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.

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Banach algebra structure

The norm of a function $f \in A (T)$ 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 $n$ th 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}}(n-m){\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}}(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 $l 1 (Z)$ , 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

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

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.

Notes

^ Weisstein, Eric W.; Moslehian, M.S. "Wiener algebra". MathWorld.

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), Nouvelle Série, 9 (51): 3–24, MR 0004726
Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. (Mat. Sbornik), Nouvelle Série, 9 (51): 51–66, MR 0004727
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, MR 0365002
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, MR 0983891

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