Английская Википедия:Beurling algebra

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

<math>f(x)=\sum a_ne^{inx}</math>

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

<math>c_k=\sup_{|n|\ge k} |a_n|</math>

of the Fourier coefficients a_n are summable. In other words

<math>\sum_{k\ge 0} c_k<\infty.</math>

Example We may consider a weight function w on <math>\mathbb{Z}</math> such that

<math>w(m+n)\leq w(m)w(n),\quad w(0)=1</math>

in which case <math>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}))</math> is a unitary commutative Banach algebra.

These algebras are closely related to the Wiener algebra.

References

• Шаблон:Eom
• Шаблон:Citation