Beurling algebra

In mathematics, the term Beurling algebra is used for different algebras introduced by Arne Beurling (1949), usually it is an algebra of periodic functions with Fourier series

${\displaystyle f(x)=\sum a_{n}e^{inx}}$

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

${\displaystyle c_{k}=\sup _{|n|\geq k}|a_{n}|}$

of the Fourier coefficients a_n are summable. In other words

${\displaystyle \sum _{k\geq 0}c_{k}<\infty .}$

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

${\displaystyle w(m+n)\leq w(m)w(n),\quad w(0)=1}$

in which case ${\displaystyle A_{w}(\mathbb {T} )=\{f:f(t)=\sum _{n}a_{n}e^{int},\,\|f\|_{w}=\sum _{n}|a_{n}|w(n)<\infty \}\,(\sim \ell _{w}^{1}(\mathbb {Z} ))}$ is a unitary commutative Banach algebra.

These algebras are closely related to the Wiener algebra.

References

• Belinsky, E.S.; Liflyand, E.R. (2001) [1994], "Beurling algebra", Encyclopedia of Mathematics, EMS Press
• Beurling, Arne (1949), "On the spectral synthesis of bounded functions", Acta Math., 81 (1): 225–238, doi:10.1007/BF02395018, MR 0027891