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.

Banach algebra structure

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

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

where

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

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

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

therefore

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

Thus the Wiener algebra is a commutative unitary Banach algebra. Also, A(T) is isomorphic to the Banach algebra l₁(Z), with the isomorphism given by the Fourier transform.

Properties

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

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

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

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

More generally,

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

for ${\displaystyle \alpha >1/2}$ (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 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

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

which is equivalent to Wiener's theorem.

See also

• Wiener–Lévy theorem

Notes

1. 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) 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   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