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. Here T denotes the circle group.

Banach algebra structure

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

[math]\displaystyle{ \|f\|=\sum_{n=-\infty}^\infty |\hat{f}(n)|,\, }[/math]

where

[math]\displaystyle{ \hat{f}(n)= \frac{1}{2\pi}\int_{-\pi}^\pi f(t)e^{-int} \, dt }[/math]

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

[math]\displaystyle{ \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]\displaystyle{ \|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, 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

[math]\displaystyle{ A(\mathbb{T})\subset C(\mathbb{T}) }[/math]

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

[math]\displaystyle{ C^1(\mathbb{T})\subset A(\mathbb{T}).\, }[/math]

More generally,

[math]\displaystyle{ \mathrm{Lip}_\alpha(\mathbb{T})\subset A(\mathbb{T})\subset C(\mathbb{T}) }[/math]

for [math]\displaystyle{ \alpha\gt 1/2 }[/math] (see (Katznelson 2004)).

Wiener's 1/f theorem

Wiener (1932, 1933) proved that if f has absolutely convergent Fourier series and is never zero, then its reciprocal 1/f also has an absolutely convergent Fourier series. Many other proofs have appeared since then, including an elementary one by Newman (1975).

Gelfand (1941, 1941b) used the theory of Banach algebras that he developed to show that the maximal ideals of A(T) are of the form

[math]\displaystyle{ 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.

See also

• Wiener–Lévy theorem

References

• Hazewinkel, Michiel, ed. (2001), "A Short Course on Spectral Theory", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=10.1007/b97227 
• Gelfand, I. (1941a), "Normierte Ringe", Rec. Math. (Mat. Sbornik), Nouvelle Série 9 (51): 3–24 
• Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. (Mat. Sbornik), Nouvelle Série 9 (51): 51–66 
• Katznelson, Yitzhak (2004), An introduction to harmonic analysis (Third ed.), New York: Cambridge Mathematical Library, ISBN 978-0-521-54359-0 
• Newman, D. J. (1975), "A simple proof of Wiener's 1/f theorem", Proceedings of the American Mathematical Society 48: 264–265, doi:10.2307/2040730, ISSN 0002-9939 
• Wiener, Norbert (1932), "Tauberian Theorems", Annals of Mathematics 33 (1): 1–100, doi:10.2307/1968102 
• Wiener, Norbert (1933), The Fourier integral and certain of its applications, Cambridge Mathematical Library, Cambridge University Press, doi:10.1017/CBO9780511662492, ISBN 978-0-521-35884-2