# 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 $f:[0,2\pi [\u27f6\u2102$ 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:

$${({a}_{k})}_{k\in \mathbb{Z}}\u27f7f(t)=\sum _{n=-\mathrm{\infty}}^{+\mathrm{\infty}}{a}_{k}{e}^{int}$$ |

## 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 |