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 convergesabsolutely, 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 Banachalgebra 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 theconvolution 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 anabsolutely convergent Fourier series.

Proof : We want to prove that $f$ is invertible in $W$. As $W$ is commutative, that is the same as provingthat $f$ does not belong to any maximal ideal of $W$. Therefore we only need to show that $f$ is not in thekernel 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.