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

[math]\displaystyle{ f(x)=\sum a_ne^{inx} }[/math]

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

[math]\displaystyle{ c_k=\sup_{|n|\ge k} |a_n| }[/math]

of the Fourier coefficients a_n are summable. In other words

[math]\displaystyle{ \sum_{k\ge 0} c_k\lt \infty. }[/math]

Example We may consider a weight function w on [math]\displaystyle{ \mathbb{Z} }[/math] such that

[math]\displaystyle{ w(m+n)\leq w(m)w(n),\quad w(0)=1 }[/math]

in which case [math]\displaystyle{ A_w(\mathbb{T}) =\{f:f(t)=\sum_na_ne^{int},\,\|f\|_w=\sum_n|a_n|w(n)\lt \infty\} \,(\sim\ell^1_w(\mathbb{Z})) }[/math] is a unitary commutative Banach algebra.

These algebras are closely related to the Wiener algebra.

References

• Hazewinkel, Michiel, ed. (2001), "Beurling algebra", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=Main_Page 
• Beurling, Arne (1949), "On the spectral synthesis of bounded functions", Acta Math. 81 (1): 225–238, doi:10.1007/BF02395018