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

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

where

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

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

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

therefore

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

Thus the Wiener algebra is a commutative unitary Banach algebra. Also, A(T) is isomorphic to the Banach algebra l_1(Z), with the isomorphism given by the Fourier transform.

Properties

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

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

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

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

More generally,

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

for $\alpha >1/2$ (see Katznelson (2004)).

Wiener's 1/f theorem

Main article: Wiener tauberian theorem

Wiener (1932, 1933) 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. Thus the maximal ideals of A(T) are of the form

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

Gelfand (1941, 1941b) gave a different proof using the spectral theory of commutative C*-algebras. In Newman (1975) an elementary proof was given.

References

• Arveson, William (2001) [1994], "A Short Course on Spectral Theory", Encyclopedia of Mathematics, EMS Press
• Gelfand, I. (1941a), "Normierte Ringe", Rec. Math. [Mat. Sbornik] N. S., 9 (51): 3–24, MR 0004726
• Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. [Mat. Sbornik] N. S., 9 (51): 51–66, MR 0004727
• Katznelson, Yitzhak (2004), An introduction to harmonic analysis (Third ed.), New York: Cambridge Mathematical Library, ISBN 9780521543590
• Newman, D. J. (1975), "A simple proof of Wiener's 1/f theorem", Proceedings of the American Mathematical Society, 48: 264–265, ISSN 0002-9939, MR 0365002
• Wiener, Norbert (1932), "Tauberian Theorems", Annals of Math., 33 (1): 1–100
• 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, MR 0983891 {{citation}}: ISBN / Date incompatibility (help)