# 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 $f:[0,2\pi[\longrightarrow\mathbb{C}$ whose Fourier series

 $f(t)=\sum_{n=-\infty}^{+\infty}\hat{f}(n)e^{int}$

is such that $\sum_{n}|\hat{f}(n)|<\infty$ .

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

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

 $(a_{k})_{k\in\mathbb{Z}}\longleftrightarrow f(t)=\sum_{n=-\infty}^{+\infty}a_{% k}e^{int}$

## 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 $\phi$ be a multiplicative linear functional in $W$. We have that

 $\phi(f)=\phi\Big{(}\sum_{n=-\infty}^{+\infty}\hat{f}(n)e^{int}\Big{)}=\sum_{n=% -\infty}^{+\infty}\hat{f}(n)\phi(e^{int})=\sum_{n=-\infty}^{+\infty}\hat{f}(n)% \phi^{n}(e^{it})$

Since $\|\phi\|=1$ we have that

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

and

 $|\phi(e^{-it})|\leq\|\phi\|\|e^{-it}\|=\|e^{-it}\|=1$

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

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

We can conclude that

$\phi(e^{it})=e^{it_{0}}\;$ for some $t_{0}\in[0,2\pi[$

Therefore we obtain

 $\phi(f)=\sum_{n=-\infty}^{+\infty}\hat{f}(n)e^{int_{0}}=f(t_{0})$

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

We conclude that $f$ does not belong to the kernel of any multiplicative linear functional $\phi$. $\square$

## 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 WienerAlgebra 2013-03-22 17:22:55 2013-03-22 17:22:55 asteroid (17536) asteroid (17536) 10 asteroid (17536) Definition msc 46J10 msc 43A50 msc 42A20 msc 46K05 Wiener theorem