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Wiener Algebra


Suppose W is the set of all complex-valued functions f on the interval [0,2pi] of the form

 f(t)=sum_(k=-infty)^inftyalpha_ke^(ikt)
(1)

for t in [0,2pi], where the alpha_k in C and sum_(k=-infty)^(infty)|alpha_k|<infty. The set W with the usual pointwise operations and with the norm

 ||f||=sum_(k=-infty)^infty|alpha_k|
(2)

is a commutative Banach algebra and is called the Wiener algebra.

There is an isometric isomorphism between l^1(Z) and W given by f->f^~, where

 f^~(t)=sum_(k=-infty)^inftyf(k)e^(ikt)
(3)

with t in [0,2pi].


This entry contributed by Mohammad Sal Moslehian

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References

Bonsall, F. F. and Duncan, J. Complete Normed Algebras. New York: Springer-Verlag, 1973.

Referenced on Wolfram|Alpha

Wiener Algebra

Cite this as:

Moslehian, Mohammad Sal. "Wiener Algebra." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/WienerAlgebra.html

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