# Wiener algebra

In mathematics, the Wiener algebra, named after Norbert Wiener and usually denoted by A(T), is the space of absolutely convergent Fourier series.[1] Here T denotes the circle group.

## Banach algebra structure

The norm of a function f ∈ A(T) is given by

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

where

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

is the nth Fourier coefficient of f. The Wiener algebra A(T) is closed under pointwise multiplication of functions. Indeed,

{\displaystyle {\begin{aligned}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} }\left\{\sum _{m\in \mathbb {Z} }{\hat {f}}(n-m){\hat {g}}(m)\right\}e^{int},\qquad f,g\in A(\mathbb {T} );\end{aligned}}}

therefore

${\displaystyle \|fg\|=\sum _{n\in \mathbb {Z} }\left|\sum _{m\in \mathbb {Z} }{\hat {f}}(n-m){\hat {g}}(m)\right|\leq \sum _{m}|{\hat {f}}(m)|\sum _{n}|{\hat {g}}(n)|=\|f\|\,\|g\|.\,}$

Thus the Wiener algebra is a commutative unitary Banach algebra. Also, A(T) is isomorphic to the Banach algebra l1(Z), with the isomorphism given by the Fourier transform.

## Properties

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

${\displaystyle A(\mathbb {T} )\subset C(\mathbb {T} )}$

where C(T) 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

${\displaystyle C^{1}(\mathbb {T} )\subset A(\mathbb {T} ).\,}$.

More generally,

${\displaystyle {\mathrm {Lip} }_{\alpha }({\mathbb {T} })\subset A({\mathbb {T} })\subset C({\mathbb {T} })}$

## Wiener's 1/f theorem

{{#invoke:main|main}}

Template:Harvs proved that if f has absolutely convergent Fourier series and is never zero, then its inverse 1/f 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 A(T) are of the form

${\displaystyle M_{x}=\left\{f\in A({\mathbb {T} })\,\mid \,f(x)=0\right\},\quad x\in {\mathbb {T} }~,}$

which is equivalent to Wiener's theorem.

## References

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}

• {{#invoke:citation/CS1|citation

|CitationClass=citation }}