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 a**n are summable. In other words

:\sum_{k\ge 0} c_k

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) is a unitary commutative Banach algebra.

These algebras are closely related to the Wiener algebra.

References

• {{citation|mr=0027891|first=Arne |last=Beurling|title= On the spectral synthesis of bounded functions|journal= Acta Math. |volume= 81 |year=1949|issue=1|pages= 225–238