Wiener algebra explained

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

Banach algebra structure

The norm of a function is given by

\|f\| = \sum ^{inf ty} _{n =-inf ty} |\hat{f}(n)|,

where

\hat{f}(n) = (1)/(2 \pi) \int ^{\pi} _{- \pi} f(t)e ^{-int} dt

is the th Fourier coefficient of . The Wiener algebra is closed under pointwise multiplication of functions. Indeed,

\begin{align} f(t)g(t) & = \sum _{m\inZ} \hat{f}(m)e ^{imt} ⋅ \sum _{n\inZ} \hat{g}(n)e ^{int} \\ & = \sum _{n,m\inZ} \hat{f}(m)\hat{g}(n)e ^{i(m+n)t} \\ & = \sum _{n\inZ} \left\{ \sum _{m} \hat{f}(n - m)\hat{g}(m) \right\}e ^{int} , f,g\in A( T ); \end{align}

therefore

\|f g\| = \sum _{n\inZ} \left| \sum _{m} \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, is isomorphic to the Banach algebra, 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 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,

Lip _{\alpha(T)\subset} A( T )\subset C( T )

for

\alpha> 1 / 2 (see).

Wiener's 1/f theorem

See main article: Wiener tauberian theorem.

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

used the theory of Banach algebras that he developed to show that the maximal ideals of are of the form

M _{x} = \left\{ f \in A( T ) \mid f(x) = 0 \right\}, x \in T ~,

which is equivalent to Wiener's theorem.

See also

• Wiener–Lévy theorem