Beurling algebra
# 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

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

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

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

of the Fourier coefficients $a_n$ are summable. In other words

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

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

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

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

These algebras are closely related to the Wiener algebra.

