multiplicative linear functional in W. We have that


Since ϕ=1 we have that




Since 1=|ϕ(eite-it)|=|ϕ(eit)||ϕ(e-it)| we deduce that


We can conclude that

ϕ(eit)=eit0 for some t0[0,2π[

Therefore we obtain


which is non-zero by definition of f.

We conclude that f does not belong to the kernel of any multiplicative linear functional ϕ.

0.0.3 Remark

The Wiener algebra is a Banach *-algebra with the involution given by f*(t):=f(-t)¯, but it is not a C*-algebra ( under this involution.

Title Wiener algebra
Canonical name WienerAlgebra
Date of creation 2013-03-22 17:22:55
Last modified on 2013-03-22 17:22:55
Owner asteroid (17536)
Last modified by asteroid (17536)
Numerical id 10
Author asteroid (17536)
Entry type Definition
Classification msc 46J10
Classification msc 43A50
Classification msc 42A20
Classification msc 46K05
Defines Wiener theorem