Beurling algebra

In mathematics, the term Beurling algebra is used for different algebras introduced by Template:Harvs, usually it is an algebra of periodic functions with Fourier series

$f(x)=\sum a_n{e}^{inx}$

Example We may consider the algebra of those functions f where the majorants

$c_k=\underset{|n|\ge k}{\sup }|a_n|$

of the Fourier coefficients a_n are summable. In other words

$\sum _{k\ge 0}{c}_k<\mathrm{\infty }.$

Example We may consider a weight function w on $\mathbb{Z}$ such that

$w(m+n)\le w(m)w(n),w(0)=1$

in which case $A_w(𝕋)=\{f:f(t)=\sum _n{a}_n{e}^{int},\Vert f{\Vert }_w=\sum _n|a_n|w(n)<\mathrm{\infty }\}(\sim {\ell }_w^1(\mathbb{Z}))$ is a unitary commutative Banach algebra.

These algebras are closely related to the Wiener algebra.

References

• Template:Eom
• Template:Citation