Wiener algebra

In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by Template:Math, is the space of absolutely convergent Fourier series. Here Template:Math denotes the circle group.

Banach algebra structure

The norm of a function Template:Math is given by

$\Vert f\Vert =\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}|\hat{f}(n)|,$

where

$\hat{f}(n)=\frac{1}{2\pi }\underset{-\pi }{\overset{\pi }{\int }}f(t)e^{-int}dt$

is the Template:Mvarth Fourier coefficient of Template:Math. The Wiener algebra Template:Math is closed under pointwise multiplication of functions. Indeed,

$\begin{array}{cc}f(t)g(t)& =\sum _{m\in \mathbb{Z}}\hat{f}(m)e^{imt}\cdot \sum _{n\in \mathbb{Z}}\hat{g}(n)e^{int}\\ & =\sum _{n,m\in \mathbb{Z}}\hat{f}(m)\hat{g}(n)e^{i(m+n)t}\\ & =\sum _{n\in \mathbb{Z}}\{\sum _{m\in \mathbb{Z}}\hat{f}(n-m)\hat{g}(m)\}{e}^{int},f,g\in A(𝕋);\end{array}$

therefore

$\Vert fg\Vert =\sum _{n\in \mathbb{Z}}|\sum _{m\in \mathbb{Z}}\hat{f}(n-m)\hat{g}(m)|\le \sum _m|\hat{f}(m)|\sum _n|\hat{g}(n)|=\Vert f\Vert \Vert g\Vert .$

Thus the Wiener algebra is a commutative unitary Banach algebra. Also, Template:Math is isomorphic to the Banach algebra Template:Math, with the isomorphism given by the Fourier transform.

Properties

The sum of an absolutely convergent Fourier series is continuous, so

$A(𝕋)\subset C(𝕋)$

where Template:Math is the ring of continuous functions on the unit circle.

On the other hand an integration by parts, together with the Cauchy–Schwarz inequality and Parseval's formula, shows that

$C^1(𝕋)\subset A(𝕋).$

More generally,

${\mathrm{L}\mathrm{i}\mathrm{p}}_{\alpha }(𝕋)\subset A(𝕋)\subset C(𝕋)$

for $\alpha >1/2$ (see Template:Harvtxt).

Wiener's 1/f theorem

Template:Harvs proved that if Template:Math has absolutely convergent Fourier series and is never zero, then its reciprocal Template:Math also has an absolutely convergent Fourier series. Many other proofs have appeared since then, including an elementary one by Template:Harvs.

Template:Harvs used the theory of Banach algebras that he developed to show that the maximal ideals of Template:Math are of the form

$M_x=\{f\in A(𝕋)\mid f(x)=0\},x\in 𝕋,$

which is equivalent to Wiener's theorem.

See also

• Wiener–Lévy theorem

References

• Template:Springer
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