en>BattyBot: fixed CS1 errors: dates & General fixes using AWB (9816)

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fixed CS1 errors: dates & General fixes using AWB (9816)

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In mathematics, the '''Wiener algebra''', named after [[Norbert Wiener]] and usually denoted by {{math|''A''('''T''')}}, is the space of [[absolute convergence|absolutely convergent]] [[Fourier series]].<ref>{{MathWorld|title=Wiener algebra|urlname=WienerAlgebra|last=Moslehian|first=M.S.}}
</ref> Here '''T''' denotes the [[circle group]].

==Banach algebra structure==
The norm of a function {{math|''f'' ∈ ''A''('''T''')}} is given by

:<math>\|f\|=\sum_{n=-\infty}^\infty |\hat{f}(n)|,\,</math>

where

:<math>\hat{f}(n)= \frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int} \, dt</math>

is the ''n''th Fourier coefficient of {{math|''f''}}. The Wiener algebra {{math|''A''('''T''')}} is closed under pointwise multiplication of functions. Indeed,

:<math>
\begin{align}
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{align}
</math>

therefore

:<math>
\|f g\|
= \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\|.\,</math>

Thus the Wiener algebra is a commutative unitary [[Banach algebra]]. Also, {{math|''A''('''T''')}} is isomorphic to the Banach algebra {{math|''l''<sub>1</sup>('''Z''')}}, with the isomorphism given by the Fourier transform.

== Properties ==

The sum of an absolutely convergent Fourier series is continuous, so
:<math>A(\mathbb{T})\subset C(\mathbb{T})</math>
where {{math|''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

: <math>C^1(\mathbb{T})\subset A(\mathbb{T}).\,</math>.

More generally,

: <math>\mathrm{Lip}_\alpha(\mathbb{T})\subset A(\mathbb{T})\subset C(\mathbb{T})</math>

for <math>\alpha>1/2</math> (see {{harvtxt|Katznelson|2004}}).

==Wiener's 1/''f'' theorem==
{{main|Wiener tauberian theorem}}

{{harvs|txt|last=Wiener|year1=1932|year2=1933}} proved that if {{math|''f''}} has absolutely convergent Fourier series and is never zero, then its inverse {{math|1/''f''}} also has an absolutely convergent Fourier series. Many other proofs have appeared since then, including an elementary one by {{harvs|txt|last=Newman|year1=1975}}.

{{harvs|txt|last=Gelfand|year1=1941|year2=1941b}} used the theory of Banach algebras that he developed to show that the maximal ideals of {{math|''A''('''T''')}} are of the form

:<math> M_x = \left\{ f \in A(\mathbb{T}) \, \mid \, f(x) = 0 \right\}, \quad x \in \mathbb{T}~,</math>

which is equivalent to Wiener's theorem.

==Notes==
{{Reflist}}

== References ==
* {{springer
| title= A Short Course on Spectral Theory
| id= 10.1007/b97227
| last=Arveson
| first=William
}}
*{{Citation | last1=Gelfand | first1=I. | title=Normierte Ringe | year=1941a | journal=Rec. Math. (Mat. Sbornik) N.S.| volume=9 (51) | pages=3–24 | mr=0004726}}
*{{Citation | last1=Gelfand | first1=I. | title=Über absolut konvergente trigonometrische Reihen und Integrale | year=1941b | journal=Rec. Math. (Mat. Sbornik) N.S.| volume=9 (51) | pages=51–66 | mr=0004727}}
* {{citation | last=Katznelson| first= Yitzhak| title=An introduction to harmonic analysis| edition =Third | publisher =Cambridge Mathematical Library | year=2004 | location=New York | isbn=978-0-521-54359-0}}
*{{Citation | last1=Newman | first1=D. J. | title=A simple proof of Wiener's 1/''f'' theorem | year=1975 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=48 | pages=264–265 | mr=0365002}}
*{{Citation | last=Wiener|first=Norbert|title=Tauberian Theorems|journal=Annals of Math.|volume=33|issue=1|year=1932|pages=1–100}}
*{{Citation | last1=Wiener | first1=Norbert | title=The Fourier integral and certain of its applications | publisher=[[Cambridge University Press]] | series=Cambridge Mathematical Library | isbn=978-0-521-35884-2 | doi=10.1017/CBO9780511662492 | year=1933 | mr=983891}}

[[Category:Banach algebras]]
[[Category:Fourier series]]

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en>BattyBot: fixed CS1 errors: dates & General fixes using AWB (9816)