Beurling algebra

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


 * $$f(x)=\sum a_ne^{inx}$$

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


 * $$c_k=\sup_{|n|\ge k} |a_n|$$

of the Fourier coefficients an are summable. In other words


 * $$\sum_{k\ge 0} c_k<\infty.$$

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 $$A_w(\mathbb{T}) =\{f:f(t)=\sum_na_ne^{int},\,\|f\|_w=\sum_n|a_n|w(n)<\infty\} \,(\sim\ell^1_w(\mathbb{Z}))$$ is a unitary commutative Banach algebra.

These algebras are closely related to the Wiener algebra.