Wiener algebra

0.0.1 Definition and classification of the Wiener algebra

Let $W$ be the space of all complex functions on $[0,2\pi [$ whose Fourier series converges absolutely, that is, all functions $f:[0,2\pi [⟶ℂ$ whose Fourier series

$$f(t)=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}\widehat{f}(n)e^{int}$$

is such that .

Under pointwise operations and the norm $\parallel f\parallel ={\sum }_n|\widehat{f}(n)|,$ $W$ is a commutative Banach algebra of continuous functions, with an identity element. $W$ is usually called the Wiener algebra.

Theorem - $W$ is isometrically isomorphic to the Banach algebra ${\mathrm{\ell }}^1(\mathbb{Z})$ with the convolution product. The isomorphism is given by:

$${(a_k)}_{k\in \mathbb{Z}}⟷f(t)=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}{a}_k{e}^{int}$$

0.0.2 Wiener’s Theorem

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

Proof : We want to prove that $f$ is invertible in $W$. As $W$ is commutative, 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 $\varphi$ be a multiplicative linear functional in $W$. We have that

$$\varphi (f)=\varphi \left(\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}\widehat{f}(n)e^{int}\right)=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}\widehat{f}(n)\varphi (e^{int})=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}\widehat{f}(n){\varphi }^n(e^{it})$$

Since $\parallel \varphi \parallel =1$ we have that

$$|\varphi (e^{it})|\le \parallel \varphi \parallel \parallel e^{it}\parallel =\parallel e^{it}\parallel =1$$

and

$$|\varphi (e^{-it})|\le \parallel \varphi \parallel \parallel e^{-it}\parallel =\parallel e^{-it}\parallel =1$$

Since $1=|\varphi (e^{it}{e}^{-it})|=|\varphi (e^{it})||\varphi (e^{-it})|$ we deduce that

$$|\varphi (e^{it})|=1$$

We can conclude that

$\varphi (e^{it})=e^{i{t}_0}$ for some $t_0\in [0,2\pi [$

Therefore we obtain

$$\varphi (f)=\sum _{n=-\mathrm{\infty }}^{+\mathrm{\infty }}\widehat{f}(n)e^{in{t}_0}=f(t_0)$$

which is non-zero by definition of $f$.

We conclude that $f$ does not belong to the kernel of any multiplicative linear functional $\varphi$. $\mathrm{\square }$

0.0.3 Remark

The Wiener algebra is a Banach *-algebra with the involution given by $f^{*}(t):=\overline{f(-t)}$, but it is not a $C^{*}$-algebra (http://planetmath.org/CAlgebra) 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